N-DETERMINED p-COMPACT GROUPS JESPER M. MfflLLER Abstract.One of the major problems in the homotopy theory of finite loop * *spaces is the classification problem for p-compact groups. It has been proposed to use * *the maximal torus normalizer (which at an odd prime essentially means the Weyl group) as th* *e distinguishing invariant. We show here that the maximal torus normalizer does indeed cla* *ssify many p-compact groups up to isomorphism when p is an odd prime. Contents 1. Introduction 1 2. Higher limits of center functors * * 4 2.14. Relation between A(W, t) and the orbit category O(W ) * * 7 2.17. Centralizers * * 7 3. N-determinism 8 3.1. N-determined automorphisms * * 8 3.5. N-determined p-compact groups * * 9 3.11. Centers and automorphism groups of p-compact groups * * 13 3.18. Canonical factorizations * * 14 4. Cohomologically unique p-compact groups * * 14 5. The p-compact group PGL (n, C) * * 19 5.17. The action of A(GL (n, C))(T, æ) on CGL(n,C)(T, æ) * * 26 5.19. Representations of extra-special p-groups. * * 26 6. The 3-compact group F4 28 7. Polynomial p-compact groups * * 28 7.10. Construction of modular, centerless, polynomial, simple p-compact grou* *ps 32 7.13. Automorphisms of X(G(m, r, n)) * * 34 7.17. Automorphisms of other modular polynomial p-compact groups * * 35 7.18. Structure of polynomial p-compact groups * * 35 8. Proofs of Theorem 1.2 and Corollaries 1.3-1.6 * * 39 9. N-determinism of product p-compact groups * * 39 10. Maximal rank subgroups of DI2 * * 43 11. Free Zp-modules and p-discrete tori * * 48 12. Shapiro's lemma 57 13. Cellular cohomology of small categories * * 58 References 64 1. Introduction This paper addresses the classification problem at odd primes for the p-compa* *ct groups intro- duced by W.G. Dwyer and C.W. Wilkerson in their seminal paper [30] (surveyed in* * [65, 53, 26]). A p-compact group is a connected, pointed, H*Fp-local space BX such that H*(X; * *Fp) is finite ___________ Date: 15th February 2002. 1991 Mathematics Subject Classification. 55R35, 55P15. Key words and phrases. Classification of p-compact groups, automorphisms of p* *-compact groups, reflection subgroup, Quillen category, left derived functors of the inverse limit functor,* * spaces with polynomial cohomology, Lie group. 1 2 J.M. MØLLER where X = BX is the loop space [30, x2]. It is customary, though ambiguous, to* * refer to BX by the name, X, for its underlying loop space. It has been conjectured [53, 72, 26], in analogy with the classification theo* *rem for compact Lie groups [25, 83], that p-compact groups are determined by their maximal torus no* *rmalizers. The maximal torus normalizer N(X) for the p-compact group X is an extension (1.1) T (X) ! N(X) ! W (X) of the maximal torus T (X) by the Weyl group W (X) [30, 9.8], and X is said to * *be totally N- determined [68, 7.1] if oX is determined by N(X), and, othe automorphisms of X are determined by their restrictions to N(X). We show here that almost all simple p-compact groups are totally N-determined a* *t odd primes. 1.2. Theorem. Let X be a simple p-compact group, where p is an odd prime. Assum* *e that the rational Weyl group (r0W (X)) 6= (r0W (E8)) if p = 3 and (r0W (X)) 6= (r0W (Ej)* *), j = 6, 7, 8, if p = 5. Then X is totally N-determined. The Weyl group W (X) [30, 9.7] of a connected p-compact group X is a finite g* *roup of auto- morphisms of the free, finitely generated Zp-module L(X) = ß1T (X), i.e. W (X) * * GL (L(X)). The rational Weyl group, r0W (X), is the image of W (X) in GL (L(X) Zp Qp), an* *d rpW (X), the Fp-Weyl group, the image of W (X) in GL (L(X) Zp Fp). As usual, (r0W (X))* * stands for the conjugacy class of the rational Weyl group. The connected p-compact group X* * is simple if L(X) Zp Qp is an irreducible r0W (X)-module [67, 5.4]. At an odd prime p, the maximal torus normalizer extension (1.1) for a connect* *ed p-compact group splits in an essentially unique way [5] and thus N(X) is in fact complete* *ly determined by the reflection group (W (X), L(X)). This explains the first part, merely a * *reformulation of Theorem 1.2, of the the corollary below; see (4.3) for the precise meaning of t* *he other statements. 1.3. Corollary.Let X be a simple p-compact groups as in Theorem 1.2. Then X is * *determined up to (local) isomorphism by its (rational) Weyl group, and the automorphism gr* *oup Aut(X) is isomorphic to NGL(L(X))(W (X))=W (X). Furthermore, if X is centerless or simply* * connected, then X is determined by its Fp-Weyl group, and X is a cohomologically unique p-compa* *ct group. For the bigger class of connected (but not necessarily simple) p-compact grou* *ps Theorem 1.2 takes on a particularly appealing form. 1.4. Corollary.Assume that p > 5. The map æ oe æ oe Isomorphism classes of(W,L) Similarity classes of connected p-compact groups----!Zp-reflection groups is a bijection, and Aut(X) is isomorphic to NGL(L)(W )=W for the connected p-co* *mpact group X corresponding to the reflection group (W, L). In the general case, for the class of not necessarily connected p-compact gro* *ups, Theorem 1.2 takes the following form. 1.5. Corollary.Let X be a p-compact group such that all its simple factors sati* *sfy the assumptions of Theorem 1.2. Then X is totally N-determined and Out(X) ~=Out(N(X)). The simple factors of the p-compact group X are the simple, centerless p-comp* *act groups in the splitting [32, 80] of P X0 = X0=Z(X0), the adjoint form of the identity compone* *nt of X. Let me also mention the following partial classification result for connected* * finite loop spaces with maximal tori [59, 1.1]. 1.6. Corollary.(Cf. [96], [59, 1.6]) Let X be a connected finite loop spaces wi* *th a maximal torus. Assume that X has the same Weyl group as the compact, connected simple Lie grou* *p G and that no simple factor of G is locally isomorphic to E6, E7, or E8. Then (BX)[1_2] a* *nd (BG)[1_2] are homotopy equivalent spaces. N-DETERMINISM 3 In light of the observation by C. Wilkerson [97] that the Weyl group of any c* *onnected finite loop space with maximal torus must agree with the Weyl group of a compact conne* *cted Lie group, this proves the maximal torus conjecture [49, Conjecture D, p. 68] [99] away fr* *om the prime 2 in a number of particular cases. The proof that the simple p-compact groups of Theorem 1.2 are N-determined go* *es in outline as follows. Consider some connected p-compact group X with maximal torus normalize* *r j :N ! X and assume that the same extended p-compact torus N also can serve as the maxim* *al torus normalizer j0:N ! X0 for some other p-compact group X0. Starting with the confi* *guration j j0 X oo__N_ ____//_X0 our task is to construct an isomorphism f :X ! X0under N. We observe (3.7) that* * it suffices to consider the centerless form of X. According to the Homology Decomposition Theo* *rem [31, x8], BX is (the p-completion of) the homotopy colimit of the A(X)op-space of central* *izers BCX (E, ) of non-trivial elementary abelian p-subgroups :E ! X of X. For any monomorphi* *sm :E ! X it is possible to find (non-uniquely) a preferred lift ~: E ! N of such that * *the morphisms Cj Cj0 CX (E, )oo_CN_(E, ~)___//_CX0(E, 0) are again maximal torus normalizers for the centralizers of (E, ) and (E, 0) * *where 0= j0~ [69]. As the center of X is trivial, the centralizer of (E, ) will have smaller coho* *mological dimension than that of X [30, 6.14, 6.15]. Assuming, as part of an inductional argument, that * *CX (E, ) (which very well may be non-connected) is totally N-determined, there will therefore b* *e an isomorphism f(E, ~): CX (E, ) ! CX0(E,un0)der CN (E, ~). It remains to show that these lo* *cally defined isomorphisms f(E, ~) do not depend on the choice of preferred lifts and that th* *ey combine to yield a morphism f :X ! X0under N. N-determinism is actually not a property of the p-compact group X itself but * *rather a property of the extended p-compact torus N(X): If X is N-determined, so is, by the very * *nature of the concept, any other p-compact group that admits N(X) for a maximal torus normali* *zer. Most of the time the prime p will be assumed to be odd. Some modifications wi* *ll be needed to handle the case where p = 2 [60]. Even the formulation of the N-conjecture itse* *lf will have to be refined as N(O(2)) = O(2) = N(SO (3)) but O(2) and SO(3) are distinct 2-compact* * groups. Organization of the paper. In Section 3, I set up the general theory that will * *be applied in a case- by-case verification of the N-conjecture for the simple, centerless p-compact g* *roups. We deal with A-family, represented by the p-compact groups PGL (n, C) = PSL(n, C), in Sectio* *n 5, and with the polynomial case, which includes nearly all remaining compact simple Lie gro* *ups and all the exotic (non-Lie) simple p-compact groups, in Section 7. The proofs of (1.2-1.6)* * are in Section 8. Sections 2 and 13 contain material dealing with the general problem of computin* *g cohomology groups of categories. (There is no claim to originality here as the vanishing * *result of (2.4) was proved in [34] and the spectral sequence of (13.2) seems to be that of Lück [54* *, 17.28] or S_lomi'nska [87].) Notation. Write Zp for the ring of p-adic integers, Qp for the field of p-adic * *numbers, and Fp for the field with p elements. For a p-compact group X, let oT (X) denote the maximal torus of X [30, 8.9], oL(X) = ß2(BT (X)) the lattice of X, oT~(X) = L(X) Z=p1 the p-discrete maximal torus of X [30, x6], ot(X) = L(X) Z=p the maximal elementary abelian subgroup of ~T(X), oW (X) the Weyl group of X [30, 9.6], r0W (X) the rational and rpW (X) the m* *od p Weyl group of X (Section 4), oN(X) the maximal torus normalizer of X [30, 9.8], oZ(X) the center of X [31, 58], or(X) the rational rank (of the identity component) of X [30, 5.11], oAut(X) the group of invertible elements in the monoid End(X) = [BX, *; BX] * *[68, x3] of based homotopy classes of based self-maps of BX, and Out(X) = Aut(X)=ß0(* *X) the corresponding group in the un-based category, and, oA(X) the Quillen category of X. 4 J.M. MØLLER The objects (E, ) of A(X) are conjugacy classes of monomorphism :E ! X of no* *n-trivial elementary abelian p-groups E into X. The morphisms (E0, 0) ! (E1, 1) of A(X)* * consists of all group homomorphisms f :E0 ! E1such that (E0, 0) = (E0, 1f). An object (E, ) * *of A(X) is toral if :E ! X factors through the maximal torus T (X) ! X. Let oA(X) t denote the full subcategory of all toral objects, and oA(X) t the full subcategory of all objects with a morphism to some non-tora* *l object. The notation for categories is opcg is the category of p-compact groups, oGrp is the category of groups, oAb is the category of abelian groups, oSp is the category of simplicial sets, and oTop is the category of topological spaces. In [pcg], [Grp ], [Sp ] the objects are p-compact groups, groups, topological s* *paces and the mor- phisms are conjugacy classes of p-compact group morphisms, conjugacy classes of* * group homomor- phisms, homotopy classes of continuous maps. Acknowledgments. I would like to thank Kasper Andersen, Jesper Grodal, Dietrich* * Notbohm, and Antonio Viruel for several fruitful discussions and for very valuable help at m* *any points, and the referee for a long list of constructive comments. This work was partially suppo* *rted by Centre de Recerca Matem`atica. 2.Higher limits of center functors This section contains a vanishing result (2.4) for the derived limits of a ce* *rtain functor, defined in purely algebraic terms, which informs on the obstruction theory associated t* *o the Jackowski- McClure centralizer homology decomposition [45, 31] of BX. Let W be a finite group and t a non-trivial FpW -module which is finite dimen* *sional as an Fp-vector space. For non-trivial subgroups E0 and E1 of t, put __ (2.1) W (E0, E1) = {w 2 W | w(E0) E1} and W (E0) = {w 2 W | we = e for all * *e 2}E0 and note_that the set of orbits for the action of the pointwise stabilizer grou* *p W (E0) on the set W (E0, E1) is the_set of group homomorphisms of E0 ! E1 induced_by elements* * of W . The stabilizer subgroup W (E0, E0) of E0 will also be written as W (E0). 2.2. Definition.A(W, t) is the category with objects:non-trivial elementary abelian subgroups E of t, and, morphisms: group homomorphisms E0 ! E1 induced by elements of W . For any ZpW -module L, Lj:A(W, t) ! Ab, j 0, is the functor that takes the ob* *ject E t to the cohomology group*Hj(W (E); L) and the morphism E0 w-!E1 in A(W, t) to the h* *omomorphism Hj(W (E0); L) resOw----!Hj(W (E1); L). Here is a more detailed explanation_of the functors Lj: Any morphism E0 ! E1 * *in A(W, t), represented by an element w 2 W (E0, E1), can be factored E0 ! wE0 E1 into an* * isomorphism w :E0 ! wE0 followed by an inclusion. Consider the corresponding group homomorp* *hisms W (E0) c(w)---!W (E0)w W (E1) where c(w)w0 = ww0w-1 is conjugation by w and W (E0)w = wW (E0)w-1 = W (wE0) an* *d let, as usual [37, 4.1.1], w*:Hj(W (E0); L) ! Hj(W (E0)w; L)be the isomorphism induc* *ed by c(w)-1 and multiplication by w on L. Define Lj(w) as the composition * res Hj(W (E0); L) w--!Hj(W (E0)w; L) --!Hj(W (E1); L) of w* followed by the restriction morphism. Since for all w0 2 W (E0) we have t* *hat W (E0)ww0 = W (E0)w and cohomology is insensitive to inner conjugation [55, IV.5.6], Lj(ww0* *) = Lj(w) for all w0_2 W (E0) and thus this morphism is independent of the choice of representati* *ve for wW (E0) 2 W (E0, E1)=W (E0) = A(W, t)(E0, E1), cf. [45, 7.6]. N-DETERMINISM 5 For instance, for a connected p-compact group X, the functors (2.3) L(X)2-j:A(W (X), t(X)) ! Ab, j = 1, 2, take the non-trivial elementary abelian p-subgroup E of t(X) to H2-j(W (X)(E); * *L(X)). 2.4. Lemma. [34, 8.1] Lj:A(W, t) ! Ab is an acyclic functor in the sense that ( j limi(A(W, t); Lj)= H (W ; L) i = 0 0 i > 0 for all j 0. For a p-compact group X, let BCX :A(X)op! pcg and BZCX :A(X) ! Top be the f* *unctors that take the object (E, ") of A(X) to (2.5) BCX (E, ")= map(BE, BX)B" (2.6) BZCX (E, ")= map(BCX (E, "), BX)Be(") where Be("): BCX (E, ") ! BXis the evaluation map, and define (2.7) ßj(BZCX ): A(X) ! Ab, j = 1, 2, to be the composition of BZCX with the jth homotopy functor. (There is no basep* *oint problem here since only abelian p-compact groups are involved.) 2.8. Lemma. Let p be an odd prime and X a connected p-compact group. Assume tha* *t the identity component CX (E)0 of the centralizer of any non-trivial elementary abelian p-su* *bgroup E of T (X) has N-determined automorphisms [68, 3.10]. Then there is an equivalence of cate* *gories A(W (X), t(X)) ! A(X) t such that the functors ßj(BZCX ) when restricted to A(X) t correspond to the fu* *nctors L(X)2-j, j = 1, 2, of ( 2.3). Proof.Take wW (E0): E0 ! E1in A(W (X), t(X)) to the morphism w|E0:(E0, ie0) ! (* *E1, ie1) in A(X) t (where ej:Ej ! t(X), j = 0, 1, is the inclusion and i the p-compact g* *roup morphism t(X) ! T (X) ! X). This provides a functor (2.9) A(W (X), t(X)) ! A(X) t Since the natural map W \[BE, BT (X)] ! [BE, BX], induced by BT (X) ! BX, is in* *jective for any elementary abelian p-group E [67, 3.4] [32, 3.4], this functor is full and * *as it is also clearly faithful, (2.9) is an equivalence of categories. Let now ~N(X) be a discrete approximation [31, 3.12] to the maximal torus nor* *malizer N(X). For any elementary abelian p-subgroup E of ~T(X), CN~(X)(E) is a discrete approxima* *tion to CN(X)(E) and ZCN~(X)(E) is a discrete approximation to ZCN(X)(E) which is isomorphic to * *ZCX (E) (2.19) [68, 4.12]. Since the prime p is odd, ~N(X) = ~T(X) o W (X) is a semi-direct pr* *oduct [5, 2.1] and hence (2.10) CN~(X)(E) = CT~(X)oW(X)= ~T(X) o W (X)(E), Z(CN~(X)(E)) = ~T(X)W(X)(E* *), so that ßjBZCX (E) = ßj((BH0(W (X)(E); ~T(X)))^p) = H2-j(W (X)(E); L(X)) = L(X)* *2-j(E). If the Weyl group element w 2 W (X) takes the elementary abelian p-subgroup E* *0 ~T(X) into the elementary abelian p-subgroup E1 ~T(X), w represents a morphism w :E* *0 ! E1 in A(W (X), t(X)). We want_to determine the effect of w on the centralizer center* *s. Choose a lift ~w2 N~(X) of w 2 W (X)(E0, E1) W (X) = N~(X)=T~(X). Conjugation by ~w, * *given by c(w~)(n) = ~wnw~-1, n 2 ~N(X), takes E0 into E1 and conjugation by ~w-1, c(w~-1* *), takes CN~(X)(E1) into CN~(X)(E0) in such a way that the diagram -1)x1 CN~(X)(E0) x E0c(w~oo___ CN~(X)(E1) x E0 e|| |1xc(w~)| fflffl| fflffl| ~N(X)oo_______e______CN~(X)(E1) x E1 6 J.M. MØLLER where e is group multiplication, commutes up to inner automorphism of ~N(X) (as* * namely c(w~) O e O (c(w~-1) x 1) = e O (1 x c(w~))). Therefore, the diagram of adjoint maps be* *tween spaces -1) BCN~(X)(E0)oo_____Bc(w~________BCN~(X)(E1) '|| '|| fflffl| fflffl| map (BE0, BN~(X))B~1oo________ map(BE1, BN~(X))B~2 Bc(w~) is homotopy commutative. (The vertical maps are equivalences by [40, Lemma 2].)* * This shows that the map CX (E1)! CX (E0)induced by the A(X)-morphism E0 ! E1 represented b* *y w lifts __ to the map c(w~-1): CN~(X)(E1) ! CN~(X)(E0)between maximal torus normalizers. * * |__| 2.11. Corollary.Let p be an odd prime and X a connected p-compact group. Then ( limi(A(X) t, ßj(BZCX ))= ßj(BZ(X)) i = 0 0 i > 0. for j = 1, 2. In particular, lim*(A(X) t, ß*(BZCX )) = 0 if and only if X is ce* *nterless. Proof.By (2.4, 2.8, 3.12.(2)), lim0(A(X) t, ßj(BZCX ))= lim0(A(W (X), t(X)), L(X)2-j)= H2-j(W (X); L(X)) = ßj(BZ(X)) and, similarly, limi(A(X) t, ßj(BZCX )) = 0 for i > 0. * * |___| Let ßj(BZCX ) t be the sub-functor of ßj(BZCX ) which vanishes on all toral o* *bjects of A(X) and has the same value as ßj(BZCX ) on all non-toral objects of A(X). (To see t* *hat this is indeed a functor, observe that there can be no morphism from a non-toral object to a t* *oral object of the Quillen category.) 2.12. Corollary.Let p be an odd prime and X a connected p-compact group. Then t* *here is an exact sequence 0 ! lim0(A(X) t; ßj(BZCX ) t)! lim0(A(X); ßj(BZCX ))! ßj(BZ(X)) ! lim1(A(X) t; ßj(BZCX ) t)! lim1(A(X); ßj(BZCX ))! 0 while limi(A(X); ßj(BZCX )) = limi(A(X) t; ßj(BZCX ) t) for i 2. In particula* *r, lim*(A(X); ßj(BZCX ) t)~=lim*(A(X) t; ßj(BZCX )~t)=lim*(A(X); ßj(BZCX )) if and only if X is centerless. Proof.The quotient functor ß*(BZCX )=ß*(BZCX ) t vanishes on all non-toral obje* *cts so that, by (13.12), for all i 0, limi(A(X); ßj(BZCX )=ßj(BZCX )=t)limi(A(X) t; ßj(BZCX )=ßj(BZCX ) t) = limi(A(X) t; ßj(BZCX )) which was computed in (2.11). Combine this with the fact that restriction lim*(A(X) t; ßj(BZCX ) t) lim*(A(X); ßj(BZCX ) t) is an isomorphism by (13.12) again. * * |___| Let St(E) denote the Steinberg representation for GL(E). 2.13. Corollary.Let p be an odd prime, X a connected p-compact group with trivi* *al center, and let j be equal to 1 or 2. If Hom A(X)(E,()St(GL (E)), ßj(BZCX (E, ))) = 0 for * *all non-toral objects (E, ) of rank j + 1 and j + 2, then limj(A(X); ßj(BZCX )) = 0 = limj+1(A(X); ßj(BZCX )) N-DETERMINISM 7 Proof.Use (2.12) and Oliver's cochain complex [81] for computing higher limits * *over A(X). |___| For example, when (X, p) is (F4, 3) or (E8, 5) we have lim*(A(X); ßj(BZCX )) * *= 0 because the Quillen category A(X) contains, up to isomorphism, a unique non-toral object (V* *, ); this V has order p3, V ~= CX (V, ), and A(X)(V, ) = SL(V ) [41, 7.4, 10.3]. The situati* *on is much more complicated for the other members of the E-family at p = 3 [3]. 2.14. Relation between A(W, t) and the orbit category O(W ). Let O0(W ) denote * *the full subcategory of the orbit category of W generated by all objects W=G with tG 6= * *0. There are obvious functors __L_//_ A(W, t)oo___O0(W )op R given by ` ' ` ' L E0 wW(E0)-----!E1= W=W (E0) -wW(E0)-----W=W (E1), w(E0) E1, i wH j ` wW(tH) ' R W=G --! W=H = tG ------ tH , w-1Gw H. Using that G W (tG) and E tW(E) we see that L and R are adjoint functors in* * that A(W, t)(E, R(W=G)) = O0(W )op(L(E), W=G) for all objects E of A(W, t) and all objects W=G of O0(W )op. Observe also that __ G __ (2.15) NW (G) W (t ) and W (E) NW (W (E)) for all non-trivial subspaces E t and all subgroups G W . In particular, th* *e endomorphism monoid of E is the quotient __ A(W, t)(E) = W (E)=W (E) __ of the group W (E) by its normal subgroup W (E). Thus A(W, t) is an EI-category* * [54], a category in which all endomorphisms are isomorphisms. A collection is a set C of subgroups of W which is closed under conjugation. * *Let OC(W ) denote the C-orbit category, the full subcategory of O(W ) generated by all objects W=* *G with G 2 C, and AC(W, t) the full subcategory of A(W, t) generated by all objects of the form t* *G for G 2 C. The collection C is said to be subgroup-sharp for the ZpW -module L [27, 1.13* *] if ( j limi(OC(W )op; Lj)= H (W ; L) i = 0 0 i > 0 where Lj(W=G) = Hj(G; L) as in (2.14). 2.16. Corollary.If the collection C is subgroup-sharp for L and tG 6= 0 for all* * g in C then Lj restricts to an acyclic functor on AC(W, t) with lim0(AC(W, t); Lj) = Hj(W ; L). Proof.This is immediate from (13.11) as AC(W, t) = ROC(W ). * * |___| It is known [45, x5] that the collection C(p) of all p-subgroups of W is subgro* *up-sharp for any ZpW -module L and, for general reasons, tP 6= 0 for any p-group P 2 C(p). 2.17. Centralizers. I close this section with a simplified proof of the followi* *ng well-known result from [74, 3.9] which was used in connection with the mapping spaces of (2.6). Let P be a p-toral Lie group (i.e. the identity component of P is a torus and* * ß0(P ) is a finite p-group), G a compact Lie group having a finite p-group as its component group,* * and CG(P ) the Lie group centralizer, which also has a finite p-group as component group [47, * *A4], of a Lie group homomorphism f :P ! G. The standard Lie group multiplication homomorphism CG(P * *)xP ! G extending f induces a p-compact group morphism C"G(P )x ^P! G^extending ^f:^P! * *^G. (G^ denotes the p-compact group BG^pobtained by p-completing the classifying space * *of the compact Lie group G.) We shall now see that "CG(P )= CG^(P^) and in particular that [Z(* *P )= Z(P^), i.e. that centralizers and centers of p-toral Lie groups can be computed either in t* *he Lie group category or in the p-compact group category. 8 J.M. MØLLER 2.18. Lemma. [35, 101, 73] The adjoint BC"G(P )! map(BP^, BG^)Bf^ of the above standard morphism is a homotopy equivalence. In particular, BZ[(P )' map(BP^, BP^)B1 where Z(P ) is the Lie group center of P . * * S Proof.The p-toral Lie group P contains [48, 1.1] a dense p-discrete toral subgr* *oup ~P= ~Pmwhich is the union of an ascending sequence of finite p-groups ~Pm. The inclusion of * *~Pinto P induces a discrete approximation i: ~P! ^Pto the p-compact toral group ^Pand so we have* * homotopy equivalences [30, x6] map(BP^, BG^)Bf^' map(BP~, BG^)Bf^i' map(BP~m, BG^)B(f^i|P~m) for m large enough. In particular, the above mapping spaces are Fp-complete [31* *, 2.5] [30, 6.20]. Furthermore, by Dwyer-Zabrodsky [35, 1.1] and [36, 2.5] or Lannes [52], the can* *onical map BCG(P~m) ! map(BP~m, BG^)B(f^i|P~m) is an H*Fp-equivalence and here CG(P~m) ~=CG(P~) ~=CG(P ) when m is large enough and since ~Pis dense in P . * * |___| Let now G be an extended p-compact torus and ~Gits discrete approximation [31* *, 3.12]. 2.19. Lemma. Let ~: ß ! ~Gbe a homomorphism from a discrete group ß into the ex* *tended p- discrete torus ~G. 1.The group theoretic centralizer CG~(~) of ~ is a discrete approximation to * *the extended p- compact torus BCG(~) = map(Bß, BG)B~. 2.The group theoretic center Z(G~) of ~Gis a discrete approximation to the ex* *tended p-compact torus BZ(G) = map(BG, BG)B1 Proof.The maps BCG~(~) ! map(Bß, BG~)B~ ! map(Bß, BG)B~ are H*Fp-equivalences: The first map is even a homotopy equivalence [40, Lemma * *2] and the fibre of the second map is [62] a K(V, 1), for some rational vector space V , b* *ecause the fibre of BG~! BG has this form [31, 3.1]. Taking ~ to be the identity map of ~G, we obta* *in_a discrete approximation to Z(G). * *|__| 3.N-determinism This section contains comments on and further development of the material in * *[68] concerning N-determined p-compact groups. 3.1. N-determined automorphisms. Let j :N(X) ! X be the maximal torus normalize* *r for a p-compact group X. Turn this maximal torus normalizer Bj :BN(X) ! BX into * *a fibra- tion. Any automorphism f :X ! X of the p-compact group X restricts to an autom* *orphism AM (f): N(X) ! N(X) of the maximal torus normalizer, unique up to the action of* * the Weyl group W (X0) = ß1(X=N(X)) of the identity component X0 of X, such that the diag* *ram B(AM (f)) BN(X) _________//_BN(X) Bj|| Bj|| fflffl| fflffl| BX ______Bf_____//BX commutes up to based homotopy [68, x3] [1] [101, Theorem C]. The Adams-Mahmud h* *omomor- phism is the resulting homomorphism (3.2) AM :Aut (X) ! Aut(N(X))=W (X0) N-DETERMINISM 9 of automorphism groups, and X is said to have N-determined automorphisms if thi* *s homomor- phism is injective [68, 3.10]. The following lemma, collecting results from [68, 4.2, 4.3, 4.8] and (9.4), r* *educes the problem of determining which p-compact groups have N-determined automorphisms to the co* *nnected and centerless case. (The simple factors of the p-compact group X are the simple, c* *enterless p-compact groups in the splitting [32, 80] of P X0 = X0=Z(X0), the adjoint form of the id* *entity component of X.) 3.3. Lemma. Let p be any prime number. 1.The connected p-compact group X has N-determined automorphisms if its adjoi* *nt form P X does. 2.The p-compact group X has N-determined automorphisms if its identity compon* *ent X0 does. 3.The p-compact group X has N-determined automorphisms if all of its simple f* *actors do. In the connected, centerless case we use an inductive procedure based on homo* *logy decomposi- tion [31, 8.1] and preferred lifts [69]. 3.4. Proposition.[68, 4.9] Suppose that the p-compact group X is connected and * *centerless. If 1.CX (L, ) has N-determined automorphisms for each rank 1 object (L, ) of A* *(X). 