THE 2-COMPACT GROUPS IN THE A-FAMILY ARE N-DETERMINED JESPER M. MfflLLER Abstract.The 2-compact groups associated to central quotients of SU(n+1),* * n 1, are shown to be determined up to isomorphism by their maximal torus normalizers. 1. Introduction A 2-compact group is a 2-complete connected based space BX such that H*( BX; * *F2) is finite where BX is the loop space [6]. It is customary, though sometimes confusing, t* *o refer to BX by the symbol X. Any 2-compact group BX comes equipped with a maximal torus normalizer BN(X) !* * BX where BN(X) is the Borel construction BT (X) ! BN(X) ! BW (X) for the action of the Weyl group W (X) on the maximal torus T (X) [6, 9.8]. Doe* *s BN(X) determine BX? The answer to this question is ön " for the following reason. Let G be a Lie * *group and N(G) ! G its Lie group maximal torus normalizer. Assuming that the component group ß0(G)* * is a finite 2- group, BGbis a 2-compact group and BNb(G) ! BGbits 2-compact group maximal toru* *s normalizer. (For any Lie group H, BHb stands for the partial 2-completion of the classifyin* *g space BH for H.) Since there are distinct Lie groups, such as O(2n) and SO (2n + 1), with isomor* *phic maximal torus normalizers, there are also distinct 2-compact groups, such as bO(2n) and* * cSO(2n + 1), with isomorphic maximal torus normalizers. Thus we need to replace the maximal torus* * normalizer by a more delicate invariant which retains information about component groups. The* * maximal torus normalizer pair is a candidate for such a more delicate invariant. For a 2-compact group BX let BX0, the identity component of X, denote the uni* *versal covering space of BX. Since BX0 is again a 2-compact group, it has a maximal torus norma* *lizer BN(X0) ! BX0. The maximal torus normalizers of X and X0 are related by a commutative dia* *gram BN(X0) _____//_BX0 | | | | fflffl| fflffl| BN(X) _______//BX | | | | fflffl| fflffl| Bß0(X) ______Bß0(X) where the columns are fibration sequences. The fibration BN(X0) ! BN(X) ! Bß0(X* *), called the maximal torus normalizer pair associated to BX, has the built-in property t* *hat it fully informs about the component group of X. Does the maximal torus normalizer pair determin* *e the 2-compact group up to isomorphism? Focusing on the following properties for a 2-compact group X, (1)X is determined by (N(X), N(X0)) (2)Automorphisms of X are determined by their restrictions to N(X) (3)Automorphisms of X are determined by their restrictions to T (X) we shall say that ___________ Date: 23rd September 2002. 2000 Mathematics Subject Classification. 55R35, 55P15. Key words and phrases. Classification of p-compact groups at the prime p = 2. 1 2 J.M. MØLLER o X is totally N-determined if it satisfies (1) and (2) o X is uniquely N-determined if it satisfies (1) and (3) In this terminology, one might formulate the conjecture that all 2-compact grou* *ps are totally N-determined, all connected 2-compact groups even uniquely N-determined. Here i* *s an infinite family of simple 2-compact groups corroborating the conjecture. 1.1. Theorem. The simple 2-compact group PGL (n + 1, C), n 1, is uniquely N-d* *etermined and its automorphism group Aut(PGL (n + 1, C)) equals Zx2for n > 1 and Zx\Zx2for n * *= 1. It immediately follows (3.2, 4.3) that the Lie group PGL (n + 1, C) = PSL(n +* * 1, C) occuring in Theorem 1.1 can be replaced by any central quotient of SL(n + 1, C). Indeed, th* *e methods used here are not confined to simple, or semi-simple, 2-compact groups. 