2.lim1(A(X) ; ß1(BZCX )) = 0 = lim2(A(X) ; ß2(BZCX )). Then X has N-determined automorphisms. Proof.Let f :X ! X be an automorphism of X such that AM (f): N ! N is conjugate* * to the identity map of N. Then (E, f ) = (E, ) for each object (E, ) of A(X) for if * *~: E ! N is a lift of :E ! X we have f = fj~ = j O AM (f) O ~ = j O ~ = . Thus composition wit* *h f determines an automorphism Cf: CX (E, ) ! CX (E,of)each centralizer in the homology decom* *position hocolimBCX ! BX [31, x8]. In particular, when (L, ) is a rank 1 object with p* *referred lift ~: L ! T ! N [69, 4.10], we obtain a commutative diagram CN (L,M~) Cjqqqq MMMCj0M qqq MMMM xxqqq ~= M&& CX (L, )_______Cf_______//_CX (L, ) which implies, using the first assumption, that Cf is conjugate to the identity* * [68, 3.9]. But then Cf is conjugate to the identity for all (E, ) 2 Ob(A(X)). To see this, choose * *any line L < E and let __:E ! CX (L, |L)be the canonical factorization (3.18) of though the cen* *tralizer of L. Then note that under the isomorphism CCX(L, |L)(E, __(L)) ~=CX (E, ) the isomorphis* *m Cf induced by f on X corresponds (3.20) to the isomorphism CCf induced by Cf on CX (L, |L). The second assumption of the lemma assures that there are no further obstruct* *ions to conju-_ gating f to the identity [100] [68, 4.9]. * * |__| 3.5. N-determined p-compact groups. Let j :N ! X be the maximal torus normalize* *r for the p-compact group X. Suppose that N may also serve as the maximal torus norma* *lizer for some other p-compact group X0 so that we have two monomorphisms 0 (3.6) X ooj_N _j__//X0 that are both maximal torus normalizers. The p-compact group X is N-determined* * if, in this situation, there always exists an isomorphism f :X ! X0 under N, i.e. a morphis* *m f :X ! X0 such that fj and j0 are conjugate. A totally N-determined p-compact group is an* * N-determined p-compact group with N-determined automorphisms [68, 7.1]. The following lemma, collecting results from [68, 7.8, 7.10] and (9.6), reduc* *es the problem of determining which p-compact groups are N-determined to the connected and center* *less case. 3.7. Lemma. Let p be an odd prime. 1.The connected p-compact group X is N-determined if its adjoint form P X is. 2.The p-compact group X is N-determined if its identity component X0 is. 3.The p-compact group X is N-determined if all of its simple factors are. 10 J.M. MØLLER Again, in the connected, centerless case we use an inductive procedure. 3.8. Proposition.(Cf. [68, 7.17]) In the situation of ( 3.6), suppose that X is* * connected and centerless and that 1.All objects of A(X) of rank 2 have totally N-determined centralizers. 2.For each non-toral rank 2 object (V, ) of A(X) there exist a rank 2 object* * (V, 0) of A(X0) and an isomorphism f(V, ): CX (V, ) ! CX0(V,su0)ch that j0~ = 0 and CN (V, ~)M Cj rrrr MMMCj0M rrr MMMM xxrrr ~= &&M CX (V, )_____f(V,_)_____//CX0(V, 0) commutes for any of the p + 1 [68, 6.2] special preferred lifts (V, ~) of (* *V, ). 3.lim2(A(X) ; ß1(BZCX )) = 0 = lim3(A(X) ; ß2(BZCX )). Then there exists an isomorphism f :X ! X0 under N. Proof.For each rank 1 object or toral rank 2 object (V, ) of A(X), put 0= j0~* * where ~: V ! N is the preferred lift [69, 4.10], and define f(V, ): CX (V, ) ! CX0(V,to0)be * *the unique isomor- phism under CN (V, ~). Then 0equals the composition _ f(V, ) res V ______//CX (V,__)___//CX0(V, _0)___//X0 __ and f(V, )__is the canonical factorization (3.18) 0of 0. Any non-toral rank 2 object (V, ) has p+1 special preferred lifts (V, ~) ind* *exed by the set of lines in V [68, 6.2]. By assumption, neither j0~ nor the isomorphism f(V, ~): CX (V, * * ) ! CX0(V, j0~) under CN (V, ~) depend on the choice of ~. Put 0= j0~ and f(V, ) = f(V, ~) wh* *ere ~ is any of the p + 1 preferred lifts of . These morphisms f(V, ) for |V | p2 respect morphisms in A(X): Consider for* * instance a morphism ff: (V1, 1) ! (V2,fr2)om a rank 1 object to a rank 2 object. Let ~2:* *V2 ! N be the special preferred lift of 2 for which ~1 = ~2ff is the preferred lift of * *1 = 2ff. Since 01= j0~1 = j0~2ff = 02ff, the group homomorphism ff is an A(X0)-morphism (V1, 01)* * ! (V, 02). Then 02= j0~2 = j0OresNO__~2= j0OresNOCN (ff)__~2= resXOCj0OCN (ff)__~2= resXOf(V1,* * 1)OCX (ff)__2 as we see from the commutative diagram CN(ff)_~2________CN0(V1,0~1)Q_______resN___________//_N______* *______________________________________________________________________@ _____________Cjmmmmmm______QQQQCj0QQ__________________|0______* *_____________________________________________________________________ _______________mmm_________________QQQQQ_________________j|______* *__________ _______________vvmmmm_____________________((Q______________~fflffl|= V2 CX(ff)_2//_CX (V1,__1)____f(V1,_1)_______//_CX0(V1,__01)resX0//_X0 N-DETERMINISM 11 __ __ and CX0(ff) 02= f(V1, 1) O CX (ff) 2 as we see from the argument of (10.13). T* *aking centralizers of V2 we obtain the commutative diagram CN (V2, ~2) rr OO MMMM rrr |~= MMM rrr | MMM rrr | MMMM_ Cj rrrrCCN(V1,~1)(V2, CN (ff)~MMCj0M2) rr rr LLL MMM rrr rrr LLL MMM rrr rrr LLL MMM xxrrr rrrrr ~= LLLL M&& CX (V2, 2)_____r__________________________LLL_____//_CX0(V2, 02) OO rrr f(V2, 2) LLL OO ~ | rrr LLL ~| = | rrr LLLL =| | yyrrr __ ~ && | * * __ CCX(V1, 1)(V2, CX (ff)_2)_______C_=_______________//_CC 0(V1, 0)(V2, CX0(f* *f) 02) f(V1, 1) X 1 | | | | fflffl| ~= fflffl| CX (V1, 1)________________f(V1,_1)________________//_CX0(V1, 01) which shows that the isomorphism f(V2, 2): CX (V2, 2) ! CX0(V2,un02)der CN (V* *2, ~2) is in- duced from the isomorphism f(V1, 1): CX (V1, 1) ! CX0(V1,un01)der CN (V1, ~1)* *. This implies naturality as we may enlarge the commutative diagram by the morphisms CX (V2, * *2) ! CX (V1, 1) and CX0(V2, 02) ! CX0(V1, 01) induced by ff (3.20). Also, if ff 2 A(X)(V, ) GL(V ) is a Quillen automorphism of the rank 2 obj* *ect (V, ), and ~: V ! N a special preferred lift of , then ~ff is again a special preferred l* *ift of and hence 0ff = j0~ff = j0~ = 0by assumption. Thus A(X)(V, ) A(X0)(V, 0) and as CX(ff)-1 f(V, ) CX(ff) CX (V, )______//_CX (V,__)~=_//_CX0(V,__0)____//CX0(V, 0) is an isomorphism under CN (V, ~ff), it equals f(V, ) by assumption. This is n* *aturality for Quillen automorphisms of (V, ). Let now (E, ) be an object of A(X) of any rank > 2. Choose a line L < E. Def* *ine 0:E ! X0 to be the composite monomorphism _ res f(L, |L) res E ______//_CX (E,__)___//_CX (L, _|L)~=_//CX0(L, ( |L)0)___//X0 and define the isomorphism of centralizers f(E, ): CX (E, ) ! CX0(E,to0)be th* *e isomorphism Cres Cf(L, |L) C__res CX (E, o)o~=_CCX(L, |L)(E, __(L))~=//_CCX0(L, |L)(E, f(L, |L)~(L))=//_CX0(E,* * 0) induced by f(L, |L). To see that this is well-defined, let L1 < E and L2 < E be two distinct rank * *1 subgroups of E and let V < E be the subgroup generated by them. Naturality for morphisms fro* *m a rank 1 object to a rank 2 object gives a commutative diagram f(L1, |L1) CX (L1, |L1)______//_CX0(L1, ( |L1)0) _ nn77 OO OO QQ (L1)nnnn | | QQQQ nnnnn | | QQQQresQQQ nnn _(V ) | f(V, |V ) | 0res Q((Q E _________//PPPCX (V,__|V_)___//CX0(V, ( |V_)_)____//X066mm PPP | | mmmmm _(L2)PPPPPP| | mmmmm res PP'' fflffl| fflffl|mmm CX (L2, |L2)f(L2,_|L2)//_CX0(L2, ( |L2)0) showing that neither 0nor f(E, ) depend on the choice of the rank 1 subgroup * *of E. 12 J.M. MØLLER In order to show functoriality of this construction, let ff: (E1, 1) ! (E2,b* *e2)a morphism in the category A(X). Choose a rank one subgroup L1 < E1 and put ff(L1) = L2 < E2.* * Naturality for the rank 1 case gives a commutative diagram f(L1, |L1) 0 E1 ________//_CX (L1,OO1|L1)____//CX0(L1,O(O1|L1)_)___//_X0 ff|| CX(ff)|| |CX0(ff)| |||| fflffl| | | || E2 ________//_CX (L2, 1|L2)f(L2,/|L2)/_CX0(L2,_(_1|L2)0)//_X0 which shows that 01= 02ff, thus A(X)((E1, 1), (E2, 2)) A(X0)((E1, 01), (* *E2, 02)), and implies commutativity of the diagram f(E1, 1) CX (E1,OO1)__~=__//_CX0(E1,OO01) CX(ff)|| |CX(ff)| | ~= | CX (E2, 2)f(E2,_2)//_CX0(E2, 02) which is naturality. We have now constructed a collection CX (E, ) f(E,-)---!CX0(E, 0) res--!X0,* * (E, ) 2 Ob(A(X)), of homotopy A(X)-invariant centric [28] monomorphisms from the centralizers of * *the homology decomposition of BX [31, 8.1] to BX0. Because the obstruction groups are assume* *d to vanish, this collection can [100] [68, x2] be realized by a morphism Bf :BX -' hocolimBCX ! BX0 such that f(E, ) CX (E, _)______//CX0(E, 0) | | | | fflffl| fflffl| X ______f______//_X0 commutes for all (E, ) 2 Ob(A(X)). In particular, f is a morphism under the ma* *ximal torus, for f is a morphism under the maximal rank monomorphisms [31, x4] X CN (L, ~) ! X* *0 for some rank 1 object (L, ) of A(X). Thus f :X ! X0 is in fact an isomorphism [32, 5.6* *] [69, 3.11] and since f is the identity on the maximal torus T = N0, also ß0AM (f): W ! W is th* *e identity map, for W is faithfully represented as a group of operators on T [30, 9.7]. Thus ß** *(BAM (f)) is the_ identity automorphism of ß*(BN) and AM (f) is the identity of N [66, 5.2] [5, 3* *.3]. |__| Verification of the third assumption reduces to a computation involving Stein* *berg representa- tions (2.13). For the verification of the second condition we shall use the fol* *lowing lemma which may look rather specialized but in fact applies in all cases considered in this* * paper. 3.9. Lemma. Let (V, ) be a non-toral rank 2 object of A(X) with special prefer* *red lift ~: V ! N and put 0= j0~. Assume that 1.All rank 2 objects of A(X), whose centralizers are isomorphic to CX (V, ),* * are isomorphic to (V, ). 2.A(X)(V, ) = SL(V ). (Then also A(X0)(V, 0) = SL(V ).) 3.The isomorphism f(V, ~): CX (V, ) ! CX0(V,un0)der CN (V, ~) is SL(V )op-eq* *uivariant. Then j0~1 = 0and f(V, ~1) = f(V, ~): CX (V, ) ! CX0(V,fo0)r all special prefe* *rred lifts (V, ~1) of (V, ). Proof.The GL (V )-orbit (V, ) . GL(V ) contains p - 1 objects, the GL (V )-orb* *it (V, ~) . GL(V ) contains (p - 1)(p + 1) objects, and the map j :(V, ~) . GL(V ) ! (V, ) .iGL(V* *s)(p + 1)-to-1 [68, 6.2]. By assumption, the orbit (V, ~) . GL(V ) contains all special preferred l* *ifts whose centralizers in N are isomorphic to N(CX (V, )). Since X and X0 have the same special prefe* *rred lifts [68, 7.13], also j0:(V, ~) . GL(V ) ! (V, j0~) .iGL(Vs)(p + 1)-to-1. Since the orbit* * (V, j0~) . GL(V ) N-DETERMINISM 13 thus contains p - 1 objects, the stabilizer subgroup of (V, j0~) must be SL(V )* * as this in the only subgroup of GL(V ) of that index. Thus the Quillen automorphism group A(X0)(V, * * 0) = SL(V ). Any other special preferred lift of has the form ~ff for an ff in SL(V ) [6* *8, 6.2], so, clearly, j0(~ff) = 0ff = 0is independent of the choice of ff. The commutative diagram CN(ff) CN(ff) CN (V, ~ff)oo_____________CN_(V, ~)_________________//_CN (V, ~ff) qq MMMCj0MM | Cj|| Cjqqqqq MMMM |Cj0 fflffl| xxqqq MM&& fflffl| CX (V, )CX(ffCX)(V,oo)_____f(V,~)_____//CX0(V, C0)_//_CX0(V, 0) X0(ff) shows that f(V, ~ff) = CX0(ff)f(V, ~)CX (ff)-1, since CX (V, ) has N-determine* *d automorphisms_ so that f(V, ~ff) = f(V, ~) by the third assumption. * * |__| The canonical factorizations of and 0 are SL(V )op-equivariant (3.19) and * *they provide a commutative diagram (3.10) V II _ vvvv II__I0 vv IIII --vvv I$$ CX (V, _)__f(V,~)__//CX0(V, 0) which shows that the restriction of f(V, ~) to V is SL(V )op-equivariant. It i* *s a tautology that f(V, ~ff) = f(V, ~) for all ff in the Borel subgroup stabilizing ~ so it is in * *fact only necessary to check equivariance with respect to one other element (of order p) [91, 3.6.21] * *of SL(V ). 3.11. Centers and automorphism groups of p-compact groups. The following theorem collects some useful facts from various sources that will be applied several ti* *mes in this paper. 3.12. Theorem. Let p be an odd prime and X a connected p-compact group. 1.[4, 5] The semi-direct product ~N(X) = ~T(X) o W (X) is a discrete approxim* *ation [31, 3.12] to the maximal torus normalizer N(X). 2.[31, x7] The abelian group ~Z(X) given by 0 0 1 H0(W (X); ~T(X)) = H (W (X); L(X)) Q =H (W (X); L(X)) x H (W (X); L(X* *)) is a discrete approximation to the center [58, 31] of X. AM 3.[68, 7.2] Aut(X) ~= Out(N(X)) provided X is totally N-determined. The automorphism group of N(X) sits [66, 5.2] in a short exact sequence (3.13) 0 ! H1(W (X); ~T(X)) ! Aut(N(X)) -!Aut(W (X), ~T(X), e(X)) ! 1 where the normal subgroup to the left consists of all automorphisms of N(X) tha* *t induce the identity on homotopy groups, and the group to the right consists of all pairs (* *ff, `) 2 Aut(W (X))x Aut(T~(X)) such that ` is ff-linear and the induced automorphism H2(ff-1, `) [9* *5, 6.7.6] preserves the extension class e(X) 2 H2(W (X); ~T(X)). The image of W (X0) W (X) = ß0N* *(X) [58, 3.8] in Aut(N(X)) does not intersect the subgroup H1(W (X); ~T(X)) (as W (X0) i* *s represented faithfully in Aut(T~(X)) [30, 9.7]) so there is an induced short exact sequence (3.14)0 ! H1(W (X); ~T(X)) ! Aut(N(X))=W (X0) -!Aut(W (X), ~T(X), e(X))=W (X0) * *! 1 whose middle term is the target of the Adams-Mahmud homomorphism (3.2). If X is connected and p is odd, the cohomology group to the left is trivial a* *nd e(X) = 0 [5] so Aut(N(X)) ~=Aut(W (X), ~T(X), 0) ~=NGL(L(X))(W (X)) is [68, 3.5] [5, 3.3] the group of self-similarities of the Zp-reflection group* * (W (X), L(X)) (4.1), and the target of the Adams-Mahmud homomorphism (3.2) (3.15) Out(N(X)) = Aut(N(X))=W (X) ~=NGL(L(X))(W (X))=W (X) 14 J.M. MØLLER is [64, x2] the middle term of an exact sequence (3.16) 1 ! AutZp[W(X)](L(X))=Z(W (X)) ! Out(N(X)) ! Out(W (X)) of automorphism groups. An automorphism of X is exotic if its lift to N(X) [68,* * 3.7] induces a non-trivial outer automorphism of W (X). 3.17. Remark. Let p and X be as in (3.12). 1.The formula ßj(BZ(X)) = H2-j(W (X); L(X)), j = 1, 2, is an alternative version of (3.12.(2)). 2.The endomorphism monoid of X is given by ( End(X) - {0} ~= NGL(L(X))(W (X))=W (X) = Aut(X) p | |W (X)| NGL(L(X) Q)(W (X)) \ End(L(X)) =W (X)p 6 | |W (X)| provided X is totally N-determined and simple [67, 5.4]; use [67, 5.6, 5.6]* * and [66, 5.2] to see this. See [46, 47, 48] for the Lie case. 3.18. Canonical factorizations. [30, 8.2] Let :V ! Xbe a monomorphism from an* * elementary abelian p-group to the p-compact group X. The canonical factorization of thro* *ugh its centralizer is the central monomorphism___(V ): V ! CX (V,wh)ose adjoint is V x V +-!V -!* *X. The composition V -! CX (V, ) res--!X equals . If ff: (V1, 1) ! (V2,is2)a morphi* *sm in A(X) then the canonical factorizations are related by a commutative diagram _1 res (3.19) V1 _______//_CXO(V1,__1)___//_XO ff|| CX(ff)|| |||| fflffl| | || V2 ____2__//_CX (V2,__2)res//_X so that ff: (V1, __1) ! (V2, CXi(ff)__2)s a morphism in A(CX (V1, 1)). The in* *duced morphisms CX (ff) and CCX(V1, 1)(ff) can be identified in that the diagram (3.20) CCX(V1, 1)(V1, __1)Cres~=//_CX4(V1,4 1) OO jjj OO | resjjjjjjj | CCX(V1, 1)(ff)|jjjjj |CX(ff) | j C | CCX(V1, 1)(V2, CX (ff)__2)res~=//_CX (V2, 2) commutes up to conjugacy. 4. Cohomologically unique p-compact groups We shall here discuss to what extent N-determined p-compact groups are determ* *ined by their Weyl groups or their mod p cohomology algebras (4.3). The message intended is t* *hat cohomolog- ical uniqueness [36, 74, 16, 93] is incidental while N-determinism is universal* *. As the Weyl group of a connected p-compact group is a reflection subgroup of the automorphism gro* *up of the lattice we start out by introducing the category of reflection subgroups. For a commutative domain R, an element g of GL (r, R) is a reflection if the * *(r x r) matrix Ir- g has rank at most 1 where Ir is the (r x r) identity matrix. A subgroup W * *of GL(r, R) is a reflection subgroup if it is generated by the reflections contained in it. 4.1. Definition.For R = Zp, Qp, Fp, let R - Reflbe the category with objects:pairs (W, L) where L is a finitely generated free R-module and W a fi* *nite reflection subgroup of AutR(L), and, morphisms: pairs (ff, `): (W1, L1) ! (W2,wL2)here ff: W1 ! W2is a group homo* *morphism and ` 2 Hom R(L1, L2) an ff-linear R-module homomorphism. N-DETERMINISM 15 A similarity is an isomorphism in R-Refl. Two objects of Zp-Reflare R-similar i* *f they are taken to isomorphic objects of R - Reflby the functor rR :Zp- Refl! R - Reflinduced b* *y - ZpR. G0(W, L) (Gp(W, L)) is the set of similarity classes of objects of Zp - Refltha* *t are Qp-similar (Fp-similar) to the object (W, L). An object (W, L) of Zp- Reflis said to be si* *mple if L Zp Qp is a simple QpW -module. A similarity class of objects of R - Reflamounts to an integer r 0 and a co* *njugacy class (W ) of a reflection subgroup of GL(r, R). The automorphism group AutR-Refl(W, L) of* * an R-reflection subgroup is isomorphic to the normalizer NGL(L R)(W ) of W in GL(L R) [64, x2* *] [63]. In Zp - Reflwe shall often write r0 (rp) for the functor rR if R = Qp (R = Fp* *). (Of course, if R = Zp, then rR is the identity functor.) By [29, Proof of 5.2], W is a refl* *ection subgroup of GL(L) if and only if rpW is a reflection subgroup of GL(L Z=p); also, W and rpW* * are abstractly isomorphic groups as the kernel of GL (r, Zp) ! GL (r, Fp) contains no non-triv* *ial finite order elements when p is odd [57] [88, 10.7.1]. Two objects, (W1, L1) and (W2, L2), o* *f Zp - Reflare Qp-similar iff there exists a morphism (ff, `): (W1, L1) ! (W2,iL2)n Zp- Reflsu* *ch that r0(ff, `) is an isomorphism in Qp - Refl, and they are Fp-similar iff there exists a grou* *p isomorphism ff: W1 ! W2and a Zp-linear isomorphism ` :L1 ! L2such that (ff, ` ZpFp) is an * *isomorphism in Fp- Refl. All elements of G0(W, L), which is a finite set according to the J* *ordan-Zassenhauss Theorem [24, 24.2], are represented by centerings of (W, L), i.e. by objects of* * the form (W, M) where M is ZpW -submodule of L and the index [L : M] is finite. Two centerings,* * (W, M1) and (W, M2), are similar if and only if A(M1) = M2 for some A in the normalizer NGL* *(L Q)(W ) of W in GL(L Q) [84, 2.1-2.3]. 4.2. Proposition.Let (W, L) be an object of Zp- Refl. 1.G0(W, L) = {(W 0) < GL(L) | (r0W 0) = (r0W )} 2.Gp(W, L) = {(W 0) < GL(L) | rpW 0= rpW } As usual, (W ) stands for the conjugacy class of the subgroup W . Proof.1. Let A(W ) = {U 2 GL (L Q) | U-1W U GL (L)} be the set of automorph* *isms of L Q that conjugate the subgroup W GL (L) into (another) subgroup of GL (L).* * We shall define surjections {(W 0) < GL(L) | (r0W 0) = (r0W )} j A(W ) i G0(W, L) and show that the corresponding equivalence relations on A(W ) are the same. Th* *e surjection to the left simply takes U 2 A(W ) to the subgroup conjugacy class (U-1W U). The m* *ap to the right takes U 2 A(W ) to the similarity class of (W, UL). This is indeed a well-defin* *ed surjection because for U 2 GL(L Q) we have -1 U-1W U GL(L) , U W U (L) = L , W (UL) = UL meaning that UL is a ZpW -submodule of L Q if and only if U 2 A(W ). The Zp- * *Refl-objects (W, UL) and (W, V L), U, V 2 A(W ), are similar if and only if V AU-1W = W V AU* *-1 for some isomorphism of the form -1 A V UL U---!~=L -!~=L -!~=V L for an A 2 GL(L). In other words, (W, UL) and (W, V L) are similar if and only * *if U-1W U and V -1W V are conjugate as subgroups of GL(L). 2. The map {(W 0) < GL(L) | rpW 0= rpW } ! Gp(W, L) taking (W 0) to (W 0, L) is clearly well-defined and injective. To see that it * *is also surjective , let (W1, L1) be an object of Zp - Reflthat admits a similarity (ffp, `p): rp(W1, L1* *) ! rp(W,iL)n Fp - Refl. Lift the isomorphism `p to a Zp-linear isomorphism ` :L1 ! L. Then (* *W1, L1) and (W 0, L), W 0= `W1`-1, are similar and rpW 0= `prp(W1)`-1p= rpW and thus the su* *bgroup W 0_ is mapped to the element of Gp(W, L) represented by (W1, L1). * * |__| The Weyl group W (X) of a connected p-compact group X is by birth a finite re* *flection subgroup of GL(L(X)) [30, 9.7] and (W (X), L(X)), (r0W (X), L(X) Qp), and (rpW (X), L(* *X) Fp) are 16 J.M. MØLLER objects of R - Reflfor R = Zp, Qp, Fp, called the Zp-Weyl group (or just the We* *yl group), the Qp-Weyl group, and the Fp-Weyl group of X, respectively. (As to functoriality w* *e note that any toric morphism [70] between connected p-compact groups determines a morphism be* *tween the corresponding reflection subgroups.) 4.3. Definition.Let X be a connected p-compact group. 1.X is determined by its R-Weyl group if any connected p-compact group Y with* * the same R-Weyl group as X, i.e. with W (Y ) R-similar to W (X), is isomorphic to X. 2.X is a cohomologically unique p-compact group if any connected p-compact gr* *oup Y with H*(BY ; Fp) isomorphic to H*(BX; Fp) as an algebra over the mod p Steenrod * *algebra, is isomorphic to X. All p-compact tori are clearly cohomologically unique. 4.4. Corollary.Let p be an odd prime and X an N-determined connected p-compact * *group. 1.X is determined by its Zp-Weyl group W (X). 2.If G0(W (X), L(X)) = *, then X is determined by its Qp-Weyl group r0W (X). 3.If Gp(W (X), L(X)) = *, then X is determined by its Fp-Weyl group rpW (X) 4.If X is determined by its Fp-Weyl group, then X is a cohomologically unique* * p-compact group. Proof.At odd primes, the (discrete) maximal torus normalizer of a connected p-c* *ompact group, which is a split extension (3.12), is determined, up to isomorphism, by the sim* *ilarity class of the Weyl group. The next two items are immediate consequences of this, since we are* * assuming W (X) recoverable from r0W (X), respectively rpW (X). The rational rank r(X) as well as the Fp-Weyl group rpW (X) can be read off f* *rom H*(BX, Fp) thanks to Lannes theory [52]. Indeed, r(X) is the maximal r 0 for which there* * exists a monomor- phism (Z=p)r æ X whose centralizer is a p-compact torus and rpW (X) is (2.8) th* *e automorphism_ group in the Quillen category of the object t(X) ! X. * * |__| 4.5. Lemma. Let W be a finite reflection subgroup of GL(L). Put t = L=pL. 1.[84, (1) p. 248] If t is an irreducible Fp[W ]-module, then G0(W, L) = *. 2.[7, 7.1.2] If H1(rpW ; Hom(t, t)) = 0, then Gp(W, L) = *. Proof.For item 1, let M be a ZpW -submodule of L not contained in pL. Since the* * image of M in t = L=pL is non-trivial, we get L = M + pL by irreducibility and L = M by Nakay* *ama's lemma __ [86, 9.2]. The H1-condition of item 2 assures that rpW lifts uniquely to GL(r, * *Zp). |__| The sets G0(W, L) and Gp(W, L) are determined in (11.18, 11.25, 11.26) for (W* *, L) a simple reflection subgroup and p an odd prime. For a connected p-compact group X, let SX stand for the universal covering gr* *oup of X and P X = X=Z(X) the adjoint form of X [31, 58]. Recall that for (W, L) 2 Ob(Zp - R* *efl) there are associated objects (SW, SL), (P W, P L) 2 Ob(Zp- Refl) [75] (11.1). 4.6. Lemma. SSX = SX = SP X and P P X = P X = P SX for any connected p-compact * *group X. Proof.Use [58, 4.7, 5.4, 5.5] and that BSX = BX<2>is the 2-connected cover of B* *X. |___| 4.7. Proposition.Let p be an odd prime and X an N-determined connected p-compac* *t group. 1.H0(W (X); ~T(X)) = ~Z(X) and H0(W (X); L(X)) = ß1(X). 2.SL(X) = L(SX) and P L(X) = L(P X). Proof.The formula for the center of X (3.12.(2)) immediately shows that P L(X) * *= L(P X). By inspection we see that (4.8) H0(W (P X); L(P X)) = ß2(BP X) for any simple p-compact group X. (The formula is known to hold in the Lie case* * by classical results. The exotic simple p-compact groups are all centerless and polynomial (* *7.9) so in this case N-DETERMINISM 17 X = P X and ß2(BX) = 0 because H2(BX; Fp) = 0. Also H0(W (X); L(X)) = 0 by (11.* *4.3) for G0(W (X)) = * (11.18) so that L(X) = SL(X).) Therefore, SL(P X) = kerL(P X) ! H0(W (P X); L(P X)) = kerL(P X) ! ß1(P X) = L(SP X) = L(SX) for any simple X. For a generalQconnected X, the Splitting Theorem for centerl* *ess p-compact groups [32] tells us that P X = P Xiwhere Xiis simple. Consequently, Y Y Y SL(X) = SP L(X) = SL(P X) = SL(P Xi) = L(SXi) = L(SP Xi) = L(SP X) = L(SX). From the finite covering ß ! SX x Z(X)0 ! X of [58, 5.4] we obtain a short exac* *t sequence of ZpW (X)-modules (4.9) 0 ! SL(X) x H0(W (X); L(X)) ! L(X) ! ß ! 0 and, using H1(W (X); ß) = 0 = H0(W (X); SL(X)) (11.3, 11.4.3), a short exact se* *quence of Zp- modules 0 ! H0(W (X); L(X)) ! H0(W (X); L(X)) ! ß ! 0 identical to the short exact sequence for computing ß1(X). * * |___| Recall, that we write X1 X2 if there exists an isogeny X1 i X2 [67, p. 216]* * in pcg and (W1, L1) (W2, L2) if there exists an isogeny (W1, L1) ! (W2, L2) in Zp- Refl(* *11.23). 4.10. Corollary.Let p be an odd prime and X1 and X2 two connected p-compact gro* *ups. Assume that P X2 is totally N-determined. 1.X1 and X2 are locally isomorphic, (W, L)(X1) and (W, L)(X2) are Qp-similar. 2.X1 X2 , (W, L)(X1) (W, L)(X2). 3.The local isomorphism system [67, 4.7] of X2 is poset isomorphic to G0(W (X* *2), L(X2)). Proof.Write (Wi, Li) for (W (Xi), L(Xi)), i = 1, 2. It is clear from the result* *s of [67, x2-x4] that if X1 and X2 are locally isomorphic (and X1 X2) then (W1, L1) and (W2, L2) ar* *e Qp-similar (and (W1, L1) (W2, L2)). Conversely, suppose that (W1, L1) and (W2, L2) are Q* *p-similar. Then (W1, L1) ~=(W2, P~ff(SL2)) for some diagram ~ff:~ß(SL2) oe ß(L1) '-!~TH0(W2; L2* *) of Zp-modules (11.20). Since SL2 = L(SX2), this means that (W1, L1) is similar to (W (X02)L(* *X02)) for the p-compact group X02= SX2x_Z(X2)0_(ß(L 1), ') locally isomorphic to X2[67, 2.8]. But X02is totally N-determined if P X2is (3.* *3, 3.7), and therefore X1 is actually isomorphic to X02(4.4). Moreover, if (W2, P~ff(SL2)) (W2, L2) * *then (11.21) there is a commutative diagram ~ß(SL2)oo_o~ß(L1)o'_//_~TH0(W2; L2) fflffl ~=|| || || |fflffl fflffl| fflfflfflffl| ~ß(SL2)oo_o~ß(L2)o__//_~TH0(W2; L2) induced by some automorphism of SL2 and some epimorphism of ~TH0(W2; L2) = ~Z(X* *2)0 onto itself with finite kernel. But any automorphism of SL2 = L(SX2) comes from an a* *utomorphism __ of SX2 (3.12.(3)) and so the above diagram determines [67, 4.3, 4.5] an isogeny* * X1 i X2. |__| 4.11. Corollary.Let p be an odd prime. There are fibration sequences Bß(L(X)) ! BSX x B2H0(W (X); L(X)) ! BX BX ! B2LH0(W (X); ~T(X)) x BP X ! B2~ß(L(X)) for any N-determined connected p-compact group X. 18 J.M. MØLLER Proof.Write (W, L) for the reflection subgroup (W (X), L(X)) associated to X. T* *he first of these fibration sequences will follow if we can show that (4.12) Bß(L) _______//BSX | || | | fflffl| fflffl| B2H0(W ; L)____//BX is homotopy fibre square. The top horizontal map corresponds to the monomorphi* *sm ß(L) æ ~ß(SL) = H0(W ; ~T(SL)) = ~Z(SX) of (11.8.2) and the bottom one corresponds to * *the monomor- phism ~TH0(W ; L) æ H0(W ; ~T(L)) = ~Z(X) of (11.4.1). There is a fibration BH0(W ; L) ! Bß(L) ! B2H0(W ; L) induced from the short exact sequence 0 ! H0(W ; L) ! H0(W ; L) ! ß(L) ! 