1.2. Corollary. [22, 1.9] The 2-compact group GL (n, C) is uniquely N-determine* *d and its auto- morphism group Aut(GL (n, C)) equals AutZ2 n(Zn2) for n > 2 and Zx\ AutZ2 2(Z22* *) for n = 2. The methods are not even confined to the connected cases. For instance, it f* *ollows from Lemma 4.1 that the 2-compact group GL (n, C) o C2, where C2 acts on GL (n, C) b* *y complex conjugation, is totally N-determined. See [31, 33, 34] for classification results for other 2-compact groups (with * *polynomial F2- cohomology). The results for the automorphism groups are not new [18] but repro* *ved here. 2. Generalities This sections contains the fundamental definitions and the first general resu* *lts. Whereas p- compact groups are determined by their maximal torus normalizers [29, 1] when p* * > 2, a finer invariant is needed for 2-compact groups as there are examples (2.2) of distinc* *t 2-compact groups with identical maximal torus normalizers. 2.1. Maximal torus normalizer pairs. Let N0 ! N be a maximal rank normal monomo* *rphism between two extended 2-compact tori, meaning simply that there exists a short e* *xact sequence of loop spaces N0 ! N ! ß for some finite group ß. We say that (N, N0) is a maxim* *al torus normalizer pair for the 2-compact group X, and we write N(X, X0) = (N, N0), if * *there exists a morphism of loop space short exact sequences N0_____//N______//_ß j0|| |j| |~=| fflffl|fflffl| fflffl| X0 ____//_X___//_ß0(X) where j and j0 are maximal torus normalizers for X and its identity component X* *0. A maximal torus normalizer pair for X determines the maximal torus T (X), the Weyl groups* *, W (X) and W (X0), of X and X0, the component group ß0(X) = N(X)=N(X0) = W (X)=W (X0), and* * [7, 7.5] the center Z(X0) ! X0 of X0. 2.2. Example. 1. Since N(SO (2n + 1)) O(2n) ( SO(2n + 1), O(2n) and SO(2n + 1* *) have the same maximal torus normalizer. Their maximal torus normalizer pairs are distinc* *t, however, for SO(2n + 1) is connected and O(2n) disconnected. 2. More generally [14], let G be any compact connected Lie group and N(G) its m* *aximal torus normalizer. If N(G) is not maximal, there exists a compact Lie group H such tha* *t N(G) H ( G. The two compact Lie groups, G and H, have isomorphic maximal torus normalizers * *but distinct maximal torus normalizer pairs as H is non-connected. 3. The Weyl groups for SO(2n + 1) and Sp(n), n 3, are isomorphic as reflectio* *n groups but N(SO (2n + 1)) is a split and N(Sp(n)) a non-split extension [3]. Thus connect* *ed 2-compact groups can not be classified by their Weyl group alone. 2.3. The Adams-Mahmud homomorphism. For a 2-compact group (or extended 2-compact torus) X, we let End(X) = [BX, *; BX] denote the monoid of homotopy classes of * *endomorphism of X. The automorphism group Aut(X) [BX, *; BX] of X is the group of invertib* *le elements in End(X) and the outer automorphism group Out(X) = Aut(X)=ß0(X) [BX; BX] is the* * group of conjugacy classes of automorphisms of X. 2-COMPACT GROUPS 3 Let X be a 2-compact group with maximal torus normalizer pair (N, N0). Turn t* *he maximal torus normalizer Bj :BN ! BX into a fibration. Any automorphism f :X ! X of th* *e 2-compact group X restricts to an automorphism AM (f): N ! N of the maximal torus normali* *zer, unique up to the action of the Weyl group W (X0) = ß1(X=N) of the identity component X* *0 of X, such that the diagram B(AM (f)) BN _________//_BN Bj|| |Bj| fflffl| fflffl| BX ____Bf___//_BX commutes up to based homotopy [26, x3]. The Adams-Mahmud homomorphism is the re* *sulting homomorphism (2.4) AM : Aut(X) ! W (X0)\ Aut(N) of automorphism groups. The automorphism group of N sits [24, 5.2] in a short exact sequence (2.5) 0 ! H1(W (X); ~T(X)) ! Aut(N) ß*-!Aut(W (X), ~T(X), e(X)) ! 