0 of a* *belian groups. But H0(W ; L) and ß1(X) are (4.7) isomorphic abelian groups and thus the left a* *nd the right vertical maps in (4.12) have identical fibres. For the second fibration, it is enough to prove that there exists a homotopy * *fibre square (4.13) BX _____//_B2LH0(W ; ~T(L)) | | | | | fflffl| |fflffl BP X ________//_B2~ß(L) with an abelian topological group in the lower right corner. The top horizontal* * map corresponds to the epimorphism H0(W ; L) i LH0(W ; ~T(L)) of (11.4.4) and the bottom one to th* *e epimorphism H0(W ; P L) = ß(P L) i ~ß(L) of (11.8.2). There is a fibration BH0(W ; ~T(L))^p! B2LH0(W ; ~T(L)) ! B2~ß(L) obtained by applying the Fibre Lemma [14, II.5.1] to the fibration BH0(W ; ~T(L* *)) ! B2~ß(L) with BH0(W ; ~T(L)) as fibre reflecting the defining short exact sequence for ~ß(L).* * But H0(W ; ~T(L)) is (3.12.(2)) a discrete approximation to the center of X and thus the left and* * the right_vertical maps in (4.13) have identical fibres. * * |__| The N-determined connected p-compact group BX is, in other words, the quotien* *t p-compact group of BSX x B2H0(W (X); L(X)) corresponding to the subgroup ß(L(X)) (11.8.(4* *)) of the center ~ß(SL(X)) x ~TH0(W (X); L(X)) (4.7, 11.8.(1)), or, the covering p-compac* *t group [17] [58, 3.3] of B2LH0(W (X); ~T(X)) x BP X corresponding to the quotient group ~ß(L(X))* * (11.8.(3)) of the fundamental group LH0(W (X); ~T(X)) x ß(P L(X)) (4.7, 11.8.(1)). According to [74, 8.1], any "p-convenient and simply connected or pseudo simp* *ly connected" compact connected Lie group satisfies (4.5.(2)). For our purposes, however, the* * following corollary will suffice. 4.14. Corollary.Let p be an odd prime and let X be the p-compact group represen* *ted by othe product subgroup U(n1) x . .x.U(nk), n1+ . .+.nk = n, ni 0, of U(n), o* *r, othe intersection with SU(n) of such a subgroup of U(n), or, othe image in PU(n) = U(n)=U(1) of such a subgroup of U(n). Then Gp(W (X), L(X)) = *. Proof.Write t = t(U(n)), t0 = t(SU (n)), and t1 = t(PU (n)) (the dual to t0). I* *t suffices (4.5.2) to show that H1(W ; -) = 0 where W is a subgroup of the form n1x . .x. nk of W (U* *(n)) = n and the blank is any of the Fp n-modules Hom (t, t), Hom (t0, t0) or Hom (t1, t* *1). Let i be 1 or 2. From the fact that Hi( n; Fp) = 0 for all n when p is odd [5* *0, 12.2.2], we inductively deduce that also Hi(W ; Fp) = 0. But then also Hi(W ; t0) ~=Hi(W ; t) ~=Hi(W ; t1) H1(W ; Hom(t0, t0)) ~=H1(W ; Hom(t, t0)), H1(W ; Hom(t1, t1)) ~=H1(W ; Ho* *m(t, t1)) N-DETERMINISM 19 as we see from the exact sequences in cohomology induced by the short exact seq* *uences 0 ! t0 ! t +-!Fp ! 0,0 ! Fp -! t ! t1 ! 0, 0 ! t0 ! Hom (t, t0) ! Hom (t0, t0)0!!0,Hom(t1, t1) ! Hom (t, t1) ! t1 ! 0 of Fp n-modules. Since the representation t = Ind nn-1(Fp) is induced from the trivial 1-dimen* *sional representa- tion, its restriction to W , Y resWn(t) = resWnInd nn-1(Fp) = IndW\x n-1(Fp), x2W\ n= n-1 is a product of representations induced from trivial 1-dimensional representati* *ons. But W \x n-1, the intersection of W with aQconjugate of n-1 = 1x n-1, is again a subgroup * *of W -type, so it follows that Hi(W ; t) = Hi(W \ x n-1; Fp) = 0. Furthermore, Hom (t, -) = * *Ind nn-1(-) so that, by the same argument, H1(W ; Hom(t, -)) = 0 where the blank can be t, t0 * *or t1. |___| 5.The p-compact group PGL (n, C) In this section we show N-determinism for the A-family of p-compact groups wh* *ere p is an odd prime. See Broto and Viruel [16, 15] for an alternative proof and [68, 7.19] f* *or a prototype of Theorem 5.1. 5.1. Theorem. The p-compact group PGL (n, C) is totally N-determined for all n * * 1 and all odd primes p. As a consequence (5.3) of this theorem also GL(n, C), for instance, is totall* *y N-determined so that we may conclude from (3.12.(3), 3.16) that Aut(GL (n, C)) = AutZpW(GL(n,C))(L(GL (n, C))) = AutZp[ n](Znp) when n > 2. 5.2. Corollary.Let X be a p-compact group whose Qp-Weyl group r0W (X) is in Cla* *rk-Ewing family 1 and assume that p is odd. Then: 1.X is totally N-determined. 2.X is determined by its Zp-Weyl group. 3.For n > 2 ( End (X) ~= Zp n < p Zxp[ {0} n p while End(SL(2, C)) = Zp=Zx. 4.If ß1(X) = 1, or Z(X) = 1, or n < p3, then X is determined by its Fp-Weyl g* *roup and X is a cohomologically unique p-compact group. Proof.This is immediate from (3.3, 3.7) and (3.17.(2), 4.4, 11.18). In connecti* *on with the appli- cation of (3.17.(2)), observe that the outer automorphism of the symmetric grou* *p 6 [91, 2.2.18, 2.2.20] can not be lifted to an automorphism of N(X) because all such automorph* *isms_take_reflec- tions to reflections. * * |__| 5.3. Corollary.Let X be the p-compact group represented by othe product subgroup GL(n1, C) x . .x.GL(nk, C), n1+ . .+.nk = n, ni 0, of* * GL(n, C), othe intersection of such a subgroup with SL(n, C), or, othe image of such a subgroup in PGL (n, C). Then X is totally N-determined, X is determined by its R-Weyl group for R = Zp,* * Fp, and X is a cohomologically unique p-compact group (p odd). Proof.That X is totally N-determined follows from (5.1) together with (3.3, 3.7* *). Apply_(4.14, 4.4) for the other properties of X. * * |__| 20 J.M. MØLLER We shall prove (5.1) by inductively verifying that BPGL (n, C) satisfies the * *sufficient criteria (3.4, 3.8, 3.9) for total N-determinism. For this process, it is crucial (2.12) to ha* *ve information about the non-toral elementary abelian p-subgroups of PGL (n, C) = GL(n, C)=Cx and th* *eir centralizers. Thus we shall start out by identifying the non-toral elementary abelian p-subgr* *oups of PGL (n, C), their Quillen automorphism groups, and their centralizers. Non-toral elementary abelian p-subgroups of PGL (n, C) can be constructed fro* *m extra-special p-groups contained in GL (n, C) as follows: Let P be an extra-special p-subgro* *up (this means [P, P ] = Z(P ) is of order p [85, 5.3]) and E an elementary abelian p-subgroup* * of GL(n, C) such that Z(P ) Z(GL (n, C)), [P, E] = {1} = P \ E, where Z(P ) is the center of P and Z(GL (n, C)) = Cx the center of GL (n, C). T* *hen T = P E is a non-abelian subgroup of GL(n, C) that maps onto a non-trivial non-toral el* *ementary abelian p-subgroup, V , of PGL (n, C) with kernel Z(P ). (V is non-toral because the pr* *e-image of toral subgroup of PGL (n, C) is toral in GL(n, C).) In fact, all non-trivial non-toral elementary abelian p-subgroups of PGL (n, * *C) have this form. 5.4. Lemma. [41, 3.1] Let V be a non-trivial non-toral elementary abelian p-sub* *group of PGL (n, C). Then op divides n, and, othere is an inclusion morphism of short exact sequences of groups 1____//_Z(P_)_____//_T_________//_V_______//1 | |æ æ| | | | fflffl| fflffl| fflffl| 1_____//_Cx____//_GL(n, C)__//_PGL(n, C)__//_1 where T = P E is the direct product of an extra-special p-group P GL (n* *, C) and an elementary abelian p-group E GL(n, C) such that P \ E = {1} = [P, E]. The* * extra-special p-group P can be chosen to have exponent p. Proof.If n is not divisible by p, then all elementary abelian p-subgroups of PG* *L (n, C) are toral. Assume now that p divides n. As H2(V ; Z=p) maps onto Hom (H2(V ); Cx) = H2(V, * *Cx), there is a subgroup R GL(n, C) that maps onto V with a kernel that is cyclic of order * *p and central in GL(n, C). If R were abelian, then R and V would be toral subgroups. The commutator subgroup [R, R] is cyclic of order p for it is non-trivial and* * contained in the kernel of the epimorphism R ! V . Thus V = R=[R, R]. Let P be a normal subgr* *oup of R such that P=[R, R] is complementary to Z(R)=[R, R]. Then R = P Z(R) and P is ex* *tra-special as Z(P ) = P \ Z(R) = [R, R] = [P Z(R), P Z(R)] = [P, P ]. The commutative diagram P x Z(R)__________//_GL(n, C) | | | | |fflffl fflffl| P=[R, R] x Z(R)=[R,_R]__//PGL(n, C) has an adjoint diagram Z(R) ___________//CGL(n,C)(P_)_____//_GL(n, C) | | | | | | fflffl| fflffl| fflffl| Z(R)=[R, R]____//_CPGL(n,C)(P=[R,_R])0//_PGL(n, C) where the horizontal maps are inclusions and the two rightmost vertical maps ar* *e epimorphisms with kernel Cx. The centralizer of P in GL(n, C) is a product of general linear* * groups [82, Propo- sition 4] and Z(R) is included here as an abelian, hence toral, subgroup. There* *fore, Z(R)=[R, R] is included as a toral subgroup in (the identity component) of the centralizer * *of P=[R, R] in PGL (n, C). It follows that Z(R)=[R, R] is the isomorphic image of an elementar* *y abelian p-group E CGL(n,C)(P ) GL(n, C). N-DETERMINISM 21 By construction, [P, E] = {1} = E \Z(P ) = E \P , so P and E permute, T = P E* * is a subgroup of GL(n, C) that maps onto Vpwith_kernel Z(P ). For any extra-special p-group P* *- GL(n, C) of exponent p2 with Z(P-) = p1 there is an extra-special p-group P+ GL(n, C) * *of exponent p that has the same center, the same centralizer, and the same image in PGL (n, C* *) as P-_(5.19). Therefore we can assume P = P+ has exponent p. * * |__| The commutatorpsubgroup_and the center of the covering group T = P E of V are [* *T, T ] = [P, P ] = Z(P ) = p1 and Z(T ) = Z(P )E [T, T ]. By taking commutators in T we get an alternating bilinear form (5.5) f :V x V ! [T, T ] on V = T=[T, T ], i.e. f(u1, u2) = [__u1, __u2] where __u2 T is a lift of u 2 V* * . This bilinear form may be degenerate in that V ?= Z(T )=[T, T ] = E and we obtain a non-degenerate alternating bilinear form __ (5.6) f:V=V ?x V=V ?! [T, T ] by factoring out V ? or, equivalently, by restricting to the subspace P=[P, P ]* * ~=V=V ?~=T=Z(T ) of V . Define Isom(V, f)= {ff 2 Aut(V ) | f(ff(u1), ff(u2)) = f(u1, u2)} __ ? __ __ Aut(f)= {(A, a) 2 Aut(V=V ) x Aut(Z(T )) | fO (A x A) = a O f} to be the group of all isometries of (V, f) and, respectively, the group of all* * pairs of automorphisms (A, a) 2 Aut(V=V ?) x Aut(Z(T )) that make _f " V=V ?x V=V ?_____//[T,ØT_]_//Z(T ) AxA || |a| fflffl| " fflffl| V=V ?x V=V ?___f_//[T,ØT_]_//Z(T ) commutative. Any outer automorphism ff of T induces an an automorphism_a(ff) of Z(T ) and * *an automorphism A(ff) of T=Z(T ) = V=V ?such that (A(ff), a(ff)) 2 Aut(f). 5.7. Lemma. For odd p there is a short exact sequence __ 1 ! Hom (V=V ?, V ?) ! Out(T ) (A,a)---!Aut(f) ! 1 for the outer automorphism group of T . Proof.The 2-cocycle for the extension Z(T ) ! T ! T=Z(T ) = V=V ?is c where __u __ ____ 1. u2= c(u1, u2)u1u2, u1, u2 2 T=Z(T ), where __u2 T is a lift of u 2 T=Z(T ). Since __ __ __ __ __ ____ -1 __ __ ____ -1 __ __ ____ -1 f(u1, u2)c(u2, u1) = [u1, u2] u2. u1 u2u1 = u1. u2 u2u1 = u1. u2 u1u2 = c(u1, u2) __ for all u1, u2 2 T=Z(T ), the 2-cochain_fmeasures the failure of the 2-cocycle * *c in being symmetric. If the pair (A, a) is in Aut(f), then the 2-cochain d = (A*c)-1(a*c) is symme* *tric, for __ __ -1 -1 f(Au1, Au2) = af(u1, u2) , c(Au1, Au2) ac(u1, u2) = c(Au2, Au1) ac(u2* *, u1), and hence 2d = ffiq where q is the associated quadratic form, q(u) = d(u, u), v* *iewed as a 1-cocycle. Thus (A, a) can be lifted to an automorphism of T . The kernel of the map ff ! (A(ff), a(ff)) is easily determined as follows. Th* *ere is a surjection __ Hom(T=Z(T ), Z(T )) i ker Out(T ) ! Aut(f) 22 J.M. MØLLER taking the homomorphism ': T=Z(T ) ! Z(Tt)o the automorphism t ! '(t)t of T . * *This au- tomorphism is inner precisely when '(t) = [u, t] for some u 2 T . Since any ho* *momorphisms T ! [T, T ] is of this form, the kernel is isomorphic to Hom(T=Z(T ), Z(T ))= H* *om(T=Z(T_), [T, T ]) ~= Hom (T=Z(T ), Z(T )=[T, T ]) ~=Hom(V=V ?, V ?). * * |__| * * __ For example, if P is an extra-special p-group then Out(P ) is isomorphic to t* *he group Aut(f) (when p is odd). We shall next determine the Quillen automorphism groups of the subgroups T * *GL(n, C) and V PGL (n, C) of (5.4). 5.8. Definition.For a homomorphism æ: H ! Gof a (finite) group H into a Lie gro* *up G, define the Quillen automorphism group A(G)(H, æ) as A(G)(H, æ) = {ff 2 Out(H) | (æff) = (æ)} where (æ) denotes the representation (æ) 2 Rep(H, G) = Hom (H, G)=G represented* * by the homo- morphism æ. If the target of æ is G = GL(n, C), in particular, then A(G)(H, æ) = {ff 2 Out(H) | tr(æff) = tr(æ)} by complex representation theory. 5.9. Lemma. Let T = P E and V = T=[T, T ] be as in ( 5.4) and assume that T has* * exponent p. Then the homomorphism A(GL (n, C))(T, æ) ! A(PGL (n, C))(V, æ) is surjective. Proof.Suppose that BCx normalizes V in PGL (n, C) for some B 2 GL(n, C). Then T* * B T Cx. But if g 2 T and gB = zh for some z 2 Cxpand_some h 2 T , then z must have orde* *r p since g and__ h have order p. Thus z is an element of p 1= [T, T ] T . Consequently, T B= T* * . |__| In the situation of (5.4), consider first the special case where E is trivial* * and T = P is an extra- special p-group whose center is central in GL(n, C). The extra-special p-group * *P has |P :[P, P ]| = p2d characters of degree 1 and p - 1 irreducible characters of degree pd (descr* *ibed in (5.19)). These irreducible representations of degree pd are faithful and they are [43, V* *.16.14] in bijective correspondence with the non-trivial linear forms ~: Z(P ) ! Cx; the representat* *ion corresponding to ~ is the representation ~P induced from any extension of ~ to a linear form * *on a maximal normal abelian subgroup of P . Thus the representation æ of P has the form X X æ = ~P + Ø for some non-trivial linear forms ~ on Z(P ) and some homomorphisms Ø: V ! Cx .* * Since æ is faithful at least one ~ must appear, and, since æ takes the center of P into* * the center of GL(n, C), no Øs can occur and exactly one ~ occurs. (Observe that for non-ident* *ity g 2 Z(P ), ~P(g) 6= 1 = Ø(g).) Thus in fact æ = m~P, pdm = n, for some non-trivial homomorphism ~: Z(P ) ! Cx. From the formula [43, V.16.14]* * [44, 7.5] ( d træ(g) = p m~(g) g 2 Z(P ) 0 g 62 Z(P ) we see that the Quillen automorphism groups A(GL (n, C))(P, æ) = {ff 2 Out(P ) | a(ff) = 1}, A(PGL (n, C))(V, æ) = * *Sp(V ) consist of those outer automorphisms of P that restrict to the identity on the * *center Z(P ) and (5.7, 5.9) of all isometries of the non-degenerate space (V, f), respectively. (Note * *also that V = P=P \Cx is unique up to isomorphism as an object of A(PGL (n, C)).) In general, T = P E is the direct product of an extra-special p-group with an* * elementary abelian p-group E. Since the restriction of æ to P is of the form æ|P = m~P, as we hav* *e just seen, representation theory for products of groups [43, V.10.3] [44, 8.1] tells us th* *at the representation N-DETERMINISM 23 æ = ~P]Ø is the outer tensor product of ~P with a faithful m-dimensional repres* *entation Ø of E = V ?. From the formula ( d~(g)Ø(e) g 2 Z(P ) træ(g, e) = p 0 g 62 Z(P ) we see that the Quillen automorphism groups A(GL (n, C))(T,=æ){ff 2 Out(T ) | a(ff) 2 A(GL (n, C))(Z(T ), ~]Ø)} A(PGL (n, C))(V,=æ){ff 2 Isom(V, f) | ff|V ?2 A(PGL (n, C))(V ?, Ø)} consist of those outer automorphisms of T that restrict to Quillen automorphism* *s of the m- dimensional representation ~]Ø of Z(T ) = Z(P )E and of those isometries of the* * inner product space (V, f) whose restrictions to V ?leave the representation Ø invariant, res* *pectively. Define A(T ) Out(T ) and A(V, f) Aut(V ) to be the groups A(T )= {ff 2 Out(T ) | a(ff) = 1} A(V, f)= {ff 2 Isom(V, f) | ff is the identity}on V ? consisting of those outer automorphisms that restrict to the identity on Z(T ),* * respectively of all isometries of (V, f) that restrict to the identity on E = V ?. Then A(T ) is a* * subgroup of the Quillen automorphism group A(GL (n, C))(T, æ) (and equal to this Quillen automo* *rphism group if T is extra-special). It follows from (5.7) that A(T ), of order |Sp(V=V ?)||* *Hom (V=V ?, V ?)|, is isomorphic to A(V, f). This proves the following lemma. 5.10. Lemma. The Quillen automorphism group A(GL (n, C))(T, æ) contains A(T ) a* *nd the Quillen automorphism group A(PGL (n, C))(V, æ) contains A(V, f). If T is extra-special,* * the Quillen au- tomorphism group A(PGL (n, C))(V, æ) = Sp(V ). The final step consists in identifying the centralizers and their centers for* * the subgroups T GL(n, C) and V PGL (n, C) of (5.4). The information we need is obtained in * *(5.12) as an application of the more general, and elementary, (5.11). 5.11. Lemma. Let T be any subgroup of GL (n, C), æ: T ! GL(n, C)the inclusion, * *and Z a central subgroup of GL(n, C). 1.There is a short exact sequence of Lie groups 1 ! CGL(n,C)(T )=Z ! CGL(n,C)=Z(T ) @-!Hom(T, Z)(æ)! 1 where the group to the right is the isotropy subgroup for the action of Hom* * (T, Z) on (æ) 2 Rep(T, GL(n, C)) and @(BZ)(g) = [B, g]. 2.The connected component of CGL(n,C)=Z(T ) is CGL(n,C)=Z(T )0 = CGL(n,C)(T )=Z and the group of components ß0 CGL(n,C)=Z(T ) is isomorphic to Hom (T, Z)(æ)= {OE: T ! Z| 9B 2 GL(n, C)8g 2 T :OE(g) = [B, g]} Proof.The exact sequence of the first point is a consequence of the short exact* * sequence 1 ! CGL(n,C)(T ) ! {B 2 GL(n, C)|[B, T ] Z} @-!Hom(T, Z)(æ)! 1 because CGL(n,C)=Z(T ) is the quotient of the middle group by the central subgr* *oup Z. The second point follows from the first because the centralizer of T in GL(n, C), a produc* *t of general_linear groups [82, Proposition 4], is connected. * * |__| 5.12. Lemma. Let T and V be as in ( 5.4). 1.If T = P is extra-special, then CPGL(pdm,C)(V ) = V x PGL(m, C), Z CPGL(pdm,C)(V ) = V, where the Quillen automorphism ff 2 A(V, f) = Sp(V ) acts as ff-1 x 1 and f* *f, respectively. 2.If V ?has rank one, then the component group of Z CPGL(pdm,C)(V ) is isomo* *rphic to V=V ? or to V . 3.ß1Z CPGL(pdm,C)(V ) is a finitely generated free abelian group with trivia* *l A(V, f)-action. 24 J.M. MØLLER Proof.In the special case where T = P is extra-special, all elements OE of Hom * *(P, Cx) are of the form OE(g) = [h, g] for some h 2 P . Thus all OE preserve the representation (æ* *) and it follows from (5.11) that the natural homomorphism (5.13) V x PGL(m, C) = V x CGL(pdm,C)(P )=Cx ! CPGL(pdm,C)(V ) is an isomorphism. Use (5.17) to get the action of the Quillen automorphism gro* *up. For the second item of the lemma, suppose that T = P E where E = V ? is one-d* *imensional. Then CPGL(pdm,C)(V ) = CPGL(pdm,C)(P E) = CCPGL(pdm,C)(P)(V ?) = CP=[P,P]xPGL(m,C)(V ?) = P=[P, P ] x CPGL(m,C)* *(V ?) and consequently, ZCPGL(pdm,C)(V ) = P=[P, P ] x ZCPGL(m,C)(V ?). Here (5.14), the second factor is either connected, in which case ß0CPGL(pdm,C)(V ) = ß0ZCPGL(m,C)(V ) = P=[P, P ] = V=V ?, or disconnected, in which case the center of Z(CPGL(pdm,C)(V )) = V . * * __ Use (5.17) for the final item of the lemma. * * |__| 5.14. Lemma. For any elementary abelian p-group E PGL (n, C) of rank one, eit* *her the cen- tralizer CPGL(n,C)(E) as well as its center ZCPGL(n,C)(E) are both connected or* * ZCPGL(n,C)(E) = E. Proof.There is (5.11) an exact sequence 1 ! Cx ! CGL(n,C)(E) ! CPGL(n,C)(E) ! Hom (E, Cx)(Ø)! 1 where Ø: E ! GL(n, C)is a lift. The group to the right is either trivial or cyc* *lic of order p. If it is trivial, then CPGL(n,C)(E) = CGL(n,C)(E)= GL(1, C), Z CPGL(n,C)(E) = Z CGL(n,C)(E) = GL(* *1, C) are both connected Lie groups [58, 4.6]. Otherwise, n = rp and Ø = ræ is a dire* *ct sum of a number of copies of the regular representation æ of E. Then the centralizer CPGL(n,C)(E) = GL(r, C)p= GL(1, C) o where oe has order p and acts on GL(r, C)p by permuting the factors cyclically.* * Thus the center of the centralizer, i p j 1 Z CPGL(n,C)(E) = GL(1, C) = GL(1, C) = H (; GL(1, C)) is cyclic of order p. * * |___| The information collected so far suffices to establish the vanishing of some * *of the higher limits for the functors ßj(BZCPGL(n,C)): A(PGL (n, C)) ! Ab(2.7). We shall make use of* * the following lemma which, together with its application in the proof of (5.16), is due to J.* * Grodal. 5.15. Lemma. Let A be a subgroup and P a parabolic subgroup of G = GL (n, Fp) s* *uch that U A P where P = UL is the Levi decomposition [23, x69A]. Then Hom Fp[A](St(G), M) = Hom Fp[A=U](St(L), M) for any Fp[A]-module M which is trivial as an Fp[U]-module and finite-dimension* *al as an Fp-vector space. ~= Proof.The standard Fp[A]-module isomorphism Hom (St(G), Fp) M -! Hom(St(G), * *M)) re- stricts to an isomorphism ~= Hom(St(G), Fp)U M -! Hom Fp[U](St(G), M) N-DETERMINISM 25 of Fp[A=U]-modules. Since Steinberg modules are self-dual and St(G)U = St(L) [8* *9, 18, 42] we have Hom(St(G), Fp)U ~=St(G)U ~=St(L) ~=Hom(St(L), Fp) as Fp[P=U]-modules. Thus Hom Fp[U](St(G), M) ~=Hom (St(L), Fp) M ~= Hom(St(L* *), M) as Fp[A=U]-modules and consequently Hom Fp[A](St(G), M) ~=HomFp[U](St(G), M)A=U ~=Hom(St(L), M)A=U = Hom Fp[A=U](St(L), M) as vector spaces. * * |___| 5.16. Lemma. limi(A(PGL (n, C)), ßj(BZCPGL(n,C))) = 0 for j = 1, 2 and i = j, j* * + 1. Proof.It suffices (2.13, 5.10, 5.12) to show that the following homomorphism gr* *oups are trivial: oHom Sp(V()St(V ), V ) where dimFpV = 2, oHom A(V,f)(St(V ), V ) and HomA(V,f)(St(V ), V=V ?) where dimFpV = 3 and f * *is a non-trivial alternating bilinear form on V , oHom A(V,f)(St(V ), Zp) where dimFpV is 3 or 4, f is a non-trivial alternati* *ng bilinear form on V , and Zp carries the trivial A(V, f)-action. Note that Zp can be replaced by Fp as target module since the Steinberg module * *is finitely generated. The first of these groups is clearly trivial as Sp(V ) = SL(V ) cont* *ains -1 which acts trivially on the Steinberg module but has no non-trivial fixed points in V . Fo* *r the remaining cases, we apply (5.15). For us, n is 3 or 4, and the group A consists of the matrices ` ' Ik * 0 SL(2) where Ik is a (k x k)-identity matrix, k = 1, 2. Take P and U = Op(P ) to be th* *e subgroups of G = GL(n, Fp) consisting of matrices of the form ` ' ` ' GL(k) * Ik * 0 GL(2) , respectively,0 I2 so that L = GL(k) x GL(2). Then Hom Fp[A](St(G), Fp)= Hom Fp[SL(2)](St(GL (k)) St(GL (2)), Fp) = Hom Fp[SL(2)](St(GL (2)), HomFp(St(GL (k), Fp))) = HomFp[SL(2)](St(GL (2)), Fp) and, for n = 3, Hom Fp[A](St(G), V=V ?) = Hom Fp[SL(2)](St(GL (2)), V=V ?) Using that St(GL (2)) is an irreducible Fp[SL(2)]-module we see that both these* * groups are trivial._ Since V ?is a trivial Fp[A]-module, also Hom Fp[A](St(G), V ) must be trivial. * * |__| Proof of Theorem 5.1.PGL(n, C) is non-modular, hence totally N-determined [68, * *3.11, 7.4], for n < p. We may therefore, inductively, assume that all elementary abelian p* *-subgroups of PGL (n, C) have totally N-determined centralizers (3.3, 3.7, 5.12) [82, Proposi* *tion 4]. But then also PGL (n, C) itself has N-determined automorphisms according to (3.4, 5.16) and i* *s N-determined according to (3.8, 5.16) provided we can verify the conditions of (3.9) when n * *= pm is divisible by p. It only remains to consider the third condition as the first two have been v* *erified in (5.4, 5.10). Let j0:N(PGL (n, C)) ! Xbe the maximal torus normalizer for some p-compact gr* *oup X. Let (V, ) denote the non-toral rank 2 object of A(PGL (n, C)), ~: V ! N(PGL (n, C)* *)a preferred lift of :V ! PGL (n, C), and put 0= j0~. The object (V, 0) of A(X) does not depe* *nd on the choice 26 J.M. MØLLER of ~ (3.9). We must show that the diagram CPGL(n,C)(V,O_)f(V,~)//_CXO(V,OO0) CPGL(n,C)(ff)|| CX(ff)|| | | CPGL(n,C)(V, _)f(V,~)//_CX (V, 0) commutes for all ff 2 Sp(V ) = SL(V ) [43, II.9.12]. This will be the case if a* *pplication of the identity component functor (-)0 and the component group functor ß0(-) gives commutative * *diagrams [66, 5.3] [64, 3.4, 3.10]. The first of these derived diagrams commutes because SL(V* * )op, generated by el- ements of order p [91, 3.6.21], acts trivially on CPGL(n,C)(V, )0 = PGL (m, C)* * = CX (V, 0)0 whose automorphism group (5.2) Aut(PGL (m, C)) ~=Zxp(or Zxp=Zx if m = 2) contains no * *elements of order p. The above diagram also commutes on the level of ß0 for ß0(f(V, ~)) is * *SL(V_)op-equivariant by (3.10, 5.12). * * |__| 5.17. The action of A(GL (n, C))(T, æ) on CGL(n,C)(T, æ). Write (æ) = ~1(æ1) + * *. .+.~t(æt) as a direct sum of inequivalent irreducible characters (æ1), . .,.(æt) with mul* *tiplicities ~1, . .,.~t, respectively. Then Y Y (5.18) CGL(n,C)(T, æ) = GL (~i, C), Z CGL(n,C)(T, æ) = Z GL (~i, * *C) æi2S(æ) æi2S(æ) where S(æ) = {æ1, . .,.æt} is the set of irreducible characters occuring in æ. * * Let (S(æ)) be the group of all permutation of S(æ) and for a given integer valued function fi* * on S(æ) write (S(æ))fifor the subgroup of permutations that preserve fi. We shall need the d* *egree function d, recording the degree of æi, and the multiplicity function ~(æ), recording the m* *ultiplicity ~iof æiin æ. The elements of NGL(n,C)(T, æ) are C-linear automorphisms of Cn that are ff-* *linear for some automorphism ff 2 Aut(T ) and there is a homomorphism A(GL (n, C))(T, æ) = NGL(n,C)(T, æ)=T CGL(n,C)(T, æ) ! (S(æ))d\ (S(æ* *))~(æ) since ff permutes the irreducible representations (æi) in a degree and multipli* *city preserving way. Now, NGL(n,C)(T, æ) NGL(n,C)(CGL(n,C)(T, æ)) NGL(n,C)(ZCGL(n,C)(T, æ)) Y GL(di~i, C) o (S(æ))d~(æ) and since the first factor of the semi-direct product acts trivially on ZCGL(n,* *C)(T, æ), the action homomorphism A(GL (n, C))(T, æ) ! Out(CGL(n,C)(T, æ)) ! Aut(ZCGL(n,C)(T, æ)) factors through (S(æ))d\ (S(æ))~(æ) (S(æ))d~(æ). Observe, in particular, th* *at the subgroup A(T ) of A(GL (n, C))(T, æ) acts trivially on CGL(n,C)(T ) because any outer au* *tomorphism ff 2 A(T ) restricts to the identity on Z(T ) so that it preserves all the irreducib* *le components Øi~P of the representation æ. 5.19. Representations of extra-special p-groups. We construct explicitly the fa* *ithful irre- ducible representations of the extra-special p-groups. Let E be an elementary abelian p-group of rank d 1 and C[E] its complex gro* *up algebra, or rather, its underlying |E|-dimensional complex vector space. Then there is a co* *mmutative diagram as in (5.4) "æ P ]Ø____//GL(C[E]) | | | | fflffl|Ø " fflffl| V _æ__//_PGL(C[E]) N-DETERMINISM 27 where V = E^ x E is the product of E and its dual E^ = Hom (E, Cx), and P = P-,* * P+ is the subgroup of GL(C[E]), P- = , P+ = , generated by Ri(v) = i(v)v, Tu(v) = u + v, i 2 E^, u, v 2 E, where ! is a primitive p2th root of unity. Since the commutator [!Ri, Tu] = [Ri, Tu] = i(u), i 2 E^, u 2 V, is scalar multiplication with the complex number i(u), the group P- (P+) is ext* *ra-special of order p1+2dand exponent p2 (p). A trace computation reveals that the p - 1 fait* *hful irreducible representations of P are obtained from the inclusion æ by composing with the au* *tomorphisms (!)Ri ! (!)Rii, Tu ! Tu, 0 < i < p, of P . Observe that P- and P+ have the same* * center, the same centralizer in GL(C[E]), and the same image in PGL (C[E]). The same is tru* *e for P- and P+ considered as subgroups of GL(C[E] m ) by means of the representation mæ. An object (V, ) of A(X) is said to be d-oversize if codimker(ß0(~): V ! ß0N(X) = W (X)) d for all preferred lifts ~: V ! N(X) of :V ! X and d 0 is the biggest such n* *atural number. Thus the 0-oversize objects are the toral objects. It may be worthwhile to note* * that the A-family provides examples of highly oversized elementary abelian subgroups. 