1 where the normal subgroup to the left consists of all automorphisms of N that i* *nduce the identity on homotopy groups and the group to the right consists of all pairs (ff, `) 2 Aut(* *W (X))xAut (T~(X)) such that ` is ff-linear and the induced automorphism H2(ff-1, `) [35, 6.7.6] p* *reserves the extension class e(X) 2 H2(W (X); ~T(X)). The image of W (X0) in Aut(N) does not intersect* * the subgroup H1(W (X); ~T(X)) (as W (X0) is represented faithfully in Aut(T~(X)) [6, 9.7]) s* *o there is an induced short exact sequence (2.6) 0 ! H1(W (X); ~T(X)) ! W (X0)\ Aut(N) ß*-!W (X0)\ Aut(W (X), ~T(X), e(X)* *) ! 1 whose middle term is the target of the Adams-Mahmud homomorphism. In particula* *r, if X is connected, this short exact sequence (2.7) 0 ! H1(W (X); ~T(X)) ! Out(N) ! W (X)\ Aut(W (X), ~T(X), e(X)) ! 1 has the group Out(N) = W (X)\ Aut(N) of outer automorphisms of N as its middle * *term. The group Aut(W (X), ~T(X), 0), which is the normalizer NGL(L(X))(W (X)) of W (X) i* *n GL (L(X)), L(X) = ß2(BT (X)), fits into an exact sequence Z(W (X))\ AutZ2W(X)(L(X)) ! W (X)\NGL(L(X))(W (X)) ! Out(W (X)) where, by Schur's lemma, AutZ2W(X)(L(X)) = Zx2if X is simple. 2.8. Totally N-determined 2-compact groups. We are now ready to formulate the c* *oncept of N-determinism that will be used in this paper. 2.9. Definition. Let X be a 2-compact group with maximal torus normalizer pair * *(N, N0). (1)X has N-determined automorphisms if the Adams-Mahmud homomorphism (2.4)fo* *r X is injective and ß*(N)-determined automorphisms if AM -1(H1(W (X); ~T(X))* * is trivial. (2)X is N-determined if for any other 2-compact group X0with maximal torus n* *ormalizer pair (N, N0) there exist an isomorphism f :X ! X0and an automorphism ff 2 ß0(N* *)\ Aut(N) with ß*(Bff) = 1 such that the diagram (2.10) BN _Bff~=//_BN Bj|| Bj0|| fflffl|~=fflffl| BX _Bf_//_BX0 commutes up to based homotopy. (3)X is totally N-determined if it has N-determined automorphisms and is N-d* *etermined. A totally N-determined 2-compact group is o uniquely N-determined if it has ß*(N)-determined automorphisms (i.e. H1(W* * (X); ~T(X))\ AM (Aut(X)) = {1}) 4 J.M. MØLLER o strongly N-determined if H1(W (X); ~T(X)) AM (Aut(X)) Thus a totally N-determined p-compact group is both uniquely and strongly N-det* *ermined if H1(W (X); ~T(X)) = 0. For a compact connected Lie group G, the cohomology group H1(W (G); ~T(G)) is* * always an elementary abelian 2-group [20, 1.1]. For instance, this first cohomology grou* *p has order 2 for G = PSU(4) [19, Appendix B] (7.2), generated by an involution ff, say, of N(PSU* * (4)). The unique solution to diagram (2.10) is N(PSU (4))__ff__//N(PSU (4)) j|| j0|| fflffl| fflffl| PSU (4)___________PSU(4) when we use the morphisms j, induced by an inclusion of Lie groups, and j0 = jf* *f for maximal torus normalizers. PSU (4) is a uniquely but not strongly N-determined 2-compac* *t group. 2.11. Proposition. Suppose that the 2-compact group X is totally N-determined. (1)For fixed ff 2 Aut(N) with ß*(Bff) = 1 there is at most one isomorphism f* * :X ! X0such that diagram in 2.9.( 2) based homotopy commutes. (2)The pair (f, ff) in in 2.9.( 2),isXuniqueis uniquely N-determined. (3)It is always possible to use ff = 1,inH2.9.(12)(W (X); ~T(X)) AM.(Aut(X* *)) (4)W (X0)\ Aut(N) = H1(W (X); ~T(X)) . AM (Aut(X)) Proof.1. If (f1, ff) and (f2, ff) are two solutions to (2.10), then AM (f-12f1)* * is the identity and f1 = f2 as AM is assumed injective. 2. Suppose that the condition is satisfied and let (f1, ff1) and (f2, ff2) be t* *wo solutions to 2.9.(2). Then AM (f-12f1) = ff-12ff1 2 W (X0)\ Aut(N) belongs to both AM (Aut(X)) and H1* *(W (X); ~T(X)) and is therefore trivial. Thus AM (f-12f1) = 1 and f2 = f1 as AM is injective* *. Conversely, if AM (f) 6= 0 lies in H1(W (X); ~T(X)) for some f 2 Aut(X) then (f, AM (f)) and (* *1, 0) are two solutions to 2.9.(2) with X0= X and j0= j. 3. Let ff 2 H1(W (X); ~T(X)). If we can always find an isomorphism under N, the* *n there exists an isomorphism f 2 Aut(X) such that fj = jff. This means that AM (f) = ff. Co* *nversely, let (f, ff) be a solution to 2.9.(2). If H1(W (X); ~T(X)) AM (Aut(X)) then AM* * (g) = ff for an automorphism g 2 Aut(X). According to the commutative diagram BN oBffoBN___Bff//_BN Bj || Bj|| |Bj0| fflffl| fflffl| fflffl| BX oBgo_BX __Bf_//BX0 fg-1: X ! X0is an isomorphism under N. 4. For any automorphism g of N it is possible to find an automorphism f of X an* *d and auto- morphism ff of N with ß*(Bff) = 1 such that the diagram Bg BN _Bff//_BN____//_BN Bj|| |Bj| fflffl| fflffl| BX ______Bf_____//BX commutes up to based homotopy. Thus g = AM (f)ff. The subgroup H1(W (X); ~T(X)) is clearly normal so that W (X0)\ Aut(N) ~=H1(W (X); ~T(X)) o Aut(X), Aut(X) ~=W (X0)\ Aut(W (X), ~T(X)* *, e(X)) for a uniquely N-determined 2-compact group X. (The corresponding statement fo* *r compact connected Lie groups is true [14, 3.10]. It is already known that compact conne* *cted Lie groups perceived as 2-compact groups have ß*(N)-determined automorphisms [18, 2.5].) 2-COMPACT GROUPS 5 2.12. Lemma. Let X be a 2-compact group. Assume that the identity component X0 * *is completely reducible [23, 3.4, 3.10] and that ~Z(X0) = ~T(X0)W(X0). (1)H1(W (X); ~T(X)) \ AM (Aut(X)) = H1(W=W0; ~T W0). (2)If H1(W=W0; ~T W0) 6= 0 then X does not have ß*(N)-determined automorphis* *ms. (3)If H1(W=W0; ~T W0) = 0 and X0 has ß*(N)-determined automorphisms, so does* * X. (4)If the monomorphism inf:H1(W=W0; ~T W0) ! H1(W ;i~T)s an isomorphism, H1(* *W ; ~T) AM (Aut(X)). Proof.(1) and (2). This follows from the commutative diagram 0 ____//_H1(W=W0;"~T`W0)___//_Aut(X)______//Aut(ß0, X0)X___//_1 | | | | |AM | fflffl| fflffl| fflffl| 0 _______//_H1(W ;_~T)___//_W0\ Aut(N)__//_W0\ Aut(W, ~T;_e)//_1 with exact rows. The upper row is [24, 5.2]. (3). We must show that Aut(X) AM__//W0\ Aut(N)__//W0\ Aut(W, ~T; e) is injective. The image of this homomorphism is contained in the subgroup W0\ A* *ut(W, W0, ~T; e) where Aut(W, W0, ~T; e) consists of those pairs (ff, `) 2 Aut(W ; ~T; e) for wh* *ich ff(W0) = W0. In the commutative diagram ~= Ø " Aut(X) _______//_Aut(ß0, X0)X_______//_Aut(ß0)"x Aut(X0) QQ ` QQQQ | | QQQQ | 1xAM| QQ((Q fflffl| fflffl| W0\ Aut(W, W0, ~T;_e)_//_Aut(ß0) x W0\ Aut(W0, ~T) the slanted arrow must be injective. We know from [24, 5.2] that Aut(X) ~=Aut(ß* *0, X0)X . (4). This is clear from (2) and (3). 