5.20. Proposition.Let T and V be as in Lemma 5.4. If T = P E where P has order * *p1+2dthen (V, æ) is a d-oversize object of the Quillen category A(PGL (n, C)). Proof.We shall first consider the case where T = P = p1+2d+is extra-special and* * æ is one of the irreducible and faithful representations that we just considered. Note that P i* *s contained in the maximal torus normalizer N GL (n, C) as x ff N GL (n, C) = Ri, Toe| i 2 map(V, C ), oe 2 (V ) is generated by all the operators Ri(v) = i(v)v, Toe(v) = oe(v) for all functio* *ns i from V into Cx and all permutations oe of the elements of V . Similarly, V is contained in * *the maximal torus normalizer N PGL (n, C) = N GL (n, C) =Cx of PGL (n, C). The centralizers CN(GL(n,C))(P ) = Cx, CN(PGL(n,C))(V ) = V so that the inclusion of V into the maximal torus normalizer is a preferred lif* *t of the inclusion of V into PGL (n, C). For this preferred lift the intersection of V with the m* *aximal torus has codimension d. But the intersection of V with any maximal torus of PGL (n, C) i* *s covered by the intersection of P with a maximal torus of GL(n, C) and such a subgroup has orde* *r at most p1+d which is the order of a maximal normal abelian subgroup of the extra-special p-* *group P . Thus (V, æ) is a d-oversize rank 2d object of the Quillen category of PGL (n, C). When æ = m~P, CN(GL(n,C))(P ) = N(GL (n, C)) \ CGL(n,C)(P ) = N(GL (n, C)) \ GL(m, C) = N(GL (m, C)) so that the centralizer of V in N(PGL (n, C)) is V x N(PGL (m, C)). Again, the * *inclusion of V into N(PGL (n, C)) is a preferred lift of the inclusion of V into PGL (n, C) an* *d we conclude, as above, that (V, æ) is a d-oversize rank 2d object of A(PGL (n, C)). In general, T = P E is the direct product of an extra-special p-group and an * *elementary abelian p-group. But still the inclusion of V = P=[P, P ] x E into the maximal torus * *normalizer is a preferred lift because its adjoint E ! CN(PGL(n,C))(P=[P, P ]) ! CPGL(n,C)(P=[P, P ]) is a preferred lift as E is toral. For this preferred lift, the intersection of* * V with the maximal torus has codimension d and, as above, this is actually the minimum. Thus (V, æ) is a* * d-oversize_rank > 2d object of the Quillen category. * * |__| 28 J.M. MØLLER 6. The 3-compact group F4 We consider the 3-compact group (BF4)^3obtained by completing the classifying* * space BF4 for the exceptional Lie group F4of rank 4. 6.1. Theorem. [92] The following hold for the 3-compact group F4: 1.F4 is totally N-determined. 2.F4 is determined by its R-Weyl group for R = Zp, Qp, Fp. 3.F4 is a cohomologically unique p-compact group. 4.End (F4) - {0} = Aut(F4) = NGL(L(F4))(W (F4))=W (F4) is an abelian group is* *omorphic to Zx3=Zx x C2 where the group C2 of order 2 is generated by an exotic automor* *phism. Proof.The information provided by Griess [41, 7.4] about elementary abelian p-s* *ubgroups of the Lie group F4 shows that the 3-compact group F4 satisfies the conditions of (3.8* *); see (3.9, 3.10) and the remark below (2.13). Combined with (4.4, 11.18, 11.25) this proves the * *first three items. Direct computation shows that the normalizer x ff NGL(4,Z3)(W (F4)) = Z3, ", W (F4) where 0 1 -1 1 0 0 p ___ B 1 1 0 0 C -2" = B@ 0 0 -1 1 CA 0 0 1 1 for the Weyl group of F4 in GL(4, Z3) as described e.g. in [12]. The final item* * of the_theorem is now a consequence of (3.17.(2)). * * |__| Note that (6.1) yields a new proof of the existence of Friedlander's exceptio* *nal isogeny [39]. 7.Polynomial p-compact groups All connected Fp-local spaces with polynomial mod p cohomology are p-compact * *groups. We study these polynomial p-compact groups in this section. See also D. Notbohm [7* *2, 76, 79] for further information and for references to the literature about this classical s* *ubject. For any connected p-compact group X, the image of H*(BX; Fp) in H*(BT (X); Fp* *) is contained in the invariant ring H*(BT (X); Fp)W(X) for the action of the Weyl group on th* *e cohomology of the maximal torus. Much work, summarized in the following lemma, has been done * *to tell when H*(BX; Fp) actually equals this invariant ring. 7.1. Lemma. Let p be an odd prime and X a p-compact group. The following condi* *tions are equivalent: 1.H*(BX; Fp) is a polynomial algebra. 2.H*(BX; Fp) = H*(BT (X); Fp)W(X). 3.H*(BX; Fp) H*(BT (X); Fp). 4.H*(BX; Fp) is concentrated in even degrees. 5.H*(BX; Zp) is concentrated in even degrees and degree-wise free. 6.H*(BX; Zp) is polynomial on even degree generators. 7.H*(BX; Zp) = H*(BT (X); Zp)W(X). 8.H*(BX; Zp) H*(BT (X); Zp). If X satisfies these equivalent conditions,Qthen the rational rank r of X [30, * *5.1] equals the Krull dimension of H*(BX; Fp), and |W | = diwhere 2di, 1 i r, are the degrees o* *f the polynomial generators [88, 5.3.5, 5.5.4]. Proof.1. ) 2. is [29, 2.11] and 2. ) 3. ) 4. ) 5. is elementary. 5. ) 1., 6., * *7., 8.: As was noted in [59, 4.2], H*( BX; Zp) is degree-wise free so that Borel's argument [8* *] [88, 10.7.5] shows that H*(BX; Fp) and H*(BX; Zp) are polynomial. But then H*(BX; R) is the invari* *ant ring for __ R = Zp, Fp by [29, 2.11] again. The implications 8. ) 5., 7. ) 5., 6. ) 5. are * *elementary. |__| N-DETERMINISM 29 7.2. Definition.A p-compact group X is polynomial if its cohomology ring H*(BX;* * Fp) is a polynomial Fp-algebra. A Zp-reflection group (W, ~T) is polynomial if its invar* *iant ring H*(T~, Fp)W is a polynomial Fp-algebra. 7.3. Example.A p-compact group X is non-modular if p does not divide the order * *of W (X). A Zp-reflection group (W, ~T) is non-modular if p does not divide the order of W * *. Any non-modular p-compact group is connected [58, 3.8] and its Weyl group is a non-modular Zp-r* *eflection group. The Shephard-Todd theorem [7, 7.2.1] says that any non-modular Zp-reflection gr* *oup (W, ~T) is polynomial, and, clearly, H0(W ; ~T) = 0 if (W, ~T) is also simple. Any non-mo* *dular p-compact group X is polynomial [68, 3.12], totally N-determined [68, 3.11, 7.7], and det* *ermined by its R- Weyl group for R = Zp, Fp (4.5, 4.4); if X is also simple, then X is centerless* * (3.12.(2)) and determined by its Qp-Weyl group (11.18, 4.4). The Weyl group of any polynomial p-compact group is a polynomial Zp-reflectio* *n group but not all polynomial Zp-reflection groups are Weyl groups of polynomial p-compact* * groups (7.4). If the ring of invariants H*(T~, Fp)W for some ZpW torus ~Tis polynomial then W i* *s a Zp-reflection group [29, Proof of 5.2]. 7.4. Remark. Borel [9, 2.5] shows that for a simple compact Lie group G and p a* *n odd prime, the Bousfield Fp-localization (BG)^pof BG [13] is a non-polynomial p-compact gr* *oup BG^precisely when oG = SU(r + 1)=Z where Z is a non-trivial central p-subgroup, or, oG = F4, PE6, E6, E7, E8 and p = 3, or, oG = E8 and p = 5. Kemper and Malle [51] show that a simple Zp-reflection group (W, ~T) is non-pol* *ynomial precisely when it is the Weyl group of one of the Lie p-compact groups on Borel's list - * *with the exception that (W, ~T)(PU (3)) at p = 3 is polynomial because we are in dimension 2 [71, * *5.1]. Combining this with (7.27), we see that the invariant ring H*(T~; Zp)W with Zp-coefficien* *ts is non-polynomial precisely when (W, ~T) = (W, ~T)(G^) is the Weyl group of one of the Lie p-comp* *act groups ^Gon Borel's list. Thus the polynomial Zp-reflection group (W, ~T)(PU (3)) at p = 3* * is not the Weyl group of a polynomial p-compact group for then also the invariant ring with Zp-* *coefficients would be polynomial (7.1). (Combine the method of (7.24) with the results of [71, x4* *] [51, x5] to see that (W, ~T)(SU (r + 1)=Z) is non-polynomial when p|r + 1, n 3, and Z is a no* *n-trivial central p-group.) 7.5. Lemma. Let p be an odd prime. Let i: ~T! Xbe a loop space homomorphism fro* *m a Zp- torus ~Tto a polynomial p-compact group X. If H*(Bi; Fp) induces an isomorphism H*(BX; Fp) ~=H*(T~; Fp)W to the ring of invariants for some finite group W of automorphisms of ~T, then * *i: ~T! X is a (p-discrete) maximal torus for X and W and W (X) are Fp-similar Zp-reflection g* *roups. Proof.Since H*(Bi; Fp) makes H*(T~; Fp) a finitely generated H*(BX; Fp)-module * *[88, 2.3.1], i: ~T! Xis a monomorphism [30, 9.11]. Moreover, ~Tand T (X) have the same rank,* * the Krull dimension of H*(BX; Fp), so that i: ~T! Xis indeed a maximal torus. Also, W and* * W (X) have the same order given by the degrees of the polynomial generators. By Lannes th* *eory [52], the homomorphism t ! ~T! X is W -equivariant up homotopy because it is so on mod p * *cohomology. This means that rpW is contained in the Quillen automorphism group A(X)(t) of t* * ! X which is __ rpW (X) (2.8). But these two groups have the same order, so they must be identi* *cal. |__| If X is polynomial, then X is connected and, by Lannes theory [52], any monom* *orphism of a non-trivial elementary abelian p-group into X factors through the maximal tor* *us and hence (2.8) the Quillen category A(X) is equivalent to A(W (X), t(X)). About the cent* *ric [28] functor BCX :A(W (X), t(X)) ! [pcg](2.5) we know that, for an odd prime p, 1.H*(BCX ; Zp) = H*(T~(X); Zp)0, 2.ßj(BZCX ) = L(X)2-j, j = 1, 2. 30 J.M. MØLLER The formula in item (1) is a consequence of [33, 1.2] and (7.1, 2.10) showing t* *hat polynomiality is preserved under taking centralizers of elementary abelian subgroups. The for* *mula in item (2) follows from (2.8). (Recall that H*(T~(X); Zp)0 is (2.2) the functor given by H* **(T~(X); Zp)0(E) = H*(T~(X); Zp)W(X)(E)= H*(BT (X); Zp)W(X)(E)and, similarly (2.3), L(X)2-jis the * *functor given by L(X)2-j(E) = H2-j(W (X)(E); L(X)) for all non-trivial subgroups E of t(X).) Combined with the acyclicity result of (2.4) this leads to a very simple proo* *f of the homology decomposition for polynomial p-compact groups. 7.6. Proposition.[45, 31] Let p be an odd prime. For any polynomial p-compact g* *roup X, the evaluation map hocolimA(W(X),t(X))opBCX ! BX is an H*Zp-equivalence. Alternatively, the full subcategory ( 2.14) AC(p)(W (X)* *, t(X))op based on the collection C(p) of all p-subgroups of W , can be used for index category. Proof.By (2.4, 2.16) and one of the formulas above, the E2-page of the Bousfiel* *d-Kan spectral sequence for the cohomology of a homotopy colimit collapses onto the vertical a* *xis and_therefore the evaluation map is a H*Zp-equivalence. * * |__| In particular, if X is a polynomial p-compact group and p does not divide ord* *er of the Weyl group (i.e. X is a non-modular p-compact group (7.3)) then BX is H*Zp-equivalent to t* *he homotopy colimit of a diagram of the form __________________________________________* *_________________ BT (X)ee_W(X)op_________________________________* *__________________________________ i.e. to BN(X); this is the case treated by Clark-Ewing [20]. If p divides the o* *rder of the Weyl group exactly once, then BX is H*Zp-equivalent to the homotopy colimit of a dia* *gram of the form ___________________________________________________* *______________________________________op__________________________W(X)@ __W(X)(tS)op=W(X)(tBCXS(tS))op88____________________________________* *_____________________________BTW(X)(X)ee______________________________@ with just two nodes; this is the cases treated by Aguad'e [2]. In general, BX i* *s H*Zp-equivalent to the homotopy colimit of a diagram with nodes in one-to-one correspondence with * *the subgroups of the Sylow p-subgroup of W (X). (The objects tP, for P a subgroup of SylpW (X* *), generate a skeletal subcategory of AC(p)(W (X), t(X)).) The decomposition (7.6) is usually only helpful when X is centerless. (Any si* *mple p-compact group X for which r0W (X) is not in family 1 of the Clark-Ewing list and not eq* *ual to r0W (E6) if p = 3, is centerless (3.12.(2), 11.18)). Conversely, given a finite group W of automorphisms of a Zp-torus ~Tsuch that* * H0(W ; ~T) = 0 and the ring of invariants H*(T~; Fp)W is polynomial, does there exist a polyn* *omial p-compact group X(W ) with Zp-reflection subgroup (W, ~T) and with mod p cohomology isomo* *rphic to this invariant ring? Note that if X(W ) exists, then the Quillen category A(X(W )) * *= A(W, t), the maximal torus normalizer ~N(X(W )) = ~To W , and the functor ~NO CX(W), giving * *the maximal torus normalizers of the centralizers, is the functor ~N:A(W, t)op! [Grp ]given* * by ` -1 -1 ' N~(E0 wW(E0)-----!E1) = ~To W (E0) -(w--,c(w--))----~To W (E1) according to the considerations of the proof of (2.8). This means that if BX(E)* * denotes the value of BCX(W) on E t then there must exist homotopy commutative diagrams ~N(wW(E0)) (7.7) BN~(E0)oo_________ BN~(E1) Bj(E0)|| Bj(E1)|| fflffl| fflffl| BX(E0) ooX(wW(E0))_BX(E1) where the vertical arrows are (discrete) maximal torus normalizers. 7.8. Theorem. (Generalized Clark-Ewing construction) Let p be an odd prime and * *(W, ~T) a polynomial Zp-reflection group with H0(W ; ~T) = 0. Suppose that there exist a * *centric functor [28] N-DETERMINISM 31 BX :A(W, t)op! [pcg]and a natural transformation Bj :BN~ ! BX such that, for e* *ach non- trivial subgroup E of t, BX(E) is a polynomial p-compact group and Bj(E): BN~(E* *) ! BX(E) is a p-discrete maximal torus normalizer. Then BX determines an essentially un* *ique functor BX :A(W, t)op! Top, and H*(BX(W ); Fp) ~=H*(T~; Fp)W as unstable algebras where ^ BX(W ) = hocolimA(W,t)opBX p is the Fp-localization of the homotopy colimit. X(W ) is a centerless polynomia* *l p-compact group whose Weyl group is Fp-similar to W . If all values of the functor BX are total* *ly N-determined p-compact groups, then also X(W ) is totally N-determined. Alternatively, the full subcategory ( 2.14) AC(p)(W, t) based on the collecti* *on C(p) of all p- subgroups of W , can be used for index category. Proof.For any non-trivial subgroup E of t, the p-compact group BX(E) has p-disc* *rete center ~Z(X(E)) = Z(N~(E)) = ~T W(E)meaning (3.17.(1)) that (ßjBZX)(E) = H2-j(W (E); L* *(T~)) = L(T~)2-j(E) for j = 1, 2. Since these functors are acyclic (2.4), [28, 1.1] tel* *ls us that BX lifts, es- sentially uniquely, to a functor taking values in the category of topological s* *paces. Let BX(W ) be the (Fp-localization of the) homotopy colimit. The polynomial p-compact group B* *X(E) has coho- mology H*(BX(E); R) = H*(T~; R)W(E) = H*(T~; R)0(E), R = Fp, Zp. Since this fun* *ctor is acyclic (2.4), the Bousfield-Kan spectral sequence for the cohomology of a homotopy col* *imit [14, XII.4.5] collapses onto the vertical axis giving the cohomology of BX(W ) and so H*(BX(W* * ); Fp) = H*(T~; Fp)W . As this invariant ring is assumed to be polynomial, X(W ) is inde* *ed a polynomial p-compact group. The p-compact group morphism T (GL (n, C)) = CGL(n,C)(t) ! X(W* * ) is a max- imal torus and rp(W ) = rpW (X(W )) by (7.5). According to [68, 4.9] and (2.11,* * 3.8), X(W ) is totally N-determined provided all values of the functor BX are totally N-determ* *ined p-compact groups. We may replace the index category A(W, t) by any of its full subcategories I * *as long as lim1+j(I; L(T~)2-j) = 0 = lim2+j(I; L(T~)2-j), j = 1, 2, and H*(BT~; Zp)0 is ac* *yclic on I with lim0equal to the invariant ring. For instance, I = AC(p)(W, t), where C(p) is t* *he collection_of all p-subgroups of W is a possibility (2.16). * * |__| In particular, if p divides the order of W exactly once, we may use the full * *subcategory A(W, t){t, tS} = I(W, W (tS)) (13.10) generated by the two objects t and tS whe* *re S = SylpW is a Sylow p-subgroup of W . The Qp-Weyl group r0W (X) (4.3) of a connected p-compact group X is a reflect* *ion subgroup of Aut(L(X) Qp) [30, 9.7]. If X is simple in the sense that this Weyl group is an * *irreducible reflection group then r0W (X) must occur in the Clark-Ewing classification table [20]. Th* *e irreducible reflection groups of this table are divided into four infinite families, denote* *d 1, 2a, 2b, and 3, and 34 sporadic reflection groups G4, . .,.G37. 7.9. Theorem. Let p be an odd prime and X a simple p-compact group with Weyl re* *flection group (W (X), L(X)). Assume that or0W (X) is not in family 1, oif p = 3, then (r0W (X)) 6= (r0W (F4)), (r0W (E6)), (r0W (E7)), (r0W (E8)),* * and, oif p = 5, then (r0W (X)) 6= (r0W (E8)). Then the following hold: 1.X is a centerless, simply connected, totally N-determined, polynomial p-com* *pact group. 2.X is determined by its R-Weyl group for R = Zp, Qp, Fp. 3.X is a cohomologically unique p-compact group. 4.End (X) is given by ( 3.17.( 2)). Proof.A glance at the Clark-Ewing classification table [20] (as presented e.g. * *in [5, Table 1]) reveals that X is either a non-modular p-compact group, which certainly has the* * stated properties (7.3), or one of the modular p-compact groups treated in (7.10) in which case w* *e apply_(7.8,_5.3) together with (4.4, 11.18, 11.25.3). * * |__| 32 J.M. MØLLER 7.10. Construction of modular, centerless, polynomial, simple p-compact groups.* * We apply (7.8) to construct polynomial p-compact groups X(G) where G GL(r, Qp) i* *s either oin family 2a, or0W (G2) at p = 3 from family 2b, oone of the groups of Aguad'e [2, Table 1], or, or0W (E6) at p = 5. There is no ambiguity in pretending that G be a subgroup of Aut(T~) = GL(r, Zp)* * since G0(G) = * in each of these cases (11.18). The rings of invariants H*(T~; R)G, R = Fp, Zp,* * are polynomial rings (7.4), and from [5, 3.4] we have that H0(G; ~T) = 0. Thus it suffices to * *find a functor BX that satisfies the conditions of (7.8). Family 2a. (Cf. [76]) Let p be an odd prime and r 1, m 2, n 2 natural n* *umbers such that r|m|p - 1. Let Cm Zxpbe the order m cyclic subgroup of the p-adic u* *nits. Define G(m, r, n) = A(m, r, n) n as the subgroup of GL (n, Zp) = Aut(T~), ~T= ~T(U(n))* *, generated by the group W (U(n)) = n of monomial matrices and the abelian group A(m, r, n) o* *f diagonal matrices with entries in Cm and determinant in the index r subgroup of Cm . (* *For instance, G(2, 1, n) = W (SO (2n+1)) and G(2, 2, n) = W (SO (2n)).) The subgroup n norma* *lizes A(m, r, n) and G(m, r, n) = A(m, r, n) o n is in fact the semi-direct product of the two * *groups. The ring of invariants [71, 2.4] [88, x7.4, Example 1] H*(T~; Zp)G(m,r,n)= Zp[y1, . .,.yn-1, e], |yi| = 2im, |e| = 2m_rn, is generated by e = (x1. .x.n)m_rtogether with the n - 1 first elementary symme* *tric polyno- mials yi = oei(xm1, . .,.xmn), 1 i n - 1, in the mth powers of the coordina* *te functions xi:H2(T~; Zp) ! Zp, 1 i n, which are considered as having degree 2. Define AC(p)(G, t) where G = G(m, r, n) or G = n, t = t(U(n)), to be the ful* *l subcategory of A(G, t) generated by all objects of the form E = tP for P Sylp n = SylpG(m* *, r, n) a subgroup of a Sylow p-subgroup of n (which is also a Sylow p-subgroup of G(m, * *r, n)). These two small categories have by definition the same set of objects E, with the same po* *int-wise stabilizer subgroups G(m, r, n)(E) = n(E), and for the morphism sets (2.1) we note that __ (E ) __ G (m, r, n)(E0, E1) = A(m, r, n) n 1 x n(E0, E1) meaning that any morphism (a, oe): E0 !iE1n AC(p)(G(m, r, n), t)(E0, E1) factor* *s uniquely as a morphism oe :E0 ! E1in AC(p)( n, t)(E0, E1) followed by multiplication a: E1 ! * *E1by a diagonal matrix a 2 A(m, r, n) n(E1)= A(m, r, n)0(E1). To see this, it is convenient to * *observe that that all objects E = tP of AC(p)(G, t) are of the special form E = {(x1, . .,.xn) 2 Fnp| xi= xj iff i and j are n(E)-equivalent} for some partition n(E) of n = {1, . .,.n} into disjoint subsets. (Thus AC(p)(G* *(m, r, n), t) can be viewed as the Grothendieck construction on the functor A(m, r, n)0 from AC(p)( * *n, t) to categories with one object.) We now define the functor BX :AC(p)(G(m, r, n), t)op! [pcg]which shall serve * *as input for the generalized Clark-Ewing construction (7.8). On objects E = tP t = t(U(n)) we * *are forced to put BX(E) = BCU(n)(E) for the point-wise stabilizer group G(m, r, n)(E) = n(E) = W* * (CU(n)(E)) and CU(n)(E), a product of unitary groups [82, Proposition 4], is determined by* * its Zp-Weyl group (5.3). For each morphism E0 oe-!E1 a-!E1 in AC(p)(G(m, r, n), t) we are require* *d (7.7) to fill in the commutative diagram -1)) (a-1,1) T~o n(E0)o(1,c(oeo~To_ n(E1)oo______T~o_ n(E1) | | | | | | fflffl| fflffl| fflffl| CU(n)(E0)oo````` ` CU(n)(E1)oo````` `CU(n)(E1) of p-compact groups with discrete maximal torus normalizers. To the left we may* * put the value BCU(n)(oe): BCU(n)(E0) BCU(n)(E1) on E0 oe-!E1of the functor BCU(n):AC(p)( n,* * t) ! [pcg]. N-DETERMINISM 33 To the right there is just one possibility, denoted _a-1, for CU(n)(E1) have N-* *determined auto- morphisms (5.3). This prompts us to declare oe a BCU(n)(oe) B_a-1 BX E0 -!E1 -!~=E1 = BCU(n)(E0) ------- BCU(n)(E1) -----~=BCU(n)(E1) However, for this to be a valid_definition of a functor we need to verify that * *the relation ø O a = øaø-1 O ø, a 2 n(E0), ø 2 n(E0, E1), which holds in AC(p)(G(m, r, n), t), als* *o holds in [pcg], i.e. that the diagram a-1 CU(n)(E0)oo______CU(n)(E0)OOOO CU(n)(ø)|| |CU(n)(ø)| | | CU(n)(E1)oo_-1_-1CU(n)(E1)_ _øa ø commutes in [pcg]. This is not difficult as _øa-1ø-1and the isomorphism induce* *d by _a-1 on CU(n)(E1) have the same effect on the maximal torus normalizer, so are identica* *l. Thus the above definition indeed makes BX into a functor. BX is clearly a centric functor beca* *use BCU(n)is and we conclude from (7.8) that there exists a centerless, polynomial, totally N-de* *termined p-compact group XG(m, r, n) with reflection subgroup Fp-similar, and hence even Zp-simila* *r (11.25), to (G(m, r, n), ~T). For future reference, we now compute the centralizer of an arbitrary non-triv* *ial subgroup E of t = t(XG(m, r, n)). Suppose that E has rank r > 0. Choose an (n x r)-matrix B w* *hose columns form a basis for E. Declare i and j to be equivalent if the ith and jth rows in* * B are Cm -multiplies of each other, 1 i, j n. Let n(E) denote the partition of n = {1, 2, . .,.n* *} into equivalence classes. If there is a zero-row in B, call the corresponding equivalence class * *the null-class. Suppose that the null-class contains u0 0 elements and that there are s 1 more clas* *ses containing u1, . .,.us elements, respectively. The following lemma, describing the point-wise stabilizer subgroup G(m, r, n)* *(E), implies that the equivalence relation n(E) does not depend on the choice of basis. 7.11. Lemma. The point-wise stabilizer G(m, r, n)(E) of E is isomorphic to the * *subgroup G(m, r, u0) x u1x . . .us where the reflection subgroup ( uj, Zujp) is similar to (W, L)(U(uj)), 1 j * *s. Proof.The element (a, oe) 2 A(m, r, n) o n stabilizes E point-wise if and only* * if aiBoe(i)= Bi where Bi, i = 1, . .,.n are the rows of the matrix B. This implies that the per* *mutation oe of the n rows of B must respect the Cm -equivalence classes. Therefore the group homom* *orphism G(m, r, u0) x u1x . .!.utG(m, r, n)(E) (b, ø), oe1, . .,.oes)! (b, a1(oe1), . .,.aj(oej)), øoe1. * *.o.es where aj(oej)iBoej(i)=QBi, 1 j s, is an isomorphism. Observe in this conne* *ction that the product aj(oej)i = 1; indeed, for fixed j, the product over all aj(oej)i, whe* *re i runs through the elements of a cycle in the decomposition of oej, equals 1. Conjugate this a* *ction of uj by the diagonal matrix consisting of the first non-zero entries in the rows of B to ob* *tain_the standard permutation action. * * |__| If we take existence for granted, referring to [76], then the above lemma and* * (3.8) would suffice to show inductively that XG(m, r, n) is totally N-determined. The 3-compact group G2. Take BX to be the functor on the 2-object category A(G,* * t){tS, t}op= I(G, W (SU (3)))op, G = W (G2) = W (SU (3)) x Z(G), Z(G) = { 1}, indicated by t* *he diagram _____________________________________________________* *______________________________________________________________________@ Z(G)op__BSU9(3)9__________________________________________* *________________________BTGopbb_______________________________________@ 34 J.M. MØLLER where Z(G) acts on BSU (3) via the unstable Adams operations _ 1. The Aguad'e groups. These are the reflection groups (G12, p = 3), (G29, p = 5), (G31, p = 5), (G34, p = 7), (G36, p = 5), (G36* *, p = 7), (G37, p = 7) where the index refers to their numbering on the Clark-Ewing list [20]. Since * *p divides the order of the Weyl group only once, it suffices to specify the functor BX on the* * full subcategory A(G, t){tS, t}op= I(G, G(tS))op= I(G, W (SU (r + 1))op (13.7.6) where r denotes* * the rank. Take BX to be the functor indicated by the diagram ______________________________________________________* *______________________________________________________________________@ (7.12) Z(G)op_BSU9(r9+_1)_________________________________________* *_________________________BTGopbb______________________________________@ where Z(G), which is cyclic of order 2, 4, 4, 6, 2, 2, 2, acts on BSU (r + 1) v* *ia unstable Adams operations. See [4] for more details. (This follows Aguad'e's original construc* *tion very closely.) The 5-compact group E6. Take BX to be the functor on A(G, t){tS, t}op = I(G, G* *(tS))op, G = W (E6), indicated by the diagram ________________________________________________________* *______________________________________________________________________@ Cop2_BU(5)9x9BU(1)_________________________________________* *__________________________BTGopbb_____________________________________@ where C2 acts on U(5) x U(1) in some way. 7.13. Automorphisms of X(G(m, r, n)). [21, (2.13)][76, x7] Assume first that A(* *m, r, n) is a characteristic subgroup of G(m, r, n). Then NGL(n,Zp)(G(m, r, n)) = ZxpG(m, 1, n) because this normalizer is contained in the normalizer of A(m, r, n) which equa* *ls Zxpo n by the argument of [82, Lemma 3], and, on the other hand, a diagonal matrix diag(u1, .* * .,.un) 2 (Zxp)n normalizes G(m, r, n) if and only if it lies in ZxpA(m, 1, n). Thus (3.12.(3)), Out(X(G(m, r, n))) ~=ZxpG(m, 1, n)=G(m, r, n) ~=ZxpA(m, 1, n)=A(m, r, * *n) is an abelian group and the exact sequence (3.16) has the form (7.14) 1 ! Zxp=ZG(m, r, n) ! ZxpG(m, 1, n)=G(m, r, n) ! C(r,n)! 1 where ZG(m, r, n), the center of G(m, r, n), is cyclic of order m_r(r, n) and C* *(r,n)denotes a cyclic group of order the greatest common divisor (r, n) of r and n. Choose a primitive (p - 1)th root of unity i 2 Zxp, choose integers s and t w* *ith (r, n) = sr + tn, and put " = diag(i p-1_m, 1, . .,.1) 2 ZxpA(m, 1, n). Then Ä (m, r, n) projects* * onto a generator of the cyclic group C(r,n)and the element p-1_tD(r,n) p-1_r_E D (r,n) p-1_r_E 2 x i m i , i m(r,n)2 = i , i m(r,n) H (C(r,n); Zp=ZG(m, r,* * n)) classifies extension (7.14) because i p-1_mtA(m, r, n) = "(r,n)A(m, r, n). Cons* *equently, p-1_tD (r,n) p-1_r_E p - 1 (7.14) splits, i m 2 i , i m(r,n), _____mt 2 Z((r,n),p-1_mr_(r,n)) ` f'ifi fi , (r, n), p_-_1_mr_(r,fn)ififip_-m1t , (r, n)fifi* *p_-_1minZ(_r_(r,n)) where we at the final stage observe that _r__(r,n)and t are relatively prime si* *nce 1 = s__r_(r,n)+ t__n_(r,n). For instance,i(7.14)jsplits whenever (r, n) = (r, n2) for then (r, n) and _r__(* *r,n)are relatively prime so p-1 p-1 that (r, n), p-1_mr_(r,n)= (r, n), ___mclearly divides ___mt. More generally, ______________(p_-_1)(r,_n)__________niijoj max x2Z p - 1, p-1_mt + x (r, n), p-1_mr_(r,n) is the smallest possible order of an exotic automorphism of X(G(m, r, n)) proje* *cting onto a gener- ator of C(r,n). 7.15. Lemma. [76, x6] A(m, r, n) is a characteristic subgroup of G(m, r, n) if * *and only if (m, r, n) 62 {(2, 1, 2), (4, 2, 2), (3, 3, 3), (2, 2, 4)}. N-DETERMINISM 35 Proof.For n > 4, A(m, r, n) is the Fitting subgroup of G(m, r, n). (Consult e.g* *. [85] for general group theoretic information.) For 2 n 4, Fit(G(m, r, n)) = A(m, r, n) o F * *where F is a subgroup of Fit( n) which is elementary abelian of order n. If A(m, r, n) is no* *t characteristic in G(m, r, n), it is not characteristic in Fit(G(m, r, n)) and then (7.16) n = 2:m_rm|2m_r(r, 2) so that (m, r) = (2, 1), (2, 2), (4, 2) or (4, 4), n = 3:m_rm2|3m_r(r, 3) so that (m, r) = (3, 3), n = 4:m_rm3|4mm_r(r, 2) so that (m, r) = (2, 1) or (2, 2). Among these options, (2, 2, 2) is an illegal choice of parameters, A(4, 4, 2) ~* *=C4 is the unique cyclic subgroup of order 4 in G(4, 4, 2) ~=D8, A(2, 1, 4) is the unique elementary abe* *lian subgroup of order 16 of G(2, 1, 4), and in the remaining four cases it can be verified that* * A(m, r,_n) is not characteristic in G(m, r, n). * * |__| 7.16. Lemma. Let A o W be the semi-direct product for the action of a finite gr* *oup W on a finite abelian group A. If A is not characteristic in A o W , then |A| divides * *|Aoe||CW (oe)| for some non-trivial element oe 2 W . Proof.If A is not characteristic, some automorphism of the semi-direct product * *takes an element of A to an element (a, oe) where oe 2 W is non-trivial. As automorphisms preser* *ve centralizers up to isomorphism, we know that |A| divides |CAoW (a, oe)|. The exact sequence 1 ! Aoe! CAoW (a, oe) ! CW (oe) shows that |CAoW (a, oe)| divides |Aoe||CW (oe)|. * * |___| Since G(2, 1, 2) is conjugate to G(4, 4, 2) [21, 2.5], its normalizer was fou* *nd above. In the remaining three cases, there are exact sequences (3.16) 1 ! Zxp=ZG(4, 2, 2) ! NGL(2,Zp)(G(4, 2, 2))=G(4, 2, 2) ! 3 ! 1 1 ! Zxp! NGL(3,Zp)(G(3, 3, 3))=G(3, 3, 3) ! A4 ! 1 1 ! Zxp=ZG(2, 2, 4) ! ZxpW (F4)=G(2, 2, 4) ! 3 ! 1 describing the automorphism groups of XG(4, 2, 2), XG(3, 3, 3), and XG(2, 2, 4). 7.17. Automorphisms of other modular polynomial p-compact groups. If W is one o* *f the Aguad'e reflection groups or W = W (G2) and p = 3, then (3.17.(2)) End(X(W )) - {0} = Out(X(W )) = Zxp=Z(W ) for NGL(r,Zp)(W ) = ZxpW according to [4, 5.7]. The 3-compact group BX(G12) is also denoted BDI2 for, since G12 GL(2, Z3) ma* *ps isomor- phically onto GL(2, F3) [11, p. 272] [88, 10.7.1], the mod 3 cohomology algebra H*(BX(G12); F3) = H*(BT~; F3)GL(2,3)= F3[x12, x16] is the rank 2 mod 3 Dickson algebra [88, 8.1.5]: A polynomial algebra on a gen* *erator x12 in degree 12 and a generator x16 = P 1x12 in degree 16. DI2has the potential of c* *ontaining all other connected 3-compact groups of rank 2 as this is certainly true on the lev* *el of Weyl groups. Section 10 elaborates on this aspect of DI2. 7.18. Structure of polynomial p-compact groups. (Cf. [79]) We start by noting t* *hat polyno- miality of a connected p-compact group is determined by the universal covering * *p-compact group and the fundamental group. 7.19. Lemma. Let X be connected p-compact group with universal covering p-compa* *ct group SX [58, 3.3] and fundamental group ß1(X). Then 1.X is polynomial, SX is polynomial and ß1(X) is a free Zp-module. 2.If X is polynomial, then H*(BX; R) ! H*(BSX; R) is surjective and the kerne* *l is the ideal generated by the degree 2 cohomology classes, R = Fp, Zp, Qp. 3.If X is polynomial, then H*(BX; R) ~=H*(ß2(BX), 2; R) H*(BSX; R) as grade* *d algebras, R = Fp, Zp, Qp. 36 J.M. MØLLER Proof.If X is polynomial, H*(BX; Zp) is (7.1) concentrated in even degrees and * *is degree-wise free so that, in particular, the second homology group H2(BX; Zp) = ß2(BX) is a* * free Zp-module. The Serre spectral sequence for the Postnikov fibration K(ß1(X), 1) ! BSX ! BX * *collapses at the E3-page to yield E3 = R H*(ß2(BX),2;R)H*(BX; R) = H*(BSX; R). Conversely, if SX is polynomial and ß1(X) is free, then the Serre spectral sequ* *ence for the fibra- tion BSX ! BX ! K(ß2(BX), 2) collapses at the E2-page for degree reasons and sh* *ows that __ H*(BX; Fp) is concentrated in even degrees and is degree-wise free. * * |__| Q Let Y = Yibe a product of finitely many simple, simplyQconnected p-compact * *groups Yi, ß a (finite) subgroup of the p-discrete center ~Z(Y ) = ~Z(Yi) of Y , and ': ß ! * *~Sa homomorphism into the discrete approximation ~Sto a p-compact torus S. Define [67, x2] the p* *-compact group X = Y x S=(ß, ') by the short exact sequence ß (incl,')----!Y x S ! X. Any connected p-compact g* *roup has this form with Y = SX and S = Z(X)0 [58, 5.4]. 7.20. Corollary.X = Y x S=(ß, ') is polynomial if and only if ': ß ! ~SisQa mon* *omorphism and each simple factor Yi in the universal covering p-compact group Y = Yi eq* *uals SU(n) for some n or is one of the p-compact groups from ( 7.9). Proof.We use criterion (7.19.1). Elementary homological algebra performed on th* *e short exact sequence 0 ! ß1(S) ! ß1(X) ! ß ! 0 shows that ß1(X) is a free Zp-module if and * *only if ': ß ! ~Sis injective. (OneQmay also use the functor ~Tof x11 to see this.)* * The universal covering p-compact group Y = Yi is polynomial iff each simple factor Yi is po* *lynomial (7.28). According to (7.9) and Borel [9, 2.5], Yi is polynomial iff Yi = SU(n) for some* * n or r0W (Yi) 6=_ r0W (F4), r0W (E6), r0W (E7), r0W (E8) if p = 3 and r0W (Yi) 6= r0W (E8) if p =* * 5. |__| In greaterQdetail, any polynomial p-compact group X is of the form X = X1 x X* *2 where X1 = SU (ni) x S =(ß, ') is a polynomial p-compact group whose universal cov* *ering is a product of special unitary p-compact groups and X2 is a product of some of the * *simple, simply connected, centerless, polynomial p-compact groups of (7.9). 7.21. Corollary.All polynomial p-compact groups are totally N-determined. Q Proof.Since all simple factors of P X = P Y = P Yiare totally N-determined (7* *.20, 5.2,_7.9), X is totally N-determined (3.3, 3.7). * * |__| 7.22. Corollary.Let Y be a simply connected, polynomial p-compact group. Then 1.Y is determined by its Fp-Weyl group, and, 2.BY is cohomologically unique among Fp-local spaces. Proof.Since Y is totally N-determined (7.21), it suffices (4.4, 4.5) to show th* *at the cohomology groupQH1(W (Y ); Hom(t(Y ), t(Y ))) isQtrivial.Q But that is proved by Notbohm * *in [79, 6.2]: Let Y = Yi be as in (7.20). Let S = Si W (Yi) be the product subgroup wit* *h factors Si = W (Yi) in case Yi = SU(n) and Si = SylpW (Yi) in case Yi is one of the p-c* *ompact groups from (7.9). Then |W (Y ): S| is prime to p. The natural homomorphism CY (t(YQ* *)S) ! Y is a monomorphism of maximal rank [31, 4.3], and, by inspection, CY (t(Y )S) = CYi* *(t(Yi)Si) is isomorphic to a product of SU(n)s and U(n)s. Thus H1(W (t(Y )S); Hom(t(Y ), t(Y* * ))) = 0 by [74, 8.2]. But then also the cohomology group H1(W (Y ); Hom(t(Y ), t(Y ))) = 0 by a* * transfer argument_ because W (CY (t(Y )S)) = W (t(Y )S) S has index prime to p in W (Y ). * * |__| 7.23. Corollary.Any polynomial p-compact group is determined up to local isomor* *phism by its mod p cohomology algebra considered as an unstable algebra over the Steenrod al* *gebra. Proof.The mod p cohomological dimension as well as H*(BSX; Fp) (7.19), and henc* *e (7.22) BSX, can be read off from H*(BX; Fp) if this is a polynomial algebra. But this* * information_ is the local isomorphism class of X [67, 2.6]. * * |__| N-DETERMINISM 37 Two locally isomorphic p-compact groups, X1 = Y x S=(ß1, '1) and X2 = Y x S=(* *ß2, '2) are isomorphic iff there exist automorphisms g 2 Out(Y ) and h 2 Out(S) = Aut(S~) s* *uch that the diagram ~Z(Y )oooß1o_//'1//_~S | Z~(g~=)|| |~= ~=h|| fflffl| fflffl||fflffl| ~Z(Y )oooß2o_//'2//_~S commutes [67, 4.3, 4.5]. The next example shows that there are locally isomorphic but non-isomorphic p* *olynomial p- compact groups with isomorphic mod p cohomology algebras. 7.24. Lemma. Let X = Y x S=(ß, ') be a polynomial p-compact group. If ß pZ~(* *Y ), then W (X) and W (Y x S) are Fp-similar so that H*(BX; Fp) and H*(BY x BS; Fp) are i* *somorphic unstable polynomial algebras. Proof.We may assume that ß, and hence S, is non-trivial as otherwise there is n* *othing to prove. From the short exact sequence 0 ! ß ! ~T(Y ) x ~T(S) ! ~T(X) ! 0 we get the exa* *ct sequence 0 ! H1(ß; Fp) ! H2(T~(X); Fp) ! H2(T~(Y ) x ~T(S); Fp) ! H2(ß; Fp) !* * 0 of FpW (X)-modules. The map induced by ': ß ! ~S, Ext(T~(S), Fp) = H2(T~(S); Fp) ! H2(ß; Fp) = Ex* *t(ß, Fp), is surjective since ' is injective. This implies that the FpW (X)-module homom* *orphism onto H2(ß; Fp) has a right inverse. We next show that the FpW (X)-module homomorphism out of H1(ß; Fp) has a left* * inverse. Write K for the kernel of the map onto H2(ß; Fp) and apply H0(W (X); -) to the * *short exact sequence 0 ! H1(ß) ! H2(T~(X)) ! K ! 0 to get the exact sequence H1(W (X); H2(T~(X))) ! H1(W (X); K) ! H1(ß) ! H0(W (X); H2(T~(X))) where H1(W (X); K) ~=H1(W (X); H2(T~(Y ) x ~T(S))) ~=H1(W (X); H2(T~(Y ))) sinc* *e the homol- ogy groups H1(W (X); H2(ß)), H2(W (X); H2(ß)), and H1(W (X); H2(T~(S))) are tri* *vial [5, 3.2] [74, 3.1]. It suffices to show that the map into H0(W (X); H2(T~(X))) is injective o* *r, by exactness, that the map out of H1(W (X); H2(T~(X))) is surjective. The maps ~Z(Y ) ! ~Z(X) = ~Z* *(Y )xS=(ß, ') ! ~Z(Y )=ß induce dual maps Hom (Z~(Y )=ß, Z=p1 ) ! Hom (Z~(X), Z=p1 ) ! Hom (Z~(* *Y ), Z=p1 ) whose composition is surjective under the assumption of the lemma that ß pZ~(* *Y ). Thus the second of these maps, which by (11.17) can be identified to the first map i* *n the above exact sequence, must be an epimorphism, too. We now conclude that 1 2 H2(T~(X); Fp)= H1(ß; Fp) + cokerH (ß; Fp) ! H (T~(X); Fp) 2 2 = H2(ß; Fp) + kerH (T~(Y ) x ~T(S); Fp) ! H (ß; Fp) = H2(T~(Y ) x ~T(S); Fp) as FpW (X)-modules and therefore the two rings of invariants, H*(BX; Fp) ~=H*(T~(X); Fp)W(X) ~=H*(T~(Y ) x ~T(S); Fp)W(X) ~=H*(BY x BS;* * Fp), are isomorphic unstable algebras over the mod p Steenrod algebra. * * |___| 7.25. Example.[98] [74, 9.6] The p-compact groups Ui= SU(p ) x U(1)=(Z=pi, incl), 0 i , 3, from the local isomorphism system of U(p ) (11.28) are distinct, polynomial (7.* *19) p-compact groups but (7.24) the Weyl groups W (Ui) are Fp-similar and the unstable algebr* *as H*(BUi; Fp) are isomorphic for 0 i < (and distinct from H*(BU(p ); Fp)). Thus the polyn* *omial p-compact group SU(p ) x U(1) is not determined by its Fp-Weyl group, not even by its mod* * p cohomology algebra. An unstable graded algebra over the mod p Steenrod algebra is 38 J.M. MØLLER opolynomial if its underlying graded algebra over Fp is polynomial on finite* *ly many generators, otopologically realizable if it is isomorphic to the mod p cohomology of a t* *opological space. Steenrod's problem [90] asks for the determination of all topologically realiza* *ble polynomial alge- bras over the the mod p Steenrod algebra. A complete solution was found by D. N* *otbohm [76, 79] but it may still bePworthwhile to record also the following form of the answer. Write P (H, t) = (dimkHi)tifor the Poincar'e series of the graded algebra H* * over the field k. 7.26. Theorem. A polynomial unstable algebra over the mod p Steenrod algebra is* * topologically realizable if and only if it is isomorphic to H*(BX; Fp) = H*(T~(X); Fp)W(X) fo* *r some polynomial p-compact group X (as described in ( 7.20)). Moreover, the following conditions* * are equivalent 1.(W, ~T) is the Weyl group of a polynomial p-compact group, 2.(W, ST~) is polynomial and H0(W ; L(T~)) is a free Zp-module, 3.(W, ~T) is polynomial and H0(W ; L(T~)) is a free Zp-module, 4.(W, ~T) is polynomial and H1(W ; ~T) = 0, 5.(W, ~T) is polynomial and P (H*(T~; Fp)W , t) = P (H*(T~; Qp)W , t), 6.H*(T~; Zp)W is polynomial, for any Zp-reflection group (W, ~T). Proof.If BX is an Fp-local space and H*(BX; Fp) is polynomial, then X is a poly* *nomial p- compact group and H*(BX; Zp) = H*(T~(X); Zp)W(X) is a polynomial ring (7.1). T* *his proves 1) 6, and 6, 5by (7.27); we proceed to show 5) 4) 3) 2) 1. If item 5 holds, then H1(W ; Hj(T~; Zp)) = 0 for all degrees j 0 (7.27). Fo* *r j = 1, in particular, H1(W ; L(T~)_) = 0 which, for general reasons (11.11.1, 11.8.8-9), is equivalen* *t to H1(W ; ~T) = 0 or to H0(W ; L(T~)) being a free Zp-module. From the (split) short exact sequen* *ce 0 ! ST~! ~T! H0(W ; ~T) ! 0 of ZpW -tori (11.8.10) we get an epimorphism H2(T~; Fp) i H2(ST~* *; Fp) of FpW - modules and therefore [71, 4.1] (W, ST~) is polynomial. In the splitting (11.15* *) of (W, ST~) into a product of simple Zp-reflection groups (Wi, ~Ti) with H0(Wi; L(T~i)) = 0, each * *factor is polynomial (7.28), i.e. not similar to the Weyl groups of F4, E6-8at p = 3 and E8at p = 5 * *(7.4). Thus all simple factors of (W, ST~) are Weyl groups of simple, simply connected, polynomial p-c* *ompact groups, (W, ST~) is the Weyl group of the product Y of these, and (W, ~T), where ~T= (S* *T~x ~S)=(ß, '), is the Weyl group of the p-compact group X = (Y x S)=(ß, '), BS = (BS~)^p, which i* *s_polynomial_ by (7.20). * *|__| The map æ oe æ oe Isomorphism classes of (W,T~) Similarity classes of polynomial polynomial p-compact groups----! Zp-reflection groups with H1 = 0 is surjective by (7.26.(4)) and injective by (7.21). 7.27. Lemma. Let (W, ~T) be a Zp-reflection group. Then H*(T~; Zp)W is polynom* *ial if and only if (W, ~T) is polynomial and P (H*(T~; Fp)W , t) = P (H*(T~; Qp)W , t). If thi* *s is the case, then H*(T~; Zp)W Z=p ~=H*(T~; Fp)W and H1(W ; H*(T~; Zp)) = 0. Proof.The Poincar'e series condition ensures that the monomorphism of H*(T~; Zp* *)W Z=p to H*(T~; Fp)W is an isomorphism. Now, if the Poincar'e series condition is satisf* *ied, and H*(T~; Zp)W Z=p = H*(T~; Fp)W is polynomial, then there exist homogeneous elements x1, . .x* *.r2 H*(T~; Zp)W , where r is the rank of ~T, that reduced mod p become polynomial generators for * *H*(T~; Fp)W . Thus the ring homomorphism Zp[x1, . .,.xr] ! H*(T~; Zp)W becomes an isomorphism mod * *p and hence it is an isomorphism by Nakayama's lemma. Conversely, if H*(T~; Zp)W is polyno* *mial, then [78, 2.3, 2.4] [77] the polynomiality condition from [11, Ch 5, x5, Exercice 5] [88,* * 5.5.4, 5.5.5] can be used to show that H*(T~; Fp)W is polynomial. The last assertion of the lemma f* *ollows from the exact sequence . ...p-!H*(T~; Zp)W i H*(T~; Fp)W -0!H1(W ; H*(T~; Zp)) .p-!H1(W ; H*(T~;* * Zp)) ! . . . where H1(W ; Hj(T~; Zp)) is a finite Zp-module for fixed degree j. * * |___| N-DETERMINISM 39 7.28. Lemma. Let A and B be finitely generated graded algebras over a field k. * *If A k B is a graded polynomial ring over k on homogeneous generators of positive degree, the* *n both factors A and B are polynomial. Proof.Since A kB is free over A (a k-basis for B provides an A-basis for A kB* *) and the global dimension of the polynomial algebra A k B is finite by Hilbert's syzygy theore* *m [7, 4.2.3], the global dimension of A is also finite. Thus A is polynomial by Serre's converse_* *[7,_6.2.3] to Hilbert's theorem. |_* *_| 8.Proofs of Theorem 1.2 and Corollaries 1.3-1.6 This small section contains the proofs of the results stated in the introduct* *ion. Proofs of Theorem 1.2, Corollary 1.3, and CorollaryT1.5.he classification of Th* *eorem 1.2 is the content of (5.2, 6.1, 7.9). To obtain Corollary 1.3 and Corollary 1.5, combine * *this with_(4.4, 4.10, 11.18, 11.25) and (3.3, 3.7, 3.12.(3)). * * |__| Proof of Corollary 1.4.Two connected p-compact groups with similar Zp-reflectio* *n groups have isomorphic maximal torus normalizers (3.12.(1)), so are isomorphic (1.2). Thus * *the map (W, L) is injective. To prove that the map (W, L) is surjective, let (W, L) be any Zp-reflection g* *roup. Then L sits as the kernel of a short exact sequence (11.5) 0 ! L ! LH0(W ; ~T) x P L ! ~ß(L) ! 0 of ZpW -modules. Choose a p-compact torus S and a centerless p-compact group P * *X such that S x P X realizes the Zp-reflection group in the middle. This is possible since* * P L is a product (11.15) of simple Zp-reflection groups, each of which is realizable (7.10). The* * Zp-reflection group (W, L) is now realized by a covering p-compact group of S x P X (4.10). * * __ The expression for the automorphism group of X is (3.12.(3), 3.15). * * |__| Proof of Corollary 1.6.Observe that that the maximal torus normalizers [59, 1.3* *] for X and G become homotopy equivalent after fibre-wise completion away from the prime 2. T* *his is because the maximal torus normalizers of the associated p-compact groups split [4] when* * p is odd, cf. [59, Proof of Proposition 5.5]. Thus there exists a space (BN)[1_2] and rational equ* *ivalences (BX)[1_2] (BN)[1_2] ! (BG)[1_2] that p-complete to maximal torus normalizers for the p-compact groups (BX)^pand* * (BG)^pat each odd prime p. In this situation, N-determinism of the p-compact group (BG)^pand * *the Arithmetic Square [14, VI.8.1], ensure the existence of a homotopy equivalence (BG)[1_2] '* * (BX)[1_2]_of spaces localized away from 2. * * |__| Within the framework of this paper, it easily can be shown (see remark below * *(2.13)) that also the simple p-compact group (E8, p = 5) [94] is totally N-determined, determined* * by its R-Weyl group for R = Zp, Qp, Fp, and is a cohomologically unique p-compact group (4.4,* * 11.18, 11.25). However, more information is needed for the other members of the E-family [6]. * *If this program goes through we can remove the exceptions from Theorem 1.2, and it will then fo* *llow that any finite family Yi of connected, simple, non-abelian, pairwise non-isomorphic p-* *compact groups is similarity free and any connected p-compact group is completely reducible in* * the (provisional) sense of [64, 3.4, 3.10] when p is odd. Thus for instance [66, 5.2] will apply * *to all (non-connected) p-compact groups G and [64, p. 381] will contain a description of the set "Q (X* *1, X2) of rational isomorphisms between any two locally isomorphic p-compact groups, X1 and X2, fo* *r p odd. 9.N-determinism of product p-compact groups We show in this section that determinacy behaves well with respect to formati* *on of (finite) products of p-compact groups. First two lemmas of a general nature. A p-compact group morphism f :X ! Y is * *said to be trivial if Bf :BX ! BY is null-homotopic. 40 J.M. MØLLER 9.1. Lemma. Let X and Y be p-compact groups and X ! Z(Y ) a p-compact group mor* *phism into the center of Y . If the composite morphism X ! Z(Y ) ! Y is trivial, then* * X ! Z(Y ) is trivial. Proof.Turn the center Bz :BZ(Y ) ! BY into a fibration (with fibre Y=Z(Y )) and* * map BX into it to obtain the fibration map(BX, Y=Z(Y )) ! map(BX, BZ(Y ))Bz-1(B0)! map(BX, BY )B0 where the total space consists of all maps BX ! BZ(Y ) that composed with Bz be* *come null- homotopic. With the help of the Sullivan Conjecture for p-compact groups [31, 9* *.3], the fibre of this fibration identifies to Y=Z(Y ) and the base to BY . Thus the total space* * identifies_to the connected space BZ(Y ) = map(BX, BZ(Y ))B0. This shows the lemma. * * |__| 9.2. Lemma. Let f :X ! Ybe a p-compact group morphism and jp:Np(X) ! X the p-no* *rmalizer [30, 9.8] of the maximal torus of X. Then 1.[32, 5.6] f is a monomorphism, fjp is a monomorphism 2.[67, 6.6] f is trivial, fjp is trivial If X is connected, this remains true with the p-normalizer replaced by the maxi* *mal torus. Proof.Suppose that the restriction fjp of f to the p-normalizer is a monomorphi* *sm. Then [30, 9.11] H*(BNp(X)) is a finitely generated H*(BY )-module via H*(Bfjp). Since H** *(BX) is a H*(BY )-submodule of H*(BNp(X)) thanks to the transfer homomorphism [31, 9.13] * *and H*(BY ) a noetherian graded ring [30, 2.4], H*(BX) is a finitely generated H*(BY )-modu* *le via H*(Bf). The converse follows from the fact that the composition of two monomorphisms is* * a monomorphism. If X is connected, any monomorphism of Z=p to X factors through the maximal t* *orus monomor- phism i: T (X) ! X[30, 4.7, 5.6] [31, 3.11]. This implies that if ~Np(X) ! Np(X* *) ! X ! Y has a non-trivial kernel, the same is true for ~T(X) ! T (X) ! X ! Y [30, x7]; here* *, N~p(X) and ~T(X) are discrete approximations [30, 6.4]. In other words, if T (X) ! X ! Y i* *s injective, so is Np(X) ! X ! Y [30, 7.3] [31, 3.5]. * * __ The second part of the lemma is [67, 6.6, 6.7]. * * |__| We now address N-determinism of automorphisms of product p-compact groups. Le* *t X1 and X2 be p-compact groups and ooß1_ _ß2_//_ X1 __'1_//X1x X2o'2oX2_ the natural projections and inclusions. 9.3. Lemma. Let f :X1x X2 ! X1 be a p-compact group morphism such that f'1:X1 !* * X1 is an isomorphism and f'2:X2 ! X1 is trivial. Then f is conjugate to f'1ß1. Proof.We want to show that the adjoint of Bf, BX2 ! map(BX1, BX1)B(f'1), which * *maps into a space homotopy equivalent [31, 1.3] to BZ(X1), is null-homotopic. But this fo* *llows immediately from (9.1) since composition with the evaluation monomorphism to BX1 gives the * *null-homotopic_ map Bf O B'2. |* *__| 9.4. Proposition.Let X1 and X2 be two connected p-compact groups with N-determi* *ned auto- morphisms. Then also the product p-compact group X1x X2 has N-determined automo* *rphisms. Proof.Let f be an automorphism of X1 x X2 under the product N1 x N2 of the two * *maximal torus normalizers. The morphism ß1f'1:X1 ! X1 is an isomorphism for [58, 3.7] [* *31, 4.7] it is a rational equivalence [30, 9.7] and a monomorphism (9.2). As also ß1f'2:X2 ! X* *1 is trivial by (9.2), it follows from (9.3) that ß1f is conjugate to ß1f'1ß1. Similarly, ß2f i* *s conjugate to ß2f'2ß2 and thus f is conjugate to the product morphism f1x f2 where f1 = ß1f'1 and f2 * *= ß2f'2. Thus N(f) = N1(f1) x N2(f2) and, since X1 and X2 have N-determined automorphisms, it* * follows_that f1 and f2 are conjugate to identity morphisms. * * |__| Next, we address N-determinism of products. This is based on a slight reformu* *lation of the Splitting Theorem [32, 6.1] [80]. N-DETERMINISM 41 9.5. Theorem. Assume that p is odd. Let X be a connected p-compact~group and i:* * T ! Xa maximal torus with normalizer j :N ! X. For any decomposition N -=!N1 x N2 of N* * into a product of two extended p-compact tori, N1 and N2, there exist p-compact groups* *, X1 and X2, and an isomorphism s: X ! X1x X2 such that ~= N ____//_N1x N2 j|| j1xj2|| fflffl|~= fflffl| X _s__//_X1x X2 commutes up to conjugacy where j1:N1 ! X1 and j2:N2 ! X2 are normalizers of max* *imal tori. Proof.Write Ni, i = 1, 2, as an extension Ti ! Ni ! Wi of a p-compact torus Ti * *and a finite group Wi. Then the Weyl group W = ß0(N) of X is isomorphic to W1x W2 and Wiacts* * [32, 6.3] as a reflection group on ß1(Ti) Q. According to [32, 6.1], the splitting ß1(T* * ) ~=ß1(T1) x ß1(T2) as a W ~=W1x W2-module can be realized by a p-compact group splitting s: X ! X1* *x X2. This means that if N(s): N ! N01x N02, where j0i:N0i! Xiis the normalizer of the max* *imal torus Ti! Xi, i = 1, 2, is the lift [67, 5.1] of s, then the discrete approximation [* *31, 3.12] ~N(s) to N(s) determines an isomorphism 0_______//~T________//~N________//W_______//_1 ~T(s)|| N~(s)|| |W(s)| fflffl| fflffl| fflffl| 0_____//~T1x ~T2_//_N01x N02_//_W1x W2____//_1 of short exact sequences where ~T(s) is the given splitting ~T~=~T1xT~2and W (s* *) the given splitting W ~=W1x W2. Relative to the given splitting ~N~=~N1x ~N2, the middle isomorphis* *m ~N(s) takes ~N1x ~N2isomorphically to ~N01x ~N02. The composite 0 ~N1'1-!~N1x ~N2~N(s)---!~N01x ~N02ß2-!~N02, where '1 is the injection and ß02the projection, can be factored through a homo* *morphism W1 ! ~T2as it restricts to the trivial morphism ~T1! T~2. Since p is assumed to be * *odd, any such homomorphism is trivial for the reflection group W1 is generated by elements of* * order prime to p. This implies that ~N(s): ~N1x ~N2! ~N01xi~N02s the product of two isomorphisms,* * ~N1! ~N01and ~N2! ~N02. Let ji, i = 1, 2, be the composite of this isomorphism ~Ni! ~N0iwith* * ji:N~0i! Xi. |___| The assumption that p should be odd is presumably not essential. 9.6. Proposition.The product of two connected N-determined p-compact groups is * *N-determined when p > 2. Proof.This is immediate from the commutative diagram N1x N2 r HHHTTTT rrr HH TTTTT rrr 0HHH TTTTT xxrrr j H$$ ~=T)) X1x X2 X0__s__//X01x X02 where X0 is any p-compact group with maximal torus normalizer j0 and s the spli* *tting iso- morphism from (9.5). For if X1 and X2 are N-determined, we get isomorphisms f1* *:X1 ! X01 and f2:X2 ! X02under N1 and N2, respectively, and s-1 O (f1 x f2) is then an is* *omorphism_ X1x X2 ! X0 under N1x N2. |_* *_| The next step is to generalize (9.4) and (9.6) to possibly nontrivial extensi* *ons. 9.7. Theorem. Let Y ! G ! X be a short exact sequence of connected p-compact gr* *oups. 1.If the adjoint forms P X and P Y have N-determined automorphisms, so does G. 2.If the adjoint forms P X and P Y are N-determined and p > 2, so is G. 42 J.M. MØLLER Since a connected p-compact group has N-determined automorphisms or is N-dete* *rmined pro- vided this holds for its adjoint form [68, 4.8, 7.10], the proof of the above t* *heorem is an immediate consequence of (9.4, 9.6) and the lemma below. 9.8. Lemma. Let Y ! G ! X be an extension of connected p-compact groups. Then 1.G is locally isomorphic [67, 2.7] to X x Y , and, 2.the adjoint form P G is isomorphic to P X x P Y . Proof.Let SX denote the universal covering p-compact group and S = Z(X)0 the id* *entity com- ponent of the center of X. Let Y ! E1 ! SX xS be the extension obtained by pull* *ing back along the isogeny [58, 5.4] SX x S ! X. Since [66, 3.2, 3.3, 3.4] the projection of S* *X x S to S induces a ~= bijection Ext(S, Y ) -! Ext(SX x S, Y ) of equivalence classes of extensions, t* *he p-compact group E1, which is locally isomorphic to G, is isomorphic to SX x E2 for some extensi* *on Y ! E2 ! S of the p-compact torus S by Y . By [66, 2.6], E2 is locally isomorphic to S x Y* * and hence G is locally isomorphic to SX x S x Y , which is locally isomorphic to X x Y . Any connected p-compact group has the same adjoint form as its universal cove* *ring p-compact_ group (4.6). Hence P G ~=P (SX x SY ) ~=P X x P Y . * * |__| 9.9. Example.Since [66, 3.3, 3.4] Ext(PU (p), SU(p)) = [BPU (p), B2Z(SU (p))] = H2(BPU (p); Z=p) = Hom (Z=p, Z* *=p) = Z=p there are p equivalence classes of extensions of PU(p) by SU(p) in the category* * of p-compact groups. However, since the local isomorphism system of the p-compact group SU(p) x SU(p* *) [67, p. 217] SU (p) x PU(p)X ffff33f XXX,,X SU (p) x SU(p)X 3PU(p)3x PU(p) XXXX++X ffffff SU(p)o PU(p) consists of very few p-compact groups, we see from (9.10) that the middle p-com* *pact group in any of these extensions must be isomorphic to either the direct product SU(p) x PU(* *p) or the semi- direct product SU(p) o PU(p) for the conjugation action of PU (p) on SU(p). All* * the p-compact groups locally isomorphic to SU(p) x SU(p) are totally N-determined. The automo* *rphism groups are, for instance, Out(SU (p) x PU(p)) = Z*px Z*p Out(SU (p) o PU(p)) ~={(u, v) 2 Z*px Z*p| u v mod p} o Z=2 where Z=2 permutes the coordinates. Formulas like these follow from (5.1) in co* *mbination with [67, 4.3] and [64, 3.5]. 9.10. Lemma. For any short exact sequence Y -'!G ß-!X of connected p-compact gr* *oups there exists a corresponding short exact sequence Z(Y ) ! Z(G) ! Z(X) of centers. In * *particular, Y and G have isomorphic centers if X is centerless. Proof.Let z :Z(Y ) ! Ybe the center of Y . In the commutative diagram map(BZ(Y ), BY )Bz___//_map(BZ(Y ), BG)B('z)//_map(BZ(Y ), BX)B0 ' || || |'| fflffl| fflffl| fflffl| BY __________________//_BG__________________//BX the horizontal lines are fibration sequences and the vertical arrows are evalua* *tion maps. Note that the middle arrow is a homotopy equivalence since the outer two arrows are homot* *opy equivalences. This shows that 'z :Z(Y ) ! Gis central and by naturality we obtain [30, 8.3] a* * short exact sequence P Y ! G=Z(Y ) ! X N-DETERMINISM 43 of p-compact groups. This extension is equivalent to the trivial extension [66* *, 3.4] since P Y is centerless [58, 4.7] [31, 6.3]. Thus G=Z(Y ) ~=P Y x X and Z(G)=Z(Y ) ~=Z(G=Z(Y )) ~=Z(P Y x X) ~=Z(X) by [58, 4.6.(4)]. * * |___| 10.Maximal rank subgroups of DI2 This section contains some general theory for monomorphisms of between p-comp* *act groups and it is shown that DI2contains essentially unique copies of each of the 3-compact* * groups SU(2) x SU(2), U(2), Spin(5), SU(3), PU(3), and G2. Recall from the previous section that BDI2 is the homotopy colimit (at p = 3)* * of a diagram of the form ________________________________________________________* *______________________________________________________________W(SU(3))@ (10.1) (Z=2)op_BSU9(3)9______________________________________________* *_____________________BTW(SUo(3))pee___________________________________@ where Z=2 acts on BSU (3) as {_ 1} and W , the Weyl group of DI2, is the subgro* *up of GL(2, Zp) W = W (DI2) = (3))= -E generated by the Weyl groups of SU(3) and PU(3) or, alternatively, by the matri* *ces ` ' ` ' oe = 01 -1-1 and ø = -01 -10 together with scalar multiplication with -1. The semi-direct product ~N(DI2) = ~To W where ~T= ~T(SU (3)) is (3.12) the discrete approximation to the maximal torus * *normalizer N(DI2) for DI2. We start our investigation of maximal rank subgroups of DI2with some general * *remarks. Let X1 and X2 be two connected p-compact groups of the same rank. Let j1:N1 !* * X1 and j2:N2 ! X2 be normalizers of maximal tori i1:T1 ! X1and i2:T2 ! X2. Consider the map [69, 3.11] (10.2) N :Mono (X1, X2) ! Mono(N1, N2) that to any conjugacy class of a monomorphism f :X1 ! X2associates the unique c* *onjugacy class N(f): N1 ! N2such that N(f) N1_____//N2 j1|| |j2| fflffl|fflffl| X1__f__//X2 commutes up to conjugacy. Here, Mono(X1, X2) [BX1, BX2] denotes the set of co* *njugacy classes of monomorphisms of X1 into X2 and Mono(N1, N2) denotes the set of conjugacy cl* *asses of maps BN1 ! BN2 inducing monomorphisms on ß1 and isomorphisms on ß2. Note that if ~N1* *! N1 and ~N2! N2 are discrete approximations then [BN~1, BN~2] = [BN1, BN2] so that Mono (N1, N2) = Mono(N~1, ~N2)=N~2 consists of conjugacy classes of monomorphisms of N~1into N~2. For any monomor* *phism f 2 Mono(X1, X2), we let N~(f) 2 Mono (N~1, ~N2), determined up to conjugacy, denot* *e any discrete approximation to N(f). 10.3. Definition.The monomorphism f 2 Mono(X1, X2) is N-determined if the subse* *t N-1(N(f)) of Mono(X1, X2) consists of f alone. Let W1 = ß0(N1) and W2 = ß0(N2) denote the Weyl groups. 44 J.M. MØLLER 10.4. Example.If p - |W1|, then all monomorphisms are N-determined. Indeed, it * *is not difficult to see that (10.2) is bijective in this case. In case X1 = X2, the map (10.2) is the homomorphism N :Out(X1) ! Out(N1) prev* *iously encountered. We say that X1 has N-determined monomorphisms if this map is injec* *tive; if X1 is totally N-determined N is a bijection. Note that (10.2) is equivariant in the s* *ense that there is a commutative diagram Mono(X1, X2) x Out(X1)___//_Mono(X1, X2) NxN || |N| fflffl| fflffl| Mono (N1, N2) x Out(N1)__//_Mono(N1, N2) relating group actions on sets of monomorphisms. Let G and H be groups. Write {G > H} for the set of conjugacy classes of subg* *roups abstractly isomorphic to H of G. 10.5. Proposition.Let i: X1 ! X2be a monomorphism between the two p-compact gro* *ups X1 and X2 of the same rank. Then the Euler characteristic Ø(X2=iX1) = |W2:W1| and * *if oi is N-determined oX1 is totally N-determined o{N~2> ~N1} is a one-point set then the action Mono(X1, X2) x Out(X1) ! Mono(X1, X2) is transitive and all mon* *omorphisms of X1 into X2 are N-determined. Proof.The first part is [69, 3.11]. For the second part, note first that for an* *y ff 2 Out(X1), iff is an N-determined monomorphism. Suppose namely that N(f) = N(iff) = N(i)N(ff) * *for some monomorphism f :X1 ! X2. Then N(fff-1) = N(f)N(ff)-1 = N(i), so fff-1 = i and t* *herefore f = iff. Let now f :X1 ! X2be any monomorphism and ~N(f): ~N1! ~N2a representative for* * the con- jugacy class N(f). Since ~N2contains but a single copy of ~N1up to conjugacy an* *d X1 is totally N-determined, N~(f) = N~(i)N~(ff) for some automorphism ff of X1. Then N~(f) =* * N~(iff)_and f = iff. * *|__| The third condition is satisfied in case N~1= ~T1oW1, N~2= ~T2oW2 are semi-di* *rect products and the set {W1 > W2} is a one-point set. 10.6. Definition.For a monomorphism f :Y ! X of p-compact groups, let WX (f) or* * WX (Y ), the Weyl group of f, denote the component group of the Weyl space WX (Y ) [32, * *4.1, 4.3]. 10.7. Proposition.Let f :Y ! X be a monomorphism of p-compact groups. 1.If the homomorphism ß0(Z(Y )) ! ß0(CX (Y )) induced by f is surjective, the* *n the Weyl group WX (Y ) is the isotropy subgroup Out(Y )f for the action of Out(Y ) on f 2 * *Mono(Y, X). 2.If f is centric [28], then there is a short exact sequence of loop spaces [* *30, 3.2] Y ! NX (Y ) ! WX (Y ) where NX (Y ) is the normalizer of f [32, 4.4]. Proof.The monomorphism f determines a fibration a Bf_ WX (Y ) ! map(BY, BY )Bff--! map(BY, BX)Bf fOff'f where the components of the total space are indexed by the isotropy subgroup Ou* *t(Y )f and the fibre is the Weyl space. The assumptions of the proposition assure that the inc* *lusion of the fibre into the total space is a bijection on ß0. If we make the additional assumption* * that f be centric, the Weyl space becomes homotopically discrete and the exact sequence of the pro* *position_is the one from [32, 4.6] * * |__| 10.8. Lemma. Suppose that i: X1 ! X2is a monomorphism and let N(i) 2 Mono(N1, N* *2) be the induced monomorphism of normalizers. Then the stabilizer subgroup Out(N1)N(* *i)of N(i) is isomorphic to the quotient group NW2(W1)=W1. N-DETERMINISM 45 Proof.Note that there is an epimorphism NN~2(N~1)=N~1i Aut (N~1)=N~1N(i)= Out(N1)N(i) given by conjugation by elements of N~2normalizing N~1. This homomorphism is a* *ctually also injective, hence bijective, for if conjugation by, say, n2 2 NN~2(N~1) agrees w* *ith conjugation by some element n1 2 N~1, then n1 and n2 have the same image in W2, so that n2 bel* *ongs to N~1. This follows because the Weyl groups of the connected p-compact groups X1 and X* *2 are faithfully represented in their maximal tori. Consequently Out(N1)N(i)~=NN~2(N~1)=N~1 and this last group is isomorphic to the quotient group NW2(W1)=W1 by the proje* *ction_~N2iW2. |__| 10.9. Proposition.Let i: X1 ! X2 be an N-determined monomorphism between the tw* *o p- compact groups X1 and X2 inducing an epimorphism ß0(Z(X1)) ! ß0(CX2(X1)). Then WX2(X1) = NW2(W1)=W1 provided X1 is totally N-determined. Proof.The assumptions imply that the Weyl group WX2(X1) is isomorphic to the st* *abilizer sub- group Out(X1)iwhich again is isomorphic to the stabilizer subgroup Out(N1)N(i)f* *or the action Mono(N~1, ~N2)=N~2x Aut(N~1)=N~1! Mono(N~1, ~N2)=N~2 of Out(N1) on N(i) 2 Mono(N1, N2). Now apply (10.8). * * |___| 10.10. Example.By (10.4), the inclusion ~To<-ø>æ T~oW is realizable by an N-det* *ermined monomorphism i: U(2) ! DI2. The monomorphism i is centric (because BU(2) = BThZ* *=2and the centralizer CU(2)(T ) ~=T ~=CDI2(T )) so (10.5, 10.7, 10.9) Ø(DI2=U(2)) = 24 and WDI2(U(2)) ~=Z(W ) and Out(U(2)) acts transitively on Mono(U(2), DI2) since {T~oW > ~To<-ø>} is a * *one-point set. 10.11. Example.Similarly, {W > Z=2xZ=2} is a one-point set, so there is an esse* *ntially unique monomorphism i: SU(2) x SU(2) ! DI2realizing the inclusion of ~To(Z=2xZ=2) into* * ~ToW . The monomorphism i is N-determined, centric, and Ø(DI2=SU (2) x SU(2)) = 12 and WDI2(SU (2) x SU(2)) ~=Z(W ) and Out(SU (2) x SU(2)) acts transitively on Mono(SU (2) x SU(2), DI2). 10.12. Example.Also {W > D8} = {D8}, where D8 is the dihedral group of order 8.* * It follows that there exists a unique monomorphism i: Spin(5) ! DI2realizing the inclusion* * ~ToD8æ ~ToW . This monomorphism is centric (because BSpin(5) and B(T~oD8) are H*F3-equivalent* *), so Ø(DI2=Spin(5)) = 6 and WDI2(Spin(5)) ~=Z(W ) and Out(Spin(5)) acts transitively on Mono(Spin(5), DI2). In a situation where a pair of monomorphisms G ! X1 and G ! X2 are given, let* * us write mapBG (BX1, BX2) for the space of maps BX1 ! BX2 under BG up to homotopy. 10.13. Lemma. Let z :Z ! X1be a central monomorphism and i: X1 ! X2any monomorp* *hism inducing an isomorphism X1 ~=CX1(z) ! CX2(f O z). Then f induces a homotopy equ* *ivalence mapBZ(BX1, BX1) ! mapBZ (BX1, BX2) of mapping spaces. 46 J.M. MØLLER Proof.The spaces BCX1(z) = map(BZ, BX1)Bz and BCX2(f O z) = map(BZ, BX2)B(fOz)a* *re X1=Z-spaces and BCf(Z): BCX1(z) ! BCX2(f O z)is an X1=Z-map inducing a map mapBZ (BX1, BX1) = BCX1(z)h(X1=Z)! BCX2(f O z)h(X1=Z)= mapBZ (BX1, BX2) of homotopy fixed point spaces. If Cf(z) is an isomorphism, then this map is a * *homotopy_equiva- lence. |* *__| This happens for instance for V ! CX (V ) ! X so that map BV(BCX (V ), BCX (V )) ' mapBV (BCX (V ), BX) for any connected p-compact group X, any elementary abelian p-group V , and any* * monomorphism V ! X. 10.14. Example.Let i: SU(3) ! DI2denote the monomorphism arising in the constru* *ction (10.1) of BDI2 as a homotopy colimit. By (10.13), Bi induces a homotopy equival* *ence mapBZ=3(BSU (3), BSU (3)) ! mapBZ=3(BSU (3), BDI2) where Z=3 ! SU(3) is the center, and thus a bijection Out+(SU (3)) ! Mono(SU (3), DI2), where Out+(SU (3)) consists of the unstable Adams operations _u indexed by unit* *s u 2 Z*3with u 1 mod 3. We obtain a commutative diagram ~= Out+(SU (3))________________//Mono(SU (3), DI2) fflffl N |~=| |N| fflffl| fflffl| Out(T~oW (SU (3)))=W (SU (3))//_Mono(T~oW (SU (3)), ~ToW )=W and using (10.8) we see that the kernel of the composition going down and then * *right is trivial. Thus i is N-determined and [28, 4.2] centric. Consequently, Ø(DI2=SU (3)) = 8 and WDI2(SU (3)) ~=Z(W ), Out(SU (3)) acts transitively on Mono(SU (3), DI2), and all monomorphisms of SU* *(3) into DI2are N-determined. 10.15. Example.Similarly, the monomorphism i: SU(3) ! G2arising in the construc* *tion (7.12) of BG2 as a homotopy colimit is N-determined and centric. Also, Out(SU (3)) act* *s transitively on Mono(SU (3), G2) with stabilizer subgroup WG2(SU (3)) = { E}, and Ø(G2=SU (3)) * *= 2. 10.16. Example.The inclusions of the maximal torus and of SU(3) into DI2constit* *ute a homo- topy coherent set of maps out of the centralizer diagram (7.10) for BG2 into BD* *I2. Observing that both maps are centric one sees first that the Wojtkowiak obstruction groups van* *ish according to (13.7) and next that the resulting map BG2 ! BDI2 is a centric monomorphism rea* *lizing the inclu- sion ~ToW (G2) ! ~ToW (G2) of maximal torus normalizers. As also {W > W (G2)} =* * {W (G2)}, we conclude that Ø(DI2=G2) = 4 and WDI2(G2) ~={1} and that Out(G2) acts transitively on Mono(G2, DI2). 10.17. Example.BPU (3) is the homotopy colimit of a diagram of the form _______________________________________________________________* *______________________________________________________________op______@ SL(V )op_BV<<________________________________________________________* *_________________Sop//_BN3[op[______________________BTW(PU(3))bb______@ ___________________________________________* *__________________3_\SL(V_)_______________________S3 \W(PU(3)) __________________________________________* *___________ Z=2 where S3 is a Sylow 3-subgroup of SL(V ) and N3 is the 3-normalizer of the maxi* *mal torus. There is a canonical map BN3 ! BDI2 because N3 is also the 3-normalizer of the maxima* *l torus of DI2. This map BN3 ! BDI2 is centric and it respects the maps of the above diagram up* * to homotopy. N-DETERMINISM 47 The obstructions to extending BN3 ! BDI2 to a map BPU (3) ! BDI2 lie in the hig* *her limits of the A(PU (3))-module ________________________________________________________________* *______________________________________________________ SL(V )ß*(BT9(DI2)))9_________________________________________________* *___________________ß*(BZ3)ZZ______________________oo_//_ß*(BTW)(PU(3))@ SL(V )=S3 ______________________________________* *______________W(PU(3))=S3_____________________________________ ______________________________________* *_________________________ Z=2 which vanish completely (13.7).(We are here implicitly using computations of ma* *pping spaces like map(BV, BDI2)Bi = BT (DI2).) Thus there exists a unique homotopy class Bi: BPU * *(3) ! BDI2 extending the inclusion of the 3-normalizer. Also, the restriction of i to the * *3-normalizer of the maximal torus is a monomorphism, so i itself is a monomorphism (9.2), and i is * *centric because the Bousfield-Kan spectral sequence [14, XI.7.1] for map(BPU (3), BDI2)Bi shows tha* *t this mapping space is weakly contractible. As also {T~oW > ~ToW (PU (3))} is a one-point se* *t and PU (3) is totally N-determined (5.1), (10.5, 10.7, 10.9) show that Ø (DI2=PU (3))= 8 and WDI2(PU (3)) = Z(W ) and that the group Out(PU (3)) acts transitively on the set Mono(PU (3), DI2) o* *f conjugacy classes of monomorphisms. In view of [10], which says that any connected, closed subgroup of maximal ra* *nk of a compact connected Lie group is the normalizer of its center, this example is somewhat s* *urprising. There is no monomorphism of PU(3) into G2 for A(PU (3))(V ) = SL(V ) (5.10) i* *s too big to be a subgroup of A(G2)(V ) = 3x Z=2 (7.10). Indeed, no nontrivial compact, connec* *ted Lie group admits a proper, centerless subgroup of maximal rank [10]. The next example describes the normalizers of the elementary abelian subgroup* *s of DI2. Strictly speaking, these normalizers are not 3-compact groups, but rather extended 3-com* *pact groups, in that their component groups are not 3-groups. We start with a general observation. 10.18. Proposition.Let :V ! X be a monomorphism of an elementary abelian p-gr* *oup V into a p-compact group X. 1.There is a short exact sequence of groups 1 ! ß0(CX ( )=V ) ! WX ( ) ! A(X)( ) ! 1 where CX ( )=V is the standard quotient [30, 8.3]. 2.There is a short exact sequence of loop spaces CX ( ) ! NX ( ) ! A(X)( ) where NX ( ) is the normalizer of [32, 4.4]. Proof.Assuming B :BV ! BX to be a fibration, consider the induced fibration a B_ WX ( ) ! map (BV, BV )Bf --! map(BV, BX)B f2A(X)( ) where the fibre is the Weyl space [32, 4.1] of and the components, each one h* *omotopy equivalent to BV , of the total space are indexed by the automorphism group of in the Qu* *illen category. The homotopy exact sequences of this fibration and of its sub-fibration CX ( )=V ! BV ! BCX ( ) give the exact sequence of groups and show that B(CX ( )=V ) is the regular cov* *ering space of BWX ( ) corresponding to the normal subgroup ß0(CX ( )=V ) . WX ( ). Thus there* * is a pull-back diagram BCX ( )_______//BNX ( ) | | | | fflffl| fflffl| B(CX ( )=V )___//_BWX ( ), where the horizontal maps are regular covering spaces. * * |___| 48 J.M. MØLLER 10.19. Example.For any monomorphism ~: Z=3 ! DI2there is (10.18) a short exact * *sequence of loop spaces SU(3) ! NDI2(~) ! Z(W ) where Z(W ) ~=Z=2 acts on SU(3) as {_ 1}. Thus NDI2(~) = SU(3)o Z(W ) where B(SU (3)o Z(W )) denotes the total space of the unique [66, 3.3, 3,7] BSU* * (3)-fibration over BZ(W ) realizing the given monodromy action. (It is not essential in [66, x3] t* *hat the component group ß0(X) be a p-group.) Since the homotopy fixed point space BZ(SU (3))hZ(W)* *is contractible, the inclusion ~ToW (SU (3))æ SU (3) extends uniquely to a short exact sequence * *morphism T~oW (SU (3))___//_NT~oW(~)__//_Z(W ) fflffl| | || | | || fflffl| fflffl| || SU (3)________//_NDI2(~)___//_Z(W ) __ where NT~oW(~) = ~ToW(~) = ~To(W (SU (3)) x Z(W )). For any monomorphism :(Z=3)2 ! DI2there is a short exact sequence of loop s* *paces T ! NDI2( ) ! W so NDI2( ) is an extended p-compact torus with ~ToW as discrete approximation [* *31, 3.12]. 10.20. Example.The normalizers of the 3-compact subgroups of DI2are (10.7.2) NDI2(G2) = G2 and NDI2(X) = Xo Z(W ) for X = U(2), SU(2)xSU (2), Spin(5), SU(3), PU(3) where Z(W ) acts on X as {_ 1* *}. In each case there is a unique short exact sequence morphism connecting the normalizer in ~T* *oW of NX (T ) and the normalizer in DI2of X. For X = PU(3), for instance, the picture is T~oW (PU (3))___//_NT~oW(T~oW (PU (3)))_//Z(W ) fflffl | | || | | || fflffl| fflffl| || PU (3)____________//NDI2(PU (3))_____//_Z(W ) where NT~oW(T~oW (PU (3))) = T~o(W (PU (3)) x Z(W )). It seems likely that thi* *s is another instance of N-determinism. 11.Free Zp-modules and p-discrete tori Nearly all material of this section is present, in one form or another, in [7* *5]. A Zp-module which is isomorphic to Zrpfor some finite r will be called a Zp-l* *attice and a Zp-module which is isomorphic to (Z=p1 )r = (Qp=Zp)r for some finite r will be * *called a Zp-torus. Let ~Tand L denote the endo-functors of the category Ab of abelian groups gi* *ven by ~T= Z=p1 - and L = Hom (Z=p1 , -). Then Hom Ab(T~(A), B) = Hom Ab(A, L(B)) so (T~* *, L) is a pair of adjoint functors. The left adjoint functor ~Tis right exact, ~Tvanishes* * on finite Zp-modules, turns Zp-lattices into Zp-tori, and its left derived functor ~T1= Tor(Z=p1 , -)* * preserves finite Zp-modules and vanishes on Zp-lattices. The right adjoint functor L is left exa* *ct, L vanishes on finite Zp-modules, turns Zp-tori into Zp-lattices, and its right derived functo* *r L1 = Ext(Z=p1 , -) preserves finite Zp-modules and vanishes on Zp-tori. In symbols: ~T0 ! S ! L ! H ! 0 = 0 ! ~T1(L) ! ~T1(H) ! ~T(S) ! ~T(L) ! ~T(H) ! 0 L 0 ! H ! ~T! ~P! 0 = 0 ! L(H) ! L(T~) ! L(P~) ! L1(H) ! L1(T~) ! 0 where S is a Zp-lattice, ~Pis a Zp-torus, and L, H, and ~Tare Zp-modules. In fa* *ct the pair (T~, L) provides adjoint equivalences [14, p. 181] between the full subcategories of (t* *he underlying abelian groups of) Zp-lattices and (the underlying abelian groups of) Zp-tori. A ZpW -module whose underlying Zp-module is a Zp-lattice will be called a ZpW* * -lattice and ZpW -module whose underlying Zp-module is a Zp-torus will be called a ZpW -toru* *s. N-DETERMINISM 49 11.1. Definition.[75, 1.1.4, 1.1.5] For a ZpW -lattice L and a Zp-torus ~T, put SL = kerL ! H0(W ; L) P L = L(P ~T(L)) 0 P ~T= cokerH (W ; ~T) ! ~T ST~= ~T(SL(T~)) In plain language, SL is simply the ZpW -submodule of L generated by the unio* *n of the subsets (1 - w)L, w 2 W , and ~T(P L) is the quotient of ~T(L) by the invariants ~T(L)W* *for the W -action. We have short exact sequences (11.2) 0 ! SL ! L ! H0(W ; L) ! 0, 0 ! H0(W ; ~T(L)) ! ~T(L) ! ~T(P L) ! 0 defining SL and P L. (SL could perhaps be called the root lattice and P L the w* *eight lattice of L.) It simplifies matters a great deal to assume that W is generated by elements * *of order prime to p (as are Zp-reflection subgroups for odd primes p). 11.3. Lemma. Suppose that W is generated by elements of order prime to p. Then * *H1(W ; H) = 0 = H1(W ; H) for any Zp-module H with trivial W -action. Proof.Observe that the abelianization H1(W ; Z) is a finite abelian group gener* *ated by elements_ of order prime to p and apply universal coefficients. * * |__| 11.4. Lemma. Let L be a ZpW -lattice. 1.The Zp-module homomorphism H0(W ; L) ! L is split injective and the Zp-modu* *le homo- morphisms ~TH0(W ; L) ! H0(W ; ~T(L)), H0(W ; L) ! H0(W ; L) are injective. 2.coker H0(W ; L) ! H0(W ; L) is finite. 3.H0(W ; SL) = 0 = H0(W ; ~T(SL)) and H0(W ; SL) is finite. If W is generated* * by elements of order prime to p, then H0(W ; SL) = 0. 4.The Zp-module homomorphism ~T(L) ! H0(W ; ~T(L)) is split surjective and th* *e Zp-module homomorphisms H0(W ; L) ! LH0(W ; ~T(L)), H0(W ; ~T(L)) ! H0(W ; ~T(L)) are* * surjective. 5.ker H0(W ; ~T(L)) ! H0(W ; ~T(L)) is finite. 6.H0(W ; ~T(P L)) = 0 = H0(W ; P L) and H0(W ; ~T(P L)) is finite. If W is ge* *nerated by ele- ments of order prime to p, then H0(W ; ~T(P L)) = 0. 7.H0(W ; L) ~=LH0(W ; ~T(L)) and H0(W ; ~T(L)) ~=~TH0(W ; L). 8.H0(W ; L) = 0 , H0(W ; L) is finite, H0(W ; ~T(L)) = 0 , H0(W ; ~T(L)) is f* *inite. 9.H0(W ; L) = 0 , H0(W ; ~T(L)) = 0 = H1(W ; ~T(L)) , H0(W ; L Zp Z=p) = 0. 10.H0(W ; ~T(L)) = 0 , H0(W ; L) = 0 = H1(W ; L) , H0(W ; Hom(Z=p, ~T(L))) = 0. Proof.The inclusion H0(W ; L) æ L has a right inverse because its cokernel is a* * torsion-free, hence free, Zp-module. Then also ~TH0(W ; L) ! H0(W ; ~T(L)) ~T(L) is injecti* *ve by functoriality. Since the first homology group H1(W ; L=H0(W ; L)) is finite, the long exact co* *efficient sequence in homology shows that H0(W ; L) ! H0(W ; L) is injective. The QpW -module L * *Qp contains H0(W ; L Qp) as a direct summand, so H0(W ; L Qp) H0(W ; L Qp), and it * *contains H0(W ; L Qp) as a direct summand, so H0(W ; L Qp) H0(W ; L Qp). Thus t* *he vector spaces H0(W ; L) Qp ~=H0(W ; L Qp) and H0(W ; L) Qp ~=H0(W ; L Qp) have* * the same dimension. This shows that the cokernel of the monomorphism H0(W ; L) æ H0(W ; * *L) is finite. Apply the left exact functor H0(W ; -) to (11.2) and, using item 1, conclude th* *at H0(W ; SL) = 0. Apply the right exact functor to (11.2) and conclude that H0(W ; SL) is finite * *(and, using (11.3), trivial if W is generated by elements of order prime to p). This proves this f* *irst three items and the next three ones are proved in a dual fashion. For item 7, take the shor* *t exact sequence 0 ! Zp ! Qp ! Z=p1 ! 0 of Zp-modules. Apply H0(W ; -) O (L -) and H0(W ; L) O* * - to it and compare the results H0(W ; L)___//_H0(WO;OL Qp)_//_H0(WO;O~T(L))//_0 || ~ | | || = | | || | | H0(W ; L)___//_H0(W ; L) _Qp__//_~TH0(W ;_L)_//_0 50 J.M. MØLLER to see that ~TH0(W ; L) ~=H0(W ; ~T(L)). Dually, compare the values of H0(W ; -* *) O Hom(-, ~T(L)) and Hom (-, H0(W ; ~T(L))) applied to the same short exact sequence and conclud* *e that H0(W ; L) and LH0(W ; ~T(L)) are isomorphic. Combine these isomorphisms with items 2 and * *5 to obtain item 8. To get the formulas of items 10 and 9, simply apply the right exact fun* *ctor H0(W ; -) to the short exact sequence 0 ! L .p-!L ! L Z=p ! 0 and the left exact functor H* *0(W ; -) to the short exact sequence 0 ! Hom (Z=p, ~T) ! ~T.p-!~T! 0 where L Z=p = Hom (Z=p, * *~T). |___| From the commutative diagrams with exact rows 0 ____//_SL_________//_L________//_H0(WO;_L)_//_0OhhP || | PP OO| || | P P | || OO| PP OO| || 0 ____//_SL___//_SL x H0(W ;_L)_//_H0(W ;_L)_//_0 0_____//H0(W ; ~T(L))________//~T(L)_________//_~T(P_L)_//_0 k k | k kk | || fflfflfflffl||uukkk fflfflfflffl|| |||| 0_____//H0(W ; ~T(L))_//H0(W ; ~T(L)) x ~T(P_L)//_~T(P_L)//_0 the Snake Lemma produces exact sequences of ZpW -modules (11.5)0 ! SL x H0(W ; L) ! L ! ß(L) ! 0 0 ! ß(L) ! ~T(SL) x ~TH0(W ; L) ! ~T(L) !* * 0 0 ! ~ß(L) ! ~T(L) ! H0(W ; ~T(L)) x ~T(P L) ! 0 0 ! L ! LH0(W ; ~T(L)) x P L ! ~ß(L) !* * 0 where ß(L) and ~ß(L) are the finite groups defined by the short exact sequences (11.6) 0 ! H0(W ; L) ! H0(W ; L) ! ß(L) ! 0 (11.7) 0 ! ~ß(L) ! H0(W ; ~T(L)) ! H0(W ; ~T(L)) ! 0 of abelian groups. We have thus constructed functors ~s:ZpW - mod ! (ZpW - mod )o o!o s :ZpW - mod ! (ZpW - mod )o!o o '(L) 0 ~'(L) ~s(L)= ~T(SL) oe ß(L) ---! ~TH (W ; L)s(L)= P L i ~ß(L) --- LH0(W ; ~T(L)) from the category of ZpW -modules into the category of push-out (pull-back) dia* *grams of ZpW - modules. Since we can recover L from the value of these functors in that colim* *~s(L) = ~T(L) and L = lims(L), the classification of ZpW -modules has been reduced to the cla* *ssification of ZpW -modules L with ß(L) = 0 or ~ß(L) = 0. 11.8. Lemma. Let L be a ZpW -lattice and assume that W is generated by elements* * of order prime to p. 1.ß(P L) = H0(W ; P L), ~ß(SL) = H0(W ; ~T(SL), ~ß(P L) = 0 = ß(SL), and ß(P * *L) ~=~ß(SL). 2.ß(L) ! H0(W ; ~T(SL)) = ~ß(SL) is injective and ß(P L) = H0(W ; P L) ! ~ß(L* *) is surjective. 3.0 ! H0(W ; L) ! LH0(W ; ~T(L)) x ß(P L) ! ~ß(L) ! 0 is an exact sequence. 4.0 ! ß(L) ! ~ß(SL) x ~TH0(W ; L) ! H0(W ; ~T(L)) ! 0 is an exact sequence. 5.SSL = SL = SP L and P P L = P L = P SL. 6.ß(L) = 0 , SL x H0(W ; L) = L and ~ß(L) = 0 , L = LH0(W ; ~T(L)) x P L. 7.If H0(W ; L) = 0, then ß(L) = H0(W ; L), ~ß(L) = H0(W ; ~T(L)), and there i* *s a short exact sequence 0 ! ß(L) ! ~ß(SL) ! ~ß(L) ! 0. 8.'(L) 2 Hom (ß(L), ~TH0(W ; L)) ~=Ext(ß(L), H0(W ; L)) classifies the above * *abelian extension ( 11.6) and ~'(L) 2 Hom (LH0(W ; ~T(L)), ~ß(L)) ~=Ext(H0(W ; ~T(L)), ~ß(L))* * classifies ( 11.7). 9.H0(W ; ~T(L)) ~=coker'(L), H1(W ; ~T(L)) ~=ker'(L), H0(W ; L) ~=ker~'(L), H* *1(W ; L) ~= coker~'(L). 10.0 ! H0(W ; L) ! L ! P L ! H1(W ; L) ! 0 and 0 ! H1(W ; ~T(L)) ! ~T(SL) ! ~T* *(L) ! H0(W ; ~T(L)) ! 0 are exact sequences. N-DETERMINISM 51 Proof.Item 1 is true because H0(W ; P L) = 0 = H0(W ; ~T(P L)) by (11.4.6). For* * item 2, note that there is a commutative diagram (ff,fi) 0____//_ß(L)PP_//~T(SL) x ~TH0(W_;_L)//_~T(L)_//_077n PPPP | nnnnn PPPP | nnn+n PP''Pfflffl|nnn T~(L) x ~T(L) for some homomorphisms ff and fi. For all elements x of ß(L), ff(x) + fi(x) = * *0 in ~T(L). If ff(x) = 0 in ~T(SL), then also ff(x) = 0 in ~T(L) so fi(x) = 0 in ~T(L). But th* *is means that the monomorphism (ff, fi) takes x to 0, so x = 0. Thus ff is a monomorphism. Appl* *y the functor H0(W ; -) to a short exact sequence from (11.5) to obtain the commutative diagr* *am 0______//L_________//LH0(W ; ~T(L)) x_P_L_//~ß(L)_//0 | | || | | || fflffl| fflffl| || 0___//H0(W ; L)//_LH0(W ; ~T(L)) x H0(W ;_P/L)/_~ß(L)//_0 with a bottom row that is exact according to (11.3). Conclude that SP L = SL. U* *sing the exact sequence of item 3 and (11.4.7) we see that there is short exact sequence 0 ! kerH0(W ; L) ! LT~H0(W ; L) ! ß(P L) ! ~ß(L) ! 0 for any ZpW -module L. Applied to SL, this gives ß(P L) ~=~ß(SL). For items 9 and 10 apply the left exact functor H0(W ; -) to one of the short* * exact sequences from (11.5) and get the commutative diagram 0________//L_______//LH0(WO;O~T(L))Ox_P_L_______//~ß(L)__________//0OOO | | | | | | OO| OO| ~'(L) |OO 0_____//H0(W ;_L)____//LH0(W ; ~T(L))____//ker(~ß(L) ! H1(W ;_L))//_0 using H0(W ; P L) = 0 = H1(W ; P L) (11.4.10). Now apply the Snake Lemma. * * |___| The group W acts on the dual Zp-lattice L_ = Hom (L, Zp) according to the rul* *e (w . ')(x) = '(w-1x), w 2 W , ' 2 L_, x 2 L. The W -equivariant duality pairing (11.9) ~T(L) x L_ ! Z=p1 obtained from the identification L_ = Hom (L, L(Z=p1 ) = Hom (T~(L), Z=p1 ) ind* *uces pairings (11.10) H*(W ; ~T(L)) x H*(W ; L_) ! Z=p1 , H*(W ; ~T(L)) x H*(W ; L_) ! Z=* *p1 relating homology and cohomology groups. (A duality pairing of Zp-modules is a* * bilinear map A x B ! C of Zp-modules such that the adjoint homomorphisms A ! Hom Zp(B, C) an* *d B ! Hom Zp(A, C) are isomorphisms.) 11.11. Lemma. Let L be a ZpW -lattice and L_ its dual. Assume that W is generat* *ed by elements of order prime to p. 1.The bilinear maps ( 11.10) are duality pairings. 2.S(L_) = (P L)_. Proof.It is immediate that H*(W ; L_) = H*(W ; Hom(T~(L), Z=p1 )) ~=Hom(H*(W ; ~T(L)), Z=p1 ) for Hom (-, Z=p1 ) is an exact functor. But then also H*(W ; ~T(L)) ~=Hom(H*(W ; L_), Z=p1 ) 52 J.M. MØLLER because A ~=Hom (Hom (A, Z=p1 ), Z=p1 ) for any Zp-torus, Zp-lattice, or finite* * Zp-module A. Apply the exact functor Hom (-, Z=p1 ) to the short exact sequence 0 ! S(L_) ! * *L_ ! H0(W ; L_) ! 0 to get the short exact sequence 0 ! H0(W ; ~T(L)) ! ~T(L) ! ~T(S(L_)_) ! 0 and conclude that P L = S(L_)_. * * |___| Suppose that the group W = W1 x . .W.nis the direct product of finitely many * *of its normal subgroups W1, . .,.Wn. For j = 1, . .,.n, let Y Wj?= Wi i6=j T denote the product of all these subgroups but Wj. Then W = Wj x Wj?and`Wj = i6* *=jWi?. Observe that H0(Wi?; L) is a ZpWi-module and also that the direct sum H0(Wi?;* * L) is a ZpW - module with a natural ZpW -module homomorphism to L given by addition. 11.12. Lemma. [32, 1.5] If H0(W ; ~T(L)) = 0 =`H0(W ; L) for a ZpW -lattice L, * *then there is a ZpW -lattice U and a short exact sequence 0 ! H0(Wi?; L) ! L ! U ! 0 of ZpW -* *lattices. Each summand H0(Wi?; L) is a ZpWi-lattice and oH0(Wi; ~TH0(Wi?; L)) = 0 provided H0(W ; ~T(L)) = 0, oH0(Wi; H0(Wi?; L)) = 0 provided H0(W ; L) = 0 and each factor group Wi is g* *enerated by elements of order prime to p. Proof.This amounts to showing that the addition maps a a H0(Wi?; L) ! L, ~TH0(Wi?; L) ! ~T(L) are injective. P Suppose that (xi),Pwith xi 2 H0(Wi?; L), satisfies xi = 0. Then, for an arb* *itrarily chosen indexTj, xj = - i6=jxi. The left hand side is fixed by Wj?and the right hand s* *ide is fixed by i6=jWi?= Wj. Thus xj is fixed by Wj?x Wj = W , so that xj 2 H0(W ; L). But H0(* *W ; L) = 0 by (11.4.8). For the other addition map, recall from (11.4.1) that ~TH0(Wi?; L)* * is contained in H0(Wi?; ~T(L)) and proceed as above. The computation H0(Wi; ~T(H0(Wi?; L))) H0(Wi; H0(Wi?; ~T(L))) = H0(Wix Wi?; ~T(L)) = H0(W ; ~T(L)) shows that H0(Wi; ~T(H0(Wi?; L))) = 0 if H0(W ; ~T(L)) = 0. If Wi is generated * *by elements of order prime to p, then H1(Wi; ß(H0(Wi?; L)) = 0 so that H0(Wi; H0(Wi?; L)) H0(Wi; H0(Wi?; L)) = H0(Wix Wi?; L) = H0(W ; L) proving the final assertion of the lemma. * * |___| We now specialize to reflection subgroups. If W Aut(L) is a group of automo* *rphisms of the Zp-lattice L, any w 2 W restricts to automorphism Sw of SL and projects to to a* *n automorphism P w of P L. If Sw is the identity on SL, then w is the identity on ~T(L) = coli* *m~s(L) so w is the identity. If w is a reflection on L, then Sw is a reflection on SL because SL=S* *L~=L=L. This means that if W is a reflection subgroup of Aut(L) then also SW (P W ) is a ref* *lection subgroup of Aut(SL) (Aut(P L). Thus the S-construction and the P -construction (11.1) ar* *e endo-functors of the category Zp- Reflof Zp-reflection subgroups (4.1). We wish to classify the elements of the category Zp - Reflup to similarity. * *The preceding general discussion implies the following first reduction of this classification* * problem. 11.13. Lemma. Let (W1, L1) and (W2, L2) be two objects of Zp- Refl. Then the fo* *llowing three statements are equivalent: 1.(W1, L1) and (W2, L2) are similar. N-DETERMINISM 53 2.The diagram ~T(SL1)oo_oß(L1)o____//~TH0(W1; L1) T~(`)~=|| ~=|| ~=~T(_*)|| fflffl| fflffl| fflffl| ~T(SL2)oo_oß(L2)o____//~TH0(W2; L2) commutes for some similarity (ff, `): (SW1, SL1) ! (SW2,,SL2)some isomorphi* *sm between ß(L1) and ß(L2), and some isomorphism _ :H0(W1; L1) ! H0(W2; L2). 3.The diagram P L1___////_~ß(L1)LH0(W1;o~T(L1))o_ `|~=| ~=|| ~=|(T~_)*| fflffl| fflffl| fflffl| P L2___////_~ß(L2)LH0(W2;o~T(L2))o_ commutes for some similarity (ff, `): (P W1, P L1) ! (P W2,,PsL2)ome isomor* *phism be- tween ~ß(L1) and ~ß(L2), and some isomorphism _ :H0(W1; ~T(L1)) ! H0(W2; ~T* *(L2)). The classification of similarity classes of objects (W, L) of Zp - Reflhas no* *w been reduced to the case where ß(L) = 0 or ~ß(L) = 0. Fortunately, this is very easy. 11.14. Theorem. [75] Let (W1, L1) and (W2, L2) be two objects of Zp - Reflwhere* * p is odd. Assume that ß(L1) = 0 = ß(L2) or ~ß(L1) = 0 = ~ß(L2) , i = 1, 2. Then (W1, L1) * *and (W2, L2) are similar if they are Qp-similar. Proof.Assume that (W1, L1) and (W2, L2) are Qp-similar objects of Zp - Reflwith* * ~ß(L1) = 0 = ~ß(L2). Since Li = P Lix LH0(Wi, ~T(Li)), i = 1, 2 (11.8.6), it suffices (1* *1.13) to show that (P W1, P L1) and (P W2, P L2) are similar. As the splitting constructed in (11.* *15) below depends on rational information only, it suffices to prove the theorem under the additi* *onal hypothesis that (Wi, Li) be simple. This is done in (11.18) below by going through the Clark-Ew* *ing_classification table [20]. * *|__| 11.15. Lemma. [32, 75] Let (W, L) be an object of Zp-Refl where p is odd. If H0* *(W ; ~T(L)) = 0 (or H0(W ; L) = 0), then Y (W, L) = (Wi, Li) splits as a product of simple objects of Zp- Reflwith H0(Wi; ~T(Li)) = 0 (or H0* *(Wi, Li) = 0) for all i. Proof.We shall only consider the case where L = P is a ZpW -lattice with H0(W ;* * ~T(P )) = 0. As W is a finite reflection`subgroup of Aut(P ) and H0(W ; P ) = 0 (11.4.8), the Q* *pW -module P ZpQp splits as a direct sum Mi~=P Zp Qp of finitely many irreducible QpW -modules* * M1, . .,.Mn. Each of these irreducible summands occurs with multiplicity one and carries a n* *on-trivial W -action [32, p. 280]. Define Wi to be the subgroup of W thatQpointwise fixes j6=iMj so* * that the action of Wi is concentrated on the summand Mi. Then W = Wi is is the direct produc* *t of these normal subgroups [32, 6.3] and, according to (11.12), P is isomorphic to the di* *rect sum of the ZpW -lattices H0(Wi?; P ). Observe that each summand H0(Wi?; L) is a ZpWi-latti* *ce and oWiis a reflection subgroup of AutZp(H0(Wi?; L)), o(Wi, H0(Wi?; L)) is simple, oP H0(Wi?; L) = H0(Wi?; L). Indeed, the first item is implicit in the proof of [32, 6.3], the second item i* *s clear because the rationalization H0(Wi?; L) Zp Qp = H0(Wi?; L Zp Qp) = Mi by construction, and* * the third_ item is contained in (11.12). * * |__| 11.16. Lemma. Let (W, L) be a Zp-reflection group. 1.(W, L) and (W, L_) are Qp-similar and ~ß(L) ~=ß(L_). 2.(W, SL) and (W, S(L_)) are Zp-similar. 3.(W, P L) and (W, P (L_)) are Zp-similar. 54 J.M. MØLLER 4.(W, SL) and (W, (P L)_) are Zp-similar. Proof.For item 2, first note that S(L_) = SP (L_) = S((SL)_) by (11.11.2). But * *SL is (11.15) a product of simpleQZp-reflection groups (Wi,QLi) with H0(Wi; Li) = 0. So (SL)_* * is isomorphic to the product (Wi, L_i) and S((SL)_) = (Wi, S(L_i)). By inspection (of re* *flection group family 1 and W (E6) at p = 3), we see that (Wi, Li) and (Wi, S(L_i)) are Zp-sim* *ilar. Thus SL and S(L_) are Zp-similar. Moreover, the isomorphisms H0(W ; L_) ~=Hom (H0(W ; ~T(L* *)), Z=p1 ) ~= Hom (T~H0(W ; L), Z=p1 ) ~=Hom (H0(W ; L), Zp) from (11.11.1) show that the lat* *tices H0(W ; L_) and H0(W ; L) have the same rank. Therefore (W, L) and (W, L_) are Qp-similar (* *11.5). Finally,_ (W, SL) ~=(W, P (L_)_) ~=(W, (P L)_) by (11.11.2) again. * * |__| 11.17. Lemma. Let (W, L) be a Zp-reflection group. Then there are natural group* * isomorphisms H0(W ; L_ Z=p) ~=Ext(H0(W ; ~T(L)), Z=p) and H1(W ; L_ Z=p) ~=Hom(H0(W ; ~T* *(L)), Z=p). Proof.Using (11.11), we get H0(W ; L_ Z=p) = H0(W ; L_) Z=p = Hom (H0(W ; ~T(L)* *), Z=p1 ) Z=p = Ext(H0(W ; ~T(L)), Z=p). In the universal coefficient exact sequence 0 ! H1(W ; L_) Z=p ! H1(W ; L_ Z=p) ! Tor(H0(W ; L_), Z=p) ! 0 the term to right identifies to Hom (H0(W ; ~T(L)), Z=p) and the term to the le* *ft is trivial_because H1(W ; L_) = Hom (H1(W ; ~T(L)), Z=p1 ) and H1(W ; ~T(L)) = 0 [5, 3.3]. * * |__| Recall that G0(W, L) stands for the set of similarity classes of reflection s* *ubgroups that are Qp-similar to (W, L) (4.1). Write PpiSU(r + 1) for the quotient SU(r + 1)=Cpiof SU(r + 1) by the central * *subgroup Cpiof order pifor 0 i p(r + 1) where p(r + 1) is the highest power of p that di* *vides r + 1. 11.18. Lemma. Let (W, L) be a simple object of Zp- Refl. Then G0(W, L) = * exce* *pt that 1.G0(W (SU (r + 1))) = {W (PpiSU(r + 1)) | 0 i p(r + 1)} contains p(r +* * 1) + 1 elements. 2.G0(W (E6)) = {W (E6), W (PE 6)} contains two elements if p = 3. Proof.The reflection subgroup rp(W, L) = (W, L Z=p) is irreducible, and hence* * G0(W ) = * (4.5.(1)), unless (W, L) is in Clark-Ewing family 1 or p = 3 and r0W is r0W (E6* *) or r0W (G2) [4, 6.2]. All the Lie cases are covered by G. Maxwell [56, Table I]. (See also [22]* * or [84, 5.1] for the A-family.) __ |__| We learn from (11.18) that two simple objects, (W1, L1) and (W2, L2), of Zp- * *Reflare similar if they are Qp-similar and either ß(L1) ~=ß(L2) or ~ß(L1) ~=~ß(L2). Combined wi* *th the splitting result (11.12), this proves (11.14). Let (W, L) be an object of Zp - Refl. We shall next describe G0(W, L) as a pa* *rtially ordered set. For given diagrams (11.19)ff: ß(P L) = H0(W ; P L) i ~ß LH0(W ; ~T(L)) ~ff:~ß(SL) = H0(W ; ~T(SL)) oe ß ! ~TH0(W* * ; L) of Zp-modules, put 0 (11.20)Sff(P L) = limP L ! ~ß LH (W ; ~T(L)) ~T(P~ff(SL)) = colim~T(SL) ß ! ~TH0(W ; * *L) so that ~ß(Sff(P L)) = ~ßand ß(P~ff(SL)) = ß. There are defining short exact se* *quences 0 ! Sff(P L) ! LH0(W ; ~T(L)) x P L ! ~ß! 0 0 ! SL x H0(W ; L) ! P~ff(SL) ! ß ! 0 of ZpW -modules. We have previously (11.5) seen that Sß(PL)i~ß(L) H0(W;L)(P L) = L = P~ß(SL)oeß(L)!T~H0(W;L)(SL). N-DETERMINISM 55 By naturality, W is a reflection subgroup of Aut(Sff(P L)) and of Aut(P~ff(SL))* *. Also by naturality, there are morphisms Sff(P L) ! Sß(PL)i0 H0(W;L)(P L) = P L x H0(W ; L) H0(W ; L) x SL = P~ß(SL)oe0!T~H0(W;L)(SL) ! P~f* *f(SL) showing that (W, Sff(P L)) and (W, P~ff(SL)) are Qp-similar to (W, L). Convers* *ely, any element of G0(W, L) will have this form because if (W1, L1) and (W2, L2) are Qp-similar* * then (SW, SL1) and (SW, SL2) ((P W, P L1) and (P W, P L2)) are Zp-similar by (11.14) and clear* *ly H0(W1; L1) and H0(W2, L2) are isomorphic Zp-lattices. Declare two diagrams of the form considered in (11.19) to be equivalent if th* *ey can be connected by an automorphism in AutZp-Refl(W, P L) (or AutZp-Refl(W, SL)) (4.1) and an au* *tomorphism in Aut(H0(W ; L)) as in (11.13). 11.21. Lemma. For any object (W, L) of Zp - Refl, p odd, there is a bijection b* *etween the following three sets: 1.G0(W, L). 2.Equivalence classes of diagrams ß(P L) i ~ß LH0(W ; ~T(L)) of Zp-modules. 3.Equivalence classes of diagrams ~ß(SL) oe ß ! ~TH0(W ; L) of Zp-modules. Since ~ß(SL) ~=ß(P L) is a finite group (11.4.2), G0(W, L) is a finite set. O* *ur next aim is to introduce an ordering relation on G0(W, L). 11.22. Lemma. For a Zp-Refl morphism (ff, `): (W1, L1) ! (W2,tL2)he following t* *hree state- ments are equivalent: 1.r0(ff, `): r0(W1, L1) ! r0(W2,iL2)s a similarity in Qp - Refland W2 acts tr* *ivially on coker`. 2.S(ff, `): S(W1, L1) ! S(W2,iL2)s a similarity in Zp - Refland the induced m* *orphism of Zp-tori ~T((ff, `)*): ~TH0(W1; L1) ! ~TH0(W2;aL2)n epimorphism with finite * *cokernel. 3.P (ff, `): P (W1, L1) ! P (W2,iL2)s a similarity in Zp - Refland the induce* *d morphism of Zp-lattices (ff, `)*:H0(W1; L1) ! H0(W2;aL2)monomorphism with finite kernel. Proof.Assume that L1 ! L2 is injective with finite cokernel H. Then there is a* * short exact sequence 0 ! kerH0(W1; L1) ! H0(W2; L2) ! cokerSL1 ! SL2 ! kerH ! H0(W2; H) ! 0 provided by the Snake Lemma. If the middle term is trivial, then H = H0(W2; H).* * If W2 acts trivially on H, then the kernel to the left is trivial because H1(W2; H) = 0 by* * (11.3), and the kernel to the right is trivial because H = H0(W2; H). The proof for P L1 ! P L2* *_is completely dual. |_* *_| 11.23. Definition.An isogeny is a Zp-Reflmorphism (ff, `): (W1, L1) ! (W2,tL2)h* *at satisfies one of the three equivalent conditions of ( 11.22). Write (W1, L1) (W2, L2) if there exists an isogeny (W1, L1) ! (W2, L2). 11.24. Lemma. If (W1, L1) (W2, L2) (W1, L1) then (W1, L1) and (W2, L2) are * *similar ob- jects of Zp- Refl. Proof.An isogeny (W1, L1) ! (W2, L2) induces a commutative diagram ~T(SL1)oo_oß(L1)o___//_~TH0(W1; L1) Ø ~=|| |~=| Ø |fflffl fflffl| fflfflØ ~T(SL2)oo_oß(L2)o___//_~TH0(W2; L2) which can be completed [61] by a vertical isomorphism to the right. * * |___| 56 J.M. MØLLER Thus the relation induces a partial ordering relation on the set of similar* *ity classes of objects of Zp- Refl; in particular on the set G0(W, L). For any object (W, L), (W, SL x H0(W ; L)) (W, L) (W, LH0(W ; ~T(L)) x P L) by (11.5) and actually G0(W, L) = {(W 0, L0) | (W 0, L0) (W, LH0(W ; ~T(L)) x P L)} = {(W 0, L0) | (W, SL x H0(W ; L)) (W * *0, L0)} is the set of similarity classes of objects above LH0(W ; ~T(L)) x P L or below* * SL x H0(W ; L). I close this section with a few remarks about the set Gp(W, L) (4.1). 11.25. Lemma. Let (W, L) be an object of Zp- Refl. 1.If (W, L) is simple, then Gp(W, L) G0(W, L). 2.G0(W, L) \ Gp(P W, P L) = * = G0(W, L) \ Gp(SW, SL) if H0(W ; L) = 0. 3.If (W, L) is simple, then Gp(W, L) = * unless (W, L) is similar to (W (X), * *L(X)) for X = PpiSU(r + 1)), 0 < i < p(r + 1). Proof.Gp(W ) G0(W ) when W is simple because any two abstractly isomorphic gr* *oups from the Clark-Ewing list happen to have the same rank r and to be conjugate as subgroup* *s of GL(r, Qp) [4, 2.6]. When H0(W ; L) = 0, P L is the unique object of G0(W, L) with ~ß= 0 * *(11.21);_this_ condition can (11.8.6) be read off from L Z=p. * * |__| 11.26. Example.Put (W, L) = (W, L)(PU (r + 1)) so that ß(L) is cyclic of order * *p where this is the highest power of p that divides r + 1. The + 1 elements of G0(W, L) ar* *e represented by the centerings (W, Li), 0 i , where Li L is the inverse image of the orde* *r pi subgroup of ß(L) (11.21) [84, 5.1]. Thus ß(Li) is cyclic of order piand L = L . Assume now * *that 0 < i < so that both ß(Li) and ~ß(Li) are non-trivial cyclic p-groups. As pointed out to m* *e by D. Notbohm, tensoring the commutative diagram of ZpW -modules with exact rows and columns 0 0 | | | | fflffl| fflffl| 0 ____//_L0____//Li____//_ß(Li)__//_0 || | | || | | || fflffl| fflffl| 0 ____//_L0____//L_____//_ß(L_)__//_0 | | | | fflffl| fflffl| ~ß(Li)_____~ß(Li)__//_0 | | | | fflffl| fflffl| 0 0 with Z=p results in the commutative diagram Tor(~ß(Li), Z=p)__Tor(~ß(Li), Z=p) | |~ | |= fflffl| fflffl| 0___//Tor(ß(Li), Z=p)//_L0 Z=p___//Li Z=p______//ß(Li) Z=p__//_0 with a split epimorphism to the right. We conclude that 0 Li Z=p ~=cokerH (W ; L0 Z=p) ! L0 Z=p H0(W ; Li Z=p), 0 < i < , as FpW -modules. (These modules are irreducible [38] and it is no coincidence [* *84, 3.3] that Li Z=p have the same irreducible constituents, namely 0 cokerH (W ; L0 Z=p) ! L0 Z=p ~=kerL Z=p ! H0(W ; L Z=p) andZ=p, for all i.) This shows that Gp(W (PpiSU(r + 1))) consists of - 2 elements for* * 0 < i < . N-DETERMINISM 57 11.27. Example.(Cf. (9.9)) For (W, L) = (W (X), L(X)), X = SU(p) x SU(p), the s* *et G0(W, L) consists of four elements corresponding to the four subgroup-orbits under the a* *ction of the auto- morphism group AutZp-Refl(W, P L) = Zxpx Zxp o Z=2 on H0(W ; P L) = Z=p x Z=p. 11.28. Example.G0(W (X), L(X)) for X = U(p ) is the poset {(i, j) 2 Z x Z | 0 * * j i } with lexicographic ordering. The point (i, j) corresponds to the diagram j Z=p Z=pi-.p-!Z=p1 where Z=pi is the subgroup of order pi of H0(W (X); ~T(SL(X))) = Z=p Z=p1 . * *U(p ) corre- sponds to ( , 0) in this formalism. If i1 i2 and j1 j2, then the commutativ* *e diagram j1 Z=p oo_oZ=pi1o.p//_Z=p1 || fflffl| | j2-j1 || | |.p || fflffl|.fflfflfflffl|pj2 Z=p oo_oZ=pi2o_//Z=p1 shows that (i1, j1) (i2, j2). 12. Shapiro's lemma The main purpose of this section is to introduce some notation to be used in * *Section 13. For any set S and any abelian group M we put M[S] = Z[S] Z M, M = Hom Z(Z[S], M) where Z[S] stands for the free abelian group with basis S. M[-] is a covariant* * and M<- >a contravariant functor from the category of sets to the category Ab of abelian * *groups. (M[-] (M<- >) is the left (right) adjoint of the forgetful functor from abelian group* *s to sets.) M can also be considered as the abelian group of all functions u: S ! M. In case S is* * a left G-set and M a left G-module for some group G, the rules g(s m) = gs gm, (gu)(s) = gu(g-1s), g 2 G, s 2 S, m 2 M, u: S !* * M, make M[S] and M into left G-modules. A special case occurs when S is the lef* *t G-set G=H of left cosets of a subgroup H of G. 12.1. Lemma. M[G=H] is isomorphic to the induced module IndGH(M) and Mis * *isomor- phic to the coinduced module CoindGH(M). Proof.Let T be a set of left coset representatives for G=H. The set T is a basis for the free right ZH-module ZG. The induced module Ind* *GH(M) = ZG ZH M is [95, 6.3.4] the sum over |T | copies t M of M with G-action g(t m) =* * s hm where gt = sh, s 2 T , h 2 H. The module M[G=H] = Z[G=H] Z M is the sum over |T | co* *pies t M of M with G-action g(t m) = s gm. The Z-linear isomorphism M[G=H] ! IndGH(M) t* *hat takes t m to t t-1m is G-linear as it takes g(t m) = y gm to y y-1gm = y ht-1m = g(t * *t-1m). The set T -1= {t-1 | t 2 T } is a basis for the free left ZH-module ZG. The * *coinduced module CoindGH(M) is [95, 6.3.4] the product over |T | of copies ßtM of M, wher* *e ßtm: ZG ! M is the H-map sending t-1 to m 2 M and z-1 to 0 for all z 6= t in T . The G-act* *ion is given by g(ßtm) = ßy(hm). The module M is the product over |T | of copies ætM o* *f M, where ætm: G=H ! M is the set map sending tH to m and zH to 0 for all z 6= t in T . * *The G-action is given by g(ætm) = æy(gm). The Z-linear isomorphism CoindGH(M) ! Mthat tak* *es ßtm to __ æt(tm) is G-linear as it takes g(ßtm) = ßy(hm) to æy(yhm) = æy(gtm) = gæt(tm). * * |__| Let ShG denote the homotopy colimit of S viewed as a functor from the categor* *y G to the category of sets. (ShG is the nerve of the small groupoid that has S for object* * set and {g 2 G | gs1 = s2} as the set of morphisms s1 ! s2.) The next lemma is just a reformulat* *ion of Shapiro's lemma. 12.2. Lemma. There are natural isomorphisms H*(G; M[S]) ~=H*(ShG; M), H*(G; M) ~=H*(ShG; M) 58 J.M. MØLLER Proof.Let X be a set of representatives for the G-orbits in S and G(x) the isot* *ropy subgroup at x 2 X. Then there are a homotopy equivalence a BG(x) ! ShG x2X and isomorphisms of G-modules a a Y Y M[S] ~= M[G=G(x)] ~= IndGG(x)(M), M ~= M ~= CoindGG(x)(M) ` induced by the isomorphism S ~= G=G(x) of G-sets. These isomorphisms combine, * *with the_help of Shapiro's lemma, to the isomorphisms of the lemma. * * |__| In other words, a Y (12.3) H*(G; M[S]) ~= H*(G(x); M), H*(G; M) ~= H*(G(x); M) where x 2 S runs through a set of representatives for the orbit set S=G. 13.Cellular cohomology of small categories The following is a general discussion of the derived functors of the inverse * *limit. Let I be a small category such that oI has only finitely many objects, oany endomorphism is an isomorphism oany isomorphism is an automorphism meaning that I is a special kind of very small ordered category [81] or EI-cate* *gory [54]. I could for instance be a skeletal subcategory of the Quillen category of a p-compact g* *roup. Write S(i, j) for the set of morphisms from the object i to the object j and * *I(i) for the group of morphisms i ! i. Under the above assumptions, the set Ob(I) of objects of I has* * the structure of a partially ordered set, a poset, where i j if there is a morphism from i to j.* * Let K(I) = Cx(Ob (I)) denote the ordered simplicial complex associated to Ob(I). The vertex set of K(* *I) is the poset Ob(I) and the p-simplices, p > 0, is the set of all strictly increasing sequenc* *es (i0. .i.p) of elements of Ob(I) (where i < j if i j and i 6= j). The ordered simplicial complex K(I)* * is d-dimensional if there exists a string i0 ! . .!.id of d morphisms between distinct objects but * *no such string of d + 1 morphisms. K(I) is again a poset with ordering given by inclusion. For any p-simplex (i0. .i.p) 2 K(I)p, put I(i0. .i.p) = I(ip) x . .I.(i0) and S(i0. .i.p) = S(ip-1, ip) x . * *.x.S(i0, i1) with the convention that for p = 0, S(i0) is understood to be a point. Form the* * homotopy orbit space ShI(i0. .i.p) = S(i0. .i.p)hI(i0...ip) for the action of the group I(i0. .i.p) on the set S(i0. .i.p) given by (ap, . .,.a0) . (ap-1p, . .,.a01) = (apap-1pa-1p-1, . .,.a1a01a-1* *0) for all aj 2 I(ij) and aj-1j2 I(ij-1, ij). This homotopy orbit space constructi* *on provides a functor ShI:K(I)op! Sp from the opposite poset of K(I) to the category Sp of simplicial sets. For any * *inclusion oe oe0of simplices, the map ShI(oe) ShI(oe0) is induced by the obvious projection I(oe* *) I(oe0) and the map S(oe) S(oe0) given by composition or omission of morphisms in the usual w* *ay. Let now M :I ! Ab be a functor. Consider the functor HqM :K(I) ! Ab that t* *akes the p-simplex (i0. .i.p) 2 K(I)p to the abelian group Hq(ShI(i0. .i.p); M(ip)) = Hq(I(i0. .i.p); M(ip) ) Define cohomology of K(I) with coefficients in HqM, H*(K(I); HqM), as the cohom* *ology of the cochain complex (C(K(I); HqM), ffi): Y ffip-1 Y (13.1) . .!. HqM(i0. .i.p-1) ---! HqM(i0. .i.p) ! . . . (i0...ip-1)2K(I)p-1 (i0...ip)2K(I)p N-DETERMINISM 59 with differential Xp ffip-1(U)(i0. .i.p) = (-1)jOEj*æ*jU(i0. .b.ij.i.p.) j=0 for all cochains U 2 Cp-1(K(I); HqM) and all p-simplices (i0. .i.p) of the simp* *licial complex K(I). Here, æj:I(ip) x . .I.(i0) ! I(ip) x . .d.I(ij)xi.s.I.(i0)the projection and th* *e homomorphisms 8 >>M(ip) if j = 0 < j D E M(ip) OE>M(ip) I(i0, . .,.bij,i.f.,.ip)0 < j < p >: M(ip-1) if j = 0 are given by 8 >u(ap-1p, . .,.ajj+1aj-1j, . .,.a01)if 0 <* * j < p :M(ap-1p)u(ap-2,p-1, . .,.a01)if j = p. It will become clear later that ffiffi = 0, i.e. that (C(K(I); HqM), ffi) is in* *deed a cochain complex. Let lim*(I; -) denote the right derived functors of the inverse limit functor* * lim:AbI ! Ab . 13.2. Theorem. [54] [87] There is a first quadrant cohomological spectral seque* *nce Epqrwith Epq1= Cp(K(I); HqM) andEpq2= Hp(K(I); HqM) converging to limp+q(I; M) This spectral sequence is associated to a descending filtration on the cochai* *n complex C(I; M) that has lim*(I; M) for cohomology groups: Let be the category of totally ordered finite sets and weakly order preserv* *ing maps. The cosimplicial replacement functor [14, XI.5] Q * I :Ab ! Ab Q* takes the abelian I-group M the cosimplicial abelian group M that in codegre* *e n is the abelian group of twisted n-cochains of I with coefficients in M, i.e. Q * n Y M = M(in) i0!...!in2N(I)n consists of all functions U from N(I)n with values U(i0 ! i1 ! . .!.in) in M(in* *). (As usual, the nerve, N(I), of I is the singular set of I: The simplicial set that in degree 0* * is the set of objects of I and in degree n > 0 is the set of all sequences i0 ! i1 ! . .!.in of n compos* *able morphisms in I.) The coface maps dj(U)(i0 ! . .!.in+1) = U(i0 ! . .!.bij! . .!.in+1), 0 j n, dn+1(U)(i0 ! . .!.in+1) = M(in ! in+1)U(i0 ! . .!.in) are the obvious ones.PDefine C(I; M) to be the underlying cochain complex whose* * differentialQis the alternating sum (-1)idi. The ith cohomotopy group of the cosimplicial abelian* * group *M, Q * i ßi M = H (C(I; M)), i 0, is defined [14, X.7.1] as the ith cohomology group of its underlying cochain co* *mplex C(I; M). 13.3. Lemma. [14, XI.6.2] [81, Lemma 2] The functors Q* i AbI- ! Ab !ß Ab, i 0, form a universal cohomological ffi-functor [95, 2.1.1] with limin degree 0. 60 J.M. MØLLER In other words, lim*= ß* O * and limi(I; M) = Hi(C(I; M)) is the ith cohomol* *ogy group of the cochain complex C(I; M) of I with (twisted) coefficients M. Define l to be the function on N(I) that is 0 on N(I)0; on N(I)1, l(i ! i) = * *0 while l(i ! j) = 1 if i and j are non-isomorphic; and in general n-1X l(i0 ! i1 ! . .!.in) = l(ii! ii+1), i=0 the function l counts the number of strict inequalities in the string i0 i1 * * . . .in. This makes the nerve into a filtered simplicial set ; = F0N(I) F1N(I) . . .FpN(I) Fp+1N(I) . . .N(I) where FpN(I) = {i0 ! i1 ! . .!.in 2 N(I) | l(i0 ! i1 ! . .!.in) < p} is the set of all strings of composable morphisms where less than p of the morp* *hisms have non- isomorphic domain and codomain. Since I has only finitely many equivalence clas* *ses of objects, the filtration is finite: Fd+1N(I) = N(I) if K(I) has dimension d. The filtration we are going to use is the induced descending filtration on th* *e cochain complex C(I; M), (13.4) C(I; M) = F0C(I; M) F1C(I; M) . . .FpC(I; M) Fp+1C(I; M) . . .{* *0} where FpC(I; M) = {U 2 C(I; M) | UFp(N(I)) = 0} consists of all cochains that vanish on FpN(I). This filtration is finite: Fd+1* *C(I; M) = {0} if K(I) is d-dimensional. Proof of Theorem 13.2.Suppose that K(I) is d-dimensional. Then the E1-page of * *the spectral sequence associated [95, 5.4.1] to the filtration (13.4) satisfies Epq1= 0 when* *ever p > d and Ed*1= H*(FdC(I; M)[-d]) where FdC(I; M)[-d] is the translated cochain complex [* *95, 1.2.8] that in degree n equals FdC(I; M)d+n. Note that M FdC(I; M)[-d] = C(i0. .i.d; M) i0...id2K(I)d splits as a direct product over the d-simplices in K(I) of the cochain complexe* *s C(i0. .i.d; M) given by * + a C(i0. .i.d; M)n = M(id) I(id)rdx I(id-1, id) x . .x.I(i0, i1) x * *I(i0)r0 r0+...+rd=n with a differential that is the restriction of the differential on C(I; M). Th* *e claim is that the cohomology of C(i0. .i.d; M) equals H*M(i0. .i.d) as defined above (13.1). The * *standard cochain complex for computing this cohomology group is (13.5) Hom I(id)x...xI(i0)(B*(I(id)) . .x.B*(I(i0)), M(id) i1)) where B*(I(id)) . .x.B*(I(i0)) as the tensor product of unnormalized bar reso* *lutions has @(ado . . .a0o) = ado . . .@a0o+ (-1)r0ado . . .@a1o a0o+ . . . + (-1)r0+...+rd-1@ado ad-1o . . .a* *0o as its differential. Here, ajo= ajrj . . .aj1where ajk2 I(aj) and rj-1X @ajo= ajrj . . .aj2+ . .a.jij+1ajij . .+.(-1)rjajrjajrj-1 . . .a1 ij=1 as usual [95, 6.5.1]. In fact, there is an isomorphism, oe, of cochain complexe* *s from the standard cochain complex (13.5) to C(i0. .i.d; M) given by oe(U)(ado, ad-1d, . .,.a1o, a01, a0o) = (-1)r1+r3+...U(ado, adoad-1d, . * *.,.a1o, a1oa01, a0o) N-DETERMINISM 61 where ajoaj-1j= ajrj. .a.j1aj-1jand the sign is (-1) raised to the power that i* *s the sum over all odd j of rj = |ajo|. I leave it to the reader to check that this isomorphis* *m oe indeed commutes with the differentials. The conclusion is that Y Edq1= Hq(FdC(I; M)[-d]) ~= HqM(i0. .i.d) (i0...id)2K(I)d is isomorphic to the degree d group of the simplicial cochain complex C(K(I); H* *qM) (13.1). This same pattern repeats itself at all stages of the filtration as Y FpC(I; M)p+q= Fp+1C(I; M)p+q C(i0. .i.p; M)p+q (i0...ip)2K(I)p and, in fact, there is an isomorphism Y FpC(I; M)[-p]=Fp+1C(I; M)[-p] ~= C(i0. .i.p; M) (i0...ip)2K(I)p of cochain complexes. So, by the above computation, Y Epq1~= HqM(i0. .i.p) (i0...ip)2K(I)p is isomorphic to Cp(K(I); HqM). It remains to compute the d1-differential. Again, it will be sufficient to c* *onsider the differ- ential dd-1q1:Ed-1q1! Edq1as similar arguments apply in general. Consider a coh* *omology class [U(i0. .i.d-1)] in Hq(id-1. .i.0; M) represented by the q-cocycle a U(i0. .i.d-1): I(id-1)rd-1x . .x.I(i0)r0! M(id-1) i1) rd-1+...r0=q and extend this to an element of Fd-1C(I; M)q+d-1by mapping the other (q + d - * *1)-simplices of N(I) to 0. The image dd-1q1[U(i0. .i.d-1)] is represented by (oe-1ffioe)U(i0. .* *i.d-1) where ffi is the zigzag-homomorphism of the short exact sequence 0 ! FdC(I; M) ! Fd-1C(I; M) ! FdC(I; M)=Fd-1C(I; M) ! 0 of cochain complexes. This means that dd-1q1[U(i0. .i.d-1)] vanishes on all (q* * + d)-simplices of N(I) except on the ones of the form aij-1i0j ai0jij (13.6) i0 ! . .!.ij-1- ---! i0j---!ij ! . .!.id-1 for some object i0jof I, where it has the value (-1)q+jU(ad-1o, ad-2d-1, . .,.ajo, aij-1i0jai0jij, aj-1o, . .,.a01* *, a0o) assuming, for simplicity, that 0 < j < d - 1. We must compare this to the homom* *orphism B*(I(id-1)) . . .B*(I(ij)) B*(I(i0j)) B*(I(ij-1)) . . .B*(I(i0)) ! * * j B*(I(id-1)) . . .B*(I(i0)) U(i0...id-1)-------!M(id-1) i1)OE! 0 0 ff M(id-1) I(id-2, id-1) x . .x.I(ij, ij) x I(ij-1, ij) x . * *.x.I(i0, i1) where the first homomorphism takes ad-1o . . .ajo a0jo aj-1o . . .a0oto ad-* *1o . . . ajo 1o aj-1o . . .a0o. Assuming U(i0. .i.d-1) to be normalized [95, 6.5.5], * *this agrees with __ the value of (oe-1ffioe)U(i0. .i.d-1) on the (q + d)-simplex (13.6) except that* * the sign is missing. |__| There is also a dual spectral sequence E2pq= Hp(K(I); HqM) ) colimp+q(I; M) where HqM(i0. .i.p) = Hq(ShI(i0. .i.p); M(i0)). 62 J.M. MØLLER 13.7. Example.1. [95, 3.5.12] If the category I is a poset S, the spectral sequ* *ence (13.2) for a functor M :S ! Ab degenerates to a cochain complex Y Y . .!. M(sp-1) ! M(sp) ! . . . (s0...sp-1)2K(S)p-1 (s0...sp)2K(S)p with cohomology lim*(S; M). 2. If the category I is a group G and M :Gop! Ab a G-module, then the spectral* * sequence 13.2 collapses onto the vertical axis in the sense that E0j1= Hj(G; M) and Eij1= 0 f* *or i > 0. 3. Suppose that I is a category S(0,1) 0___________//1 with two objects and no non-identity automorphisms. The limn(I; M) = 0 for n > * *1 and there is an exact sequence 0 ! lim0(I; M)! M(0) x M(1) ffi-!M(1) ! lim1(I; M)! 0 where ffi(m0, m1)(a) = m1- M(a)(m0) for all morphisms a 2 S(0, 1). 4. For a category I with two objects, 0 and 1, there is a long exact sequence . .!.Hj(I(0); M(0)) Hj(I(1); M(1)) d1-!Hj(E(0, 1); M(1)) ! limj+1(I; M* *)! . . . where we are assuming that I(1)xI(0) acts transitively on S(0, 1) with stabiliz* *er subgroup E(0, 1). 5. With I = A(W, t){E, t}, the full subcategory of A(W, t) containing t and one* * of its non-trivial subspaces E 6= t, we get a long exact sequence __ j d1 j __ . .!.Hj(W (E)=W (E); M(E)) x H (W ; M(t)) -! H (W (E); M(t)) ! limj+1(I; M)! . . . * * __ where the homomorphism d1 is induced from M(E) ! M(t) and from the inclusion of* * W (E) W . In case E = tS 6= t is the fixed-point space for the action of the Sylow p-su* *bgroup S of W and M(tS) and M(t) are Z(p)-modules, we conclude that there is an exact sequence __ S S __ S (13.8) 0 ! lim0(I; M)! M(tS)W (t )=W(tx)M(t)W ! M(t)W (t )! lim1(I; M)! 0 and j(__W(tS); M(t)) (13.9) limj+1(I; M)= H_____________Hj(W,; M(t))j 1. This quotient group vanishes if S has order p for N__W(tS)(S) = NW (S) (2.15) a* *nd both cohomol- ogy groups equal Hj(S; M(t))NW (S)as these are the stable elements [19, 9.1, 10* *.1] in this case. Assuming, furthermore, that M(tS) = MW(tS)and M(t) = M for some Z(p)[W ]-module* * M, we rediscover the formula ( W limj(I; M)= M if j = 0 0 if j > 0 from [2]. 6. Let H be a subgroup of the group G and I(G, H) = O(G)op{G=H, G={e}} the full* * subcategory _________G=H_____________________________________* *______________________________________________________________________@ (13.10) NG(H)=H __9G=H9_________//_______________________________* *____________________________________G={e}Gee__________________________@ of O(G)opcontaining the two objects G={e} and G=H. The limits of any functor M * *:I(G, H) ! Ab fit into a long exact sequence . .!.Hj(NG(H)=H; M(G=H)) Hj(G; M(G={e})) d1-!Hj(NG(H); M(G={e})) ! limj+1(I(G, H); M)! . . . N-DETERMINISM 63 where the homomorphism d1 is induced from M(H): M(G=H) ! M(G={e}) and from the* * inclu- sion of NG(H) into G. 7. Let I be a category with three objects, 0, 11, and 12, of the shape ____________________________________* *______________ I1o711gg__________________________________* *_____________________________7o ________________________________________________* *________________oooo ___:0:__________________________________________* *________________________OOOO I2O''O12__________________________________* *______________________________ gg__________________________________* *_______________ and let M be an I-module with M(0) = 0, M(11) = M1, and M(12) = M2. Then lim*(I; M) = lim*(I1; M1) x lim*(I2; M2) where I1 is the full subcategory generated by the objects 0 and 11, I2 the full* * subcategory generated by the objects 0 and 12, and the I(11)-module M1 is considered as an I1-module * *and the I(12)- module M2 as an I2-module. Of course, this extends to an arbitrary star shaped * *finite category with outward pointing arrows. 8. Let I be a category with three objects, 01, 02, and 1, of the shape _______________________________________________* *___ __7017_________________________________________* *____________________I1OOO OO''O1__________________________________* *___________________________________ o77dd__________________________________* *______________________________o _______________________________________________* *___oooo __7027_________________________________________* *____________________I2 and let M be an I-module with M(01) = 0 = M(02) and M(1) = M. Then there is a M* *ayer- Vietoris sequence . .!.Hj(I(1); M) ! limj(I1; M1) x limj(I2; M2) ! limj(I; M) ! Hj+1(I(1); M)* * ! . . . where I1 is the full subcategory generated by the objects 01 and 1 and I2 the f* *ull subcategory generated by the objects 02 and 1. In the next lemma, R(A) means the full subcategory containing all objects of * *the form Ra for a 2 Ob(A). 13.11. Lemma. Let __L__// IooR__J be an adjunction between two small categories, I(i, Rj) = J(Li, j) for all i 2 * *Ob(I), j 2 Ob(J), and A a full subcategory of J. Then 1.lim*(J; L*M) ~=lim*(I; M). 2.lim*(A; M) ~=lim*(R(A); L*M) ~=lim*(LR(A); M) for any functor M :J ! Ab . Proof.Since any left adjoint functor is left cofinal, the first assertion is a * *consequence of the Cofinality Theorem [14, XI.9.2, XI.7.2]. For the proof of the second assertion, where we may assume that Ob(J) = Ob(A)* * [ Ob(LRA), we consider the inclusion functors A ,! J - LRA The inclusion of LRA into J is left cofinal for the universal arrow LRa ! a is * *a terminal object in the over category LRA # a for all a 2 Ob(J). For the other inclusion, consider * *the Grothendieck spectral sequence limp(J; limq(a # A; M)) ) limp+q(A; M) If a in an object of A, the identity of a is an initial object in the under cat* *egory a # A. Otherwise, note that the restrictions __L__// Ra # RAoo___LRa # A R 64 J.M. MØLLER are adjoint functors so that ( limq(LRa # A; M) ~=limq(Ra # RA; L*M)= MLRa q = 0 0 q > 0 because the identity of Ra is an initial object in the under category Ra # RA. * *We conclude that lim*(A; M) ~=lim*(J; M) ~=lim*(LRA; M). Finally, observe that there is an indu* *ced adjoint- ness between RA and LRA so that also the two groups lim*(RA; L*M) and lim*(LRA;* *_M)_are isomorphic. |* *__| 13.12. Proposition.Let J be a full subcategory of I. If M vanishes on all objec* *ts outside J and if any object of I with a morphism to an object of J is in J, then lim*(I; M) ~* *=lim*(J; M). Proof.The cochain projection map Y Y M(in) ! M(jn) i0!...!in2N(I)n j0!...!jn2N(J)n is an isomorphism. * *|___| References [1]J. F. Adams and Z. Mahmud, Maps between classifying spaces, Inv. Math. 35 (* *1976), 1-41. MR 54 #11331 [2]J. Aguad'e, Constructing modular classifying spaces, Israel J. Math. 66 (19* *89), no. 1-3, 23-40. [3]K. Andersen, J. Grodal, J. Mfiller, and A. Viruel, The classification of p-* *compact groups for p odd, In prepa- ration. [4]Kasper K. S. Andersen, Cohomology of Weyl groups with applications to topol* *ogy, Master's thesis, Kfibenhavns Universitet, August 1997. [5]_____, The normalizer splitting conjecture for p-compact groups, Fund. Math* *. 161 (1999), no. 1-2, 1-16, Algebraic topology (Kazimierz Dolny, 1997). MR 1 713 198 [6]_____, Some computations for the exceptional groups relevant to the classif* *ication of p-compact groups, Ph.D. thesis, University of Copenhagen, January 2001. [7]David J. Benson, Polynomial invariants of finite groups, London Mathematica* *l Society Lecture Note Series, vol. 190, Cambridge University Press, Cambridge, 1994. [8]A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espaces * *homog`enes de groupes de Lie com- pacts, Ann. of Math. (2) 57 (1953), 115-207. [9]_____, Sous-groupes commutatifs et torsion des groupes de Lie compacts conn* *exes, Toh^oku Math. J. (2) 13 (1961), 216-240. [10]A. Borel and J. De Siebenthal, Les sous-groupes ferm'es de rang maximum de* *s groupes de Lie clos, Comment. Math. Helv. 23 (1949), 200-221. [11]N. Bourbaki, 'El'ements de math'ematique. Fasc. XXXIV. Groupes et alg`ebre* *s de Lie. Chapitre IV: Groupes de Coxeter et syst`emes de Tits. Chapitre V: Groupes engendr'es par des r'e* *flexions. Chapitre VI: syst`emes de racines, Hermann, Paris, 1968, Actualit'es Scientifiques et Industrielles, * *No. 1337. [12]N. Bourbaki, Groupes et alg`ebres de Lie, Chp. 9, Masson, Paris, 1982. [13]A.K. Bousfield, The localization of spaces with respect to homology, Topol* *ogy 14 (1975), 133-150. [14]A.K. Bousfield and D.M. Kan, Homotopy limits, completions and localization* *s, 2nd ed., Lecture Notes in Mathematics, vol. 304, Springer-Verlag, Berlin-Heidelberg-New York-London-P* *aris-Tokyo, 1987. [15]Carlos Broto and Antonio Viruel, Projective unitary groups are totally N-d* *etermined p-compact groups, Preprint. [16]____, Homotopy uniqueness of BPU (3), Group representations: cohomology, g* *roup actions and topology (Seattle, WA, 1996), Amer. Math. Soc., Providence, RI, 1998, pp. 85-93. MR * *99a:55013 [17]W. Browder, Torsion in H-spaces, Ann. of Math. (2) 74 (1961), 24-51. [18]Marc Cabanes, Irreducible modules and Levi supplements, J. Algebra 90 (198* *4), no. 1, 84-97. [19]Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton Universi* *ty Press, Princeton, New Jersey, 1956. [20]Allan Clark and John Ewing, The realization of polynomial algebras as coho* *mology rings, Pacific J. Math. 50 (1974), 425-434. MR 51 #4221 [21]Arjeh M. Cohen, Finite complex reflection groups, Ann. Sci. 'Ecole Norm. S* *up. (4) 9 (1976), no. 3, 379-436. [22]Maurice Craig, A characterization of certain extreme forms, Illinois J. Ma* *th. 20 (1976), no. 4, 706-717. [23]Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol* *. II, John Wiley & Sons Inc., New York, 1987, With applications to finite groups and orders, A Wiley-Intersci* *ence Publication. MR 88f:20002 [24]____, Methods of representation theory. Vol. I, John Wiley & Sons Inc., Ne* *w York, 1990, With applications to finite groups and orders, Reprint of the 1981 original, A Wiley-Intersci* *ence Publication. MR 90k:20001 [25]Morton Curtis, Alan Wiederhold, and Bruce Williams, Normalizers of maximal* * tori, Localization in group theory and homotopy theory, and related topics (Sympos., Battelle Seattle R* *es. Center, Seattle, Wash., 1974) (Berlin), Springer, 1974, pp. 31-47. Lecture Notes in Math., Vol. 418. N-DETERMINISM 65 [26]W. G. Dwyer, Lie groups and p-compact groups, Proceedings of the Internatio* *nal Congress of Mathematicians, Vol. II (Berlin, 1998), vol. 1998, pp. 433-442 (electronic). MR 99h:55025 [27]____, Sharp homology decompositions for classifying spaces of finite groups* *, Group representations: coho- mology, group actions and topology (Seattle, WA, 1996), Amer. Math. Soc., Pr* *ovidence, RI, 1998, pp. 197-220. [28]W. G. Dwyer and D. M. Kan, Centric maps and realization of diagrams in the * *homotopy category, Proc. Amer. Math. Soc. 114 (1992), no. 2, 575-584. MR 92e:55011 [29]W. G. Dwyer, H. R. Miller, and C. W. Wilkerson, Homotopical uniqueness of c* *lassifying spaces, Topology 31 (1992), no. 1, 29-45. MR 92m:55013 [30]W. G. Dwyer and C. W. Wilkerson, Homotopy fixed-point methods for Lie group* *s and finite loop spaces, Ann. of Math. (2) 139 (1994), no. 2, 395-442. MR 95e:55019 [31]____, The center of a p-compact group, The ~Cech centennial (Boston, MA, 19* *93) (Providence, RI), Amer. Math. Soc., 1995, pp. 119-157. MR 96a:55024 [32]____, Product splittings for p-compact groups, Fund. Math. 147 (1995), no. * *3, 279-300. MR 96h:55005 [33]____, Kähler differentials, the T-functor, and a theorem of Steinberg, Tran* *s. Amer. Math. Soc. 350 (1998), no. 12, 4919-4930. MR 99e:55027 [34]W.G. Dwyer and C.W. Wilkerson, A new finite loop space at the prime 2, J. A* *mer. Math. Soc. 6 (1993), 37-64. [35]W.G. Dwyer and A. Zabrodsky, Maps between classifying spaces, Algebraic Top* *ology. Barcelona 1986. Lec- ture Notes in Mathematics, vol. 1298 (Berlin-Heidelberg-New York-London-Pari* *s-Tokyo) (J. Aguad'e and R. Kane, eds.), Springer-Verlag, 1987, pp. 106-119. [36]William G. Dwyer, Haynes R. Miller, and Clarence W. Wilkerson, The homotopi* *c uniqueness of BS3, Alge- braic topology, Barcelona, 1986, Springer, Berlin, 1987, pp. 90-105. MR 89e:* *55019 [37]L. Evens, The cohomology of groups, Clarendon Press, Oxford-New York-Tokyo,* * 1991. [38]H. K. Farahat, On the natural representation of the symmetric groups, Proc.* * Glasgow Math. Assoc. 5 (1962), 121-136 (1962). MR 25 #3097 [39]Eric M. Friedlander, Exceptional isogenies and the classifying spaces of si* *mple Lie groups, Ann. Math. (2) 101 (1975), 510-520. [40]D.H. Gottlieb, Covering transformations and universal fibrations, Illinois * *J. Math 13 (1969), 432-437. [41]Robert L. Griess, Jr., Elementary abelian p-subgroups of algebraic groups, * *Geom. Dedicata 39 (1991), no. 3, 253-305. MR 92i:20047 [42]James E. Humphreys, The Steinberg representation, Bull. Amer. Math. Soc. (N* *.S.) 16 (1987), no. 2, 247-263. MR 88c:20050 [43]Bertram Huppert, Endliche Gruppen. I, Springer-Verlag, Berlin, 1967, Die Gr* *undlehren der Mathematischen Wissenschaften, Band 134. MR 37 #302 [44]____, Character theory of finite groups, Walter de Gruyter & Co., Berlin, 1* *998. MR 99j:20011 [45]Stefan Jackowski and James McClure, Homotopy decomposition of classifying s* *paces via elementary abelian subgroups, Topology 31 (1992), no. 1, 113-132. MR 92k:55026 [46]Stefan Jackowski, James McClure, and Bob Oliver, Homotopy classification of* * self-maps of BG via G-actions. I, Ann. of Math. (2) 135 (1992), no. 1, 183-226. MR 93e:55019a [47]____, Homotopy classification of self-maps of BG via G-actions. II, Ann. of* * Math. (2) 135 (1992), no. 2, 227-270. MR 93e:55019b [48]____, Self-homotopy equivalences of classifying spaces of compact connected* * Lie groups, Fund. Math. 147 (1995), no. 2, 99-126. MR 96f:55009 [49]Richard M. Kane, The homology of Hopf spaces, North-Holland Publishing Co.,* * Amsterdam, 1988. [50]Gregory Karpilovsky, Group representations. Vol. 2, North-Holland Publishin* *g Co., Amsterdam, 1993. MR 94f:20001 [51]G. Kemper and G. Malle, The finite irreducible linear groups with polynomia* *l ring of invariants, Transform. Groups 2 (1997), no. 1, 57-89. MR 98a:13012 [52]Jean Lannes, Sur les espaces fonctionnels dont la source est le classifiant* * d'un p-groupe ab'elien 'el'ementaire, Inst. Hautes 'Etudes Sci. Publ. Math. (1992), no. 75, 135-244, With an appen* *dix by Michel Zisman. MR 93j:55019 [53]____, Th'eorie homotopique des groupes de Lie (d'apr`es W. G. Dwyer et C. W* *. Wilkerson), Ast'erisque (1995), no. 227, Exp. No. 776, 3, 21-45, S'eminaire Bourbaki, Vol. 1993/94. * *MR 96b:55017 [54]Wolfgang Lück, Transformation groups and algebraic K-theory, Lecture Notes * *in Mathematics, vol. 1408, Springer-Verlag, Berlin, 1989, Mathematica Gottingensis. [55]Saunders MacLane, Homology, first ed., Springer-Verlag, Berlin, 1967, Die G* *rundlehren der mathematischen Wissenschaften, Band 114. MR 50 #2285 [56]George Maxwell, The crystallography of Coxeter groups, J. Algebra 35 (1975)* *, 159-177. MR 58 #11162a [57]H. Minkowski, Über den arithmetischen Begriff der äquivalenz und über die e* *ndlichen Gruppen linearer ganz- zahliger Substitutionen, Crelles Journal 100 (1887), 449-458. [58]J. M. Mfiller and D. Notbohm, Centers and finite coverings of finite loop s* *paces, J. Reine Angew. Math. 456 (1994), 99-133. MR 95j:55029 [59]____, Connected finite loop spaces with maximal tori, Trans. Amer. Math. So* *c. 350 (1998), no. 9, 3483-3504. MR 98k:55008 [60]Jesper M. Mfiller, N-determined 2-compact groups, In preparation. [61]____, Rational equivalences between classifying spaces, Math. Z. (to appear* *). 66 J.M. MØLLER [62]____, Spaces of sections of Eilenberg-Mac Lane fibrations, Pacific J. Math* *. 130 (1987), no. 1, 171-186. MR 89d:55048 [63]____, The normalizer of the Weyl group, Math. Ann. 294 (1992), no. 1, 59-8* *0. MR 94b:55010 [64]____, Completely reducible p-compact groups, The ~Cech centennial (Boston,* * MA, 1993), Amer. Math. Soc., Providence, RI, 1995, pp. 369-383. MR 97b:55020 [65]____, Homotopy Lie groups, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 4,* * 413-428. MR 96a:55026 [66]____, Extensions of p-compact groups, Algebraic topology: new trends in lo* *calization and periodicity (Sant Feliu de Gu'``xols, 1994), Birkhäuser, Basel, 1996, pp. 307-327. MR 98c:550* *12 [67]____, Rational isomorphisms of p-compact groups, Topology 35 (1996), no. 1* *, 201-225. MR 97b:55019 [68]____, Deterministic p-compact groups, Stable and unstable homotopy (Toront* *o, ON, 1996), Amer. Math. Soc., Providence, RI, 1998, pp. 255-278. MR 99b:55012 [69]____, Normalizers of maximal tori, Math. Z. 231 (1999), no. 1, 51-74. MR 2* *000i:55028 [70]____, Toric morphisms of p-compact groups, Cohomological methods in homoto* *py theory (Jaume Aguad'e, Carles Broto, and Carles Casacuberta, eds.), Progress in Mathematics, vol. * *196, Birkhäuser Verlag, 2001, Barcelona Euroconference on Algebraic Topology (BCAT) held in Bellaterra, J* *une 4-10, 1998, pp. 271-306. [71]Haruhisa Nakajima, Invariants of finite groups generated by pseudoreflecti* *ons in positive characteristic, Tsukuba J. Math. 3 (1979), no. 1, 109-122. MR 82i:20058 [72]D. Notbohm, Classifying spaces of compact Lie groups and finite loop space* *s, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 1049-1094. MR 96m:55029 [73]Dietrich Notbohm, Maps between classifying spaces, Math. Z. 207 (1991), no* *. 1, 153-168. MR 92b:55017 [74]____, Homotopy uniqueness of classifying spaces of compact connected Lie g* *roups at primes dividing the order of the Weyl group, Topology 33 (1994), no. 2, 271-330. MR 95e:55020 [75]____, p-adic lattices of pseudo reflection groups, Algebraic topology: New* * trends in localization and peri- odicity (Sant Feliu de Gu'``xols, 1994), Birkhäuser, Basel, 1996, pp. 337-3* *52. MR 97e:20005 [76]____, Topological realization of a family of pseudoreflection groups, Fund* *. Math. 155 (1998), no. 1, 1-31. [77]____, Erratum: öF r which pseudo-reflection groups are the p-adic polynomi* *al invariants again a polyno- mial algebra?" [J. Algebra 214 (1999), no. 2, 553-570; MR 2000a:13015], J. * *Algebra 218 (1999), no. 1, 286-287. MR 2000f:13009 [78]____, For which pseudo-reflection groups are the p-adic polynomial invaria* *nts again a polynomial algebra?, J. Algebra 214 (1999), no. 2, 553-570. MR 2000a:13015 [79]____, Spaces with polynomial mod-p cohomology, Math. Proc. Cambridge Philo* *s. Soc. 126 (1999), no. 2, 277-292. MR 1 670 237 [80]____, Unstable splittings of classifying spaces of p-compact groups, Q. J.* * Math. 51 (2000), no. 2, 237-266. MR 1 765 793 [81]Bob Oliver, Higher limits via Steinberg representations, Comm. Algebra 22 * *(1994), no. 4, 1381-1393. MR 95b:18007 [82]____, p-stubborn subgroups of classical compact Lie groups, J. Pure Appl. * *Algebra 92 (1994), no. 1, 55-78. MR 94k:57055 [83]Akimou Osse, ~-structures and representation rings of compact connected Li* *e groups, J. Pure Appl. Algebra 121 (1997), no. 1, 69-93. MR 98k:55017 [84]Wilhelm Plesken, On absolutely irreducible representations of orders, Numb* *er theory and algebra, Academic Press, New York, 1977, pp. 241-262. MR 57 #6072 [85]Derek J. S. Robinson, A course in the theory of groups, second ed., Spring* *er-Verlag, New York, 1996. MR 96f:20001 [86]R. Y. Sharp, Steps in commutative algebra, Cambridge University Press, Cam* *bridge, 1990. [87]Jolanta S_lomi'nska, Homotopy colimits on E-I-categories, Algebraic topolo* *gy Pozna'n 1989, Springer, Berlin, 1991, pp. 273-294. MR 92g:55023 [88]Larry Smith, Polynomial invariants of finite groups, Research Notes in Mat* *hematics, vol. 6, A K Peters Ltd., Wellesley, MA, 1995. [89]Stephen D. Smith, Irreducible modules and parabolic subgroups, J. Algebra * *75 (1982), no. 1, 286-289. MR 83g:20043 [90]Norman Steenrod, Polynomial algebras over the algebra of cohomology operat* *ions, H-spaces (Actes R'eunion Neuch^atel, 1970), Springer, Berlin, 1971, pp. 85-99. MR 44 #3316 [91]Michio Suzuki, Group theory. I, Springer-Verlag, Berlin, 1982, Translated * *from the Japanese by the author. MR 82k:20001c [92]Antonio Viruel, Homotopy uniqueness of classifying spaces of compact Lie g* *roups, Ph.D. thesis, Universitat Aut`onoma de Barcelona, 1996. [93]____, On the mod3 homotopy type of the classifying space of a central prod* *uct of SU(3)'s, J. Math. Kyoto Univ. 39 (1999), no. 2, 249-275. MR 2000g:55023 [94]____, E8 is a totally N-determined 5-compact group, Une d'egustation topol* *ogique [Topological morsels]: homotopy theory in the Swiss Alps (Arolla, 1999), Amer. Math. Soc., Provide* *nce, RI, 2000, pp. 223-231. MR 1 804 681 [95]Charles A. Weibel, An introduction to homological algebra, Cambridge Unive* *rsity Press, Cambridge, 1994. MR 95f:18001 [96]Clarence Wilkerson, Rational maximal tori, J. Pure Appl. Algebra 4 (1974),* * 261-272. MR 49 #8008 N-DETERMINISM 67 [97]____, Classifying spaces, Steenrod operations and algebraic closure, Topol* *ogy 16 (1977), no. 3, 227-237. MR 56 #1307 [98]____, p-compact groups with abelian Weyl groups, Preprint, 1999. [99]Clarence W. Wilkerson, Integral closure of unstable Steenrod algebra actio* *ns, J. Pure Appl. Algebra 13 (1978), no. 1, 49-55. MR 58 #24266 [100]Zdzis_law Wojtkowiak, On maps from holim-!F to Z, Algebraic topology, Barc* *elona, 1986, Springer, Berlin, 1987, pp. 227-236. MR 89a:55034 [101]Alex Zabrodsky, On spaces of functions between classifying spaces, Israel * *J. Math. 76 (1991), no. 1-2, 1-26. MR 93i:55020 Matematisk Institut, Universitetsparken 5, DK-2100 København E-mail address: moller@math.ku.dk