2.13. Example. The 2-compact group GL (2n, R) = SL(2n, R) o Z=2, n > 1, does no* *t have ß*(N)-determined automorphisms for H1(W=W0; ~T W0) = H1(Z=2; Z=2) = Z=2 is non-* *trivial. The maximal torus normalizer for GL (2n, R) is the same as the one for SL(2n + 1, R* *) so H1(W ; ~T) equals Z=2 for n = 2 and (Z=2)2 for n 3 (2.2, 7.3). The 2-compact group GL (* *2n + 1, R) = SL(2n + 1, R) x Z=2, n > 0, has ß*(N)-determined automorphisms for [24, 5.4.(1)* *] Aut(GL (2n + 1, R)) = Aut(SL(2n+1, R)) and SL(2n+1, R) has ß*(N)-determined automorphisms (a* *s does any compact connected Lie group [18, 2.5]). The 2-compact group PGL (2n, R) = PSL(2* *n, R) o Z=2 has ß*(N)-determined automorphisms since the identity component has trivial cen* *ter. In fact, H1(W (PSL (2n, R)) o Z=2; ~T) H1(Z=2; H0(W ; ~T)) + H0(Z=2; H1(W ; ~T)) = 0 (* *for n 5) since H0(W ; ~T) = 0 = H1(W ; ~T) for PSL(2n, R) by [13]. 2.14. Lemma. Let X be a connected 2-compact group with maximal torus normalizer* * j :N ! X. Then X is (uniquely) N-determined if and only if for any other connected 2-comp* *act group X0with maximal torus normalizer j0:N ! X0 there exists a (unique) morphism f :X ! X0 s* *uch that TA j|T~~~ AAj0|TAA ~~ AAA ""~~ __ X _____f_____//_X0 commutes up to conjugacy. Proof.The morphism f :X ! X0 in the above commutative diagram is in fact an iso* *morphism [8, 5.6] [27, 3.11]. The assumption of the lemma that f be a morphism under T * *means (use 6 J.M. MØLLER W \[BT, BX] = [BT, BX] [25, 3.4] [8, 3.4]) that f admits a restriction N(f) to * *N which is the identity on T , i.e. such that Bj BT _____//BN____//_BX || | | || BN(f)| Bf| || fflffl| fflffl| BT _____//BN_Bj0//_BX0 is homotopy commutative. But then also ß0N(f): W ! W is the identity map for W * *is faithfully represented as a group of operators on T [6, 9.7]. Thus ß*(BN(f)) is the identi* *ty automorphism of ß*(BN). Assume that the isomorphism f exists and is uniquely determined. In particula* *r, the identity of X is the only automorphism under T . That f 2 Aut(X) is a map under T means * *precisely that AM (f) 2 H1(W ; ~T). Thus X is uniquely N-determined by (2.11.2). Suppose, conv* *ersely, that X has this property and let f0, f1:X ! X0be two isomorphisms under T . Then f-11f* *0 2 Aut(X) is an isomorphism under T so equals the identity. 2.15. Remark. When the 2-compact group X has N-determined automorphisms, also t* *he un- based Adams-Mahmud homomorphism Out(X) = W (X)\ Aut(X) ! Out(N) = ß0(N)\ Aut(N) is injective [26, 3.7-3.9]. 2.16. LHS 2-compact groups. Let N0 ! N be maximal rank normal monomorphism betw* *een two extended 2-compact tori, i.e. a commutative diagram with rows and columns t* *hat are short exact sequences of loop spaces T _______T_____//{1} | | | | | | |fflffl fflffl| fflffl| N0 _____//N____//_W=W0 | | || | | || |fflffl fflffl| || W0 _____//W____//_W=W0 where T is a 2-compact torus and W0 a normal subgroup of the finite group W . T* *he 5-term exact sequence 0 ! H1(W=W0; ~T W0) inf--!H1(W ; ~T) res--!H1(W0; ~T)W=W0d2-!H2(W=W0; ~T W0) i* *nf--!H2(W ; ~T) is part of the Lyndon-Hochschild-Serre spectral sequence [15] converging to H*(* *W ; ~T). 2.17. Definition. The pair (N, N0) of extended 2-compact tori is LHS if the ini* *tial segment 0 ! H1(W=W0; ~T W0) inf--!H1(W ; ~T) res--!H1(W0; ~T)W=W0! 0 is a short exact sequence. A 2-compact group is LHS if its maximal torus normal* *izer pair is LHS. Here are two ways to check if a given p-compact group X is LHS (besides the e* *vident situations where ~T W0= 0 or W = W0x W=W0 is a direct product). The inflation homomorphism is the composition 2(W!W=W0) H2(W=W0; ~T W0) ! H2(W=W0; ~T) H----------!H2(W ; ~T) of a coefficient group homomorphism followed by the restriction homomorphism in* *duced by the projection of W onto the group of components W=W0. If the Weyl group W = W0o W=* *W0 is a semi-direct product, H2(W ! W=W0) is injective and therefore (2.18)H1(W ; ~T) ! H1(W0; ~T)W=W0 is surjective, H2(W=W0; ~T W0) ! H2(W=W0; ~T) is inject* *ive by exactness of the Lyndon-Hochschild-Serre spectral sequence. 2-COMPACT GROUPS 7 Another possibility is to use the description of H1(W0; ~T) from [13]. The sh* *ort exact sequence 1 ! W0 ! W ! W=W0 ! 1 of groups yields an exact sequence H2(W ) ! H2(W=W0) ! ((W0)ab)W=W0 ! Wab! (W=W0)ab! 0 of abelian groups (where H2(W=W0) = 0 if W=W0 has order two). The middle arrow * *in this exact sequence can be used to define a homomorphism Hom (W, ~T W) = Hom (Wab, (T~W0)W=W0 ) ! Hom (((W0)ab)W=W0 , (T~W0)W=W0 ) ! Hom (W0, ~T W0)W=W0 which fits into the commutative diagram (2.19) H1(WO;O~T)______//H1(W0;O~T)W=W0O | | | | | | Hom (W, ~T W)___//_Hom(W0, ~T W0)W=W0 Here, the left vertical arrow, say, takes a homomorphism W ! T~W to the cohomol* *ogy class represented by the crossed homomorphism W ! ~T W,! ~T. Since the right vertical* * arrow is an epimorphism in many cases [13, 1.2, 1.3], this can sometimes be used to show th* *at H1(W ; ~T) ! H1(W0; ~T)W=W0 is surjective. 2 2.20. Example. 1. The 2-compact group GL(m,C)_GL(1,C)o C2, m 1, where the C2-* *action switches the two GL(m, C)-factors, is LHS because (2.18) the map i ~Sx ~Sj i ~Smx ~Smj H2 C2; _____~! H2 C2; ________, S~= Z=21 , S S~ can be identified to the identity on H3(C2; ~S) = Z=2isincejH>0(C2; ~SxS~) = 0 * *= H>0(C2; ~SmxS~m) by Shapiro's lemma. Moreover, H1(W=W0; ~T W0) = H2 C2; ~Sx~S_~S= H2(C2; ~S) = 0. 2xGL(i ,C)2 2. The 2-compact group GL(i0,C)______1_GL(1,C)o C2, i0, i1 1, where C2 acts d* *iagonally by switching the two GL(i0, C)-factors and the two GL(i1, C)-factors, is LHS, again, because i ~S2x ~S2j i (S~i0)2x (S~i1)2j H2 C2; _______~! H2 C2; ____________ , S~= Z=21 , S ~S * *i ~2~2j can be identified to the identity on H3(C2; ~S). Moreover, H1(W=W0; ~T W0) = H2* * C2; SxS__~S= H2(C2; ~S) = 0. 4 3. The 2-compact group GL(m,C)_GL(1,C)o (C2 x C2), m 1, where C2 x C2 = <(12)* *(34), (13)(24)> permutes the four GL(m, C)-factors, is LHS. Again, i ~S2x ~S2j i (S~m)2x (S~m)2j H2 C2x C2; _______~! H2 C2x C2; ____________ , S~= Z=21 , S ~S identifies to the identity on H3(C2 x C2;i~S) by meansjof Shapiro's lemma and t* *he Künneth iso- x2~S2 2 2 morphism. Moreover, H1(W=W0; ~T W0) = H2 C2x C2; ~S___~S= H (C2x C2; ~S) = H (C* *2; ~S) + H1(C2; Z=2) = H1(C2; ~S) = Z=2. 4. The 2-compact group GL (2n, R) = SL(2n, R) o C2, n 2, is LHS by (2.18). T* *he homo- morphism Z=2 = H2(C2; Z=2) ! H2(C2, ~T) = Z=2 is injective because the action o* *f C2 on ~T= (Z=21 )n has (-1, 1, . .,.1) as its matrix. The 2-compact group GL(4, R) = * *SL(4, R) o C2 = (SL(2, C) O SL(2, C)) o C2, in particular, is strongly, but not uniquely N-dete* *rmined because 0 6= H1(W=W0; ~T W0) = H1(W ; ~T) (2.12, 7.3). For n > 2, GL(2n, R) can be neit* *her uniquely nor strongly N-determined. 8 J.M. MØLLER 2.21. The center of the maximal torus normalizer. We need criteria to ensure th* *at the center of the 2-compact group X agrees with the center of its maximal torus normalizer. 2.22. Proposition. [29, 4.12] Let X be a 2-compact group. If Z(X0) = Z(N(X0)) a* *nd X0 has N-determined automorphisms, then Z(X) = Z(N(X)). Assume from now on that X is a connected 2-compact group. Let N(X) ! X be the* * maximal torus normalizer and Z ! N a central monomorphism such that also the compositio* *n Z ! N(X) ! X is central. The action map BZ xBN(X) ! BN(X) induces an action [BN(X),* * BZ]x Out(N(X)) ! Out(N(X)) of the group [BN(X), BZ] ~=H1(N~(X); ~Z) on the set Out(N* *(X)). Let [BN(X), BZ](1)denote the isotropy subgroup at (1) 2 Out(N(X)). 2.23. Lemma. If Z(X) = Z(N(X)) and [BN(X), BZ](1)= 0, then Z(X=Z) = ZN(X=Z). Proof.Using [21, 4.6.4], the assumption of the lemma, and [29, 5.11], we get Z(* *X=Z) = Z(X)=Z = Z(N(X))=Z = Z(N(X)=Z) = ZN(X=Z). 2.24. Remark. Inspection shows that Z(G) = ZN(G) for any simply connected compa* *ct Lie group G; see [5, 1.4] for a conceptual proof of this fact. In fact, Z(G) = ZN(G* *) for any compact connected Lie group G containing no direct factors isomorphic to SO(2n + 1) [20* *, 1.6]. Q 2.25. Example. Let X = GL (ni, C) be a product of general linear groups and Z* * = Cx . (1, . .,.1). Then Z(X=Z) = ZN(X=Z) (2.24), unless X = GL(2, C), and, assuming t* *hat X=Z has N-determined automorphisms, Z(Xhß) = ZN(Xhß) for any 2-compact group Xhß with X* * as its identityQcomponent (2.22). Indeed, the discrete approximation to N(X) has the * *form N~(X) = (T~io ni) = ~To W . Suppose that (t, w) 2 ~N(X) is such that [(t, w), (s, 1)* *] 2 ~Z= Z=21 for all s 2 ~T. Then (w - 1)T~ ~Z, which means that w acts trivially on ~T=Z~. But* * W is faithfully represented as a group of automorphisms of this maximal torus, so w = 1. Suppo* *se therefore that t 2 T~is such that [(t, 1), (s, v)] 2 Z~for all (s, v) 2 N~(X). Then (v -* * 1)t 2 Z~for all v 2 W and v ! (v - 1)t is an element of H1(W ; ~Z) which becomesLtrivial in H1(* *W ; ~T) where it is a principalLcrossed homomorphism. Actually, H1(W ; ~Z) = H1( ni; ~Z) is is* *omorphic to the subgroup H1( ni; ~T) of H1(W ; ~T). 3.2-compact groups with N-determined automorphisms Let X be a 2-compact group with maximal torus normalizer pair N(X, X0) = (N, * *N0). 3.1. Lemma. [26, 4.2] Suppose that (1)X0 has N-determined automorphisms (2)H1 W=W0; ~Z(X0) ! H1 W=W0; Z(N~0) is injective Then X has N-determined automorphisms. 3.2. Lemma. [26, 4.8] Suppose that X is connected. If the adjoint form P X = X* *=Z(X) has ß*(N)-determined automorphisms, so does X. Proof.If f 2 Aut(X) is an automorphism under T (X), the induced automorphism P * *f 2 Aut(P X) is an automorphism under T (P X), hence equals the identity, and the induced au* *tomorphism Z(f) 2 Aut(ZX) is also the identity since the center ZX ! X factors through the* * maximal torus T (X) ! X [7, 7.5] [21, 4.3]. But then f itself is the identity for Aut(* *X) embeds into Aut(P X) x Aut(ZX) [25, 4.3]. The functor BCX :A(X) ! Top takes an object (V, ) of the Quillen category A* *(X) to its centralizer BCX (V, ) = map(BV, BX)B . The functor ßj(BZCX ): A(X) ! Abtakes (* *V, ) into the abelian group ßj(map (BCX (V, ), BX)e( )where e( ): BCX (V, ) ! BXis the * *evaluation map. 3.3. Lemma. [26, 4.9] Suppose that X is connected and centerless. If (1)CX (L, ~) has N-determined (ß*(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 (ß*(N)-determined) automorphisms. 2-COMPACT GROUPS 9 Proof.Suppose first that each line centralizer has ß*(N)-determined automorphis* *ms. Let f :X ! X be an automorphism under the maximal torus T ! X. Since any monomorphism ~: L * *! X, L = Z=2, factors through the maximal torus, the commutative diagram r9N9____//_Xr ~T rrrr | | L ____//_TL |AM(f) |f LLL | | L%fflffl||fflffl%L N ____//_X shows that f~ = ~ and gives a commutative diagram CN7(L)____//_CX7(L) oo oooo | | T O CAM(f)(L)||Cf(L) OOOO | | O'' fflffl| fflffl| CN (L)____//_CX (L) of automorphisms under T . Thus AM (Cf(L)) = CAM (f)(L): CN (L) ! CN (L). Now, * *ß*(CN (L)) is a subgroup of ß*(N) (for ß1(CN (L)) = ß1(N) and ß0(CN (L)) = W (X)(L) is [7,* * 7.6] [25, 3.2.(1)] the stabilizer subgroup at L < ~Tfor the action of W (X) on ~T) so ß*(* *CAM (f)(L)) = 1 and Cf(L) ' 1CX(L)since CX (L) has ß*(N)-determined automorphisms. For any other ob* *ject (V, ) of A(X) of rank > 1, choose a line L in V . Since the monomorphism :V ! Xcano* *nically factors through CX (L) [6, 8.2] [29, 3.18], the commutative diagram _________X00______________________________* *______________________________________________________________________@ ______________|_______________________________* *_________________oooo _______________________________________o V ____//__________|f|_____________________________* *_____________CX (L) _______________|_______________________________* *________________________OOO ___________.fflffl|.________________________* *______________________________________________________________________@ X shows that f = and the induced diagram ~ 5CX5(V ) k=kkkkkk | CCX(L)(V ) Cf(V|)| SSSSS | ~= S))S fflffl| CX (V ) that Cf(V ): CX (V ) ! CX (Vi)s conjugate to the identity. The second assumptio* *n of the lemma assures that there are no obstructions to conjugating f to the the identity now* * that we know that the restriction of f to each of the centralizers is conjugate to the identity, * *see [26, 4.9]. Suppose next that each line centralizer has N-determined automorphisms. Let f* * :X ! X be an automorphism such that the diagram r8X8r rrrr | N L f| LLL | L&&fflffl|L X commutes up to conjugacy. For each line L in T , the induced diagram CX6(L)6 mmmmmm | CN (L) |Cf(L)| QQQQ | QQ(( fflffl| CX (L) also commutes up to conjugacy. By assumption, this means (2.15) that the induce* *d automorphisms Cf(L) of line centralizers are conjugate to the identity. As above, this implie* *s that the induced 10 J.M. MØLLER map Cf(V ): CX (V ) ! CX (Vi)s conjugate to the identity for any object (V, ) * *of the Quillen category for X and that f is conjugate to the identity. 3.4. Lemma. [29, 9.4] If the two connected 2-compact groups X1 and X2 have N-de* *termined (ß*(N)-determined) automorphisms, so does the product X1x X2. Proof.Since the statement conecerning N-determined automorphisms is proved in [* *29, 9.4] we deal here only with the case of ß*(N)-determined automorphisms. Let f be an automorp* *hism under T1x T2 of the product 2-compact group X1x X2. Then f1:X1 ! X1x X2 f-!X1x X2 ! X1 f2:X2 ! X1x X2 f-!X1x X2 ! X2 are endomorphisms under the maximal tori and therefore conjugate to the respect* *ive identity maps. But f is [29, 9.3] in fact conjugate to the product morphism (f1, f2) which is * *the identity. 4. N-determined 2-compact groups Let X be a 2-compact group with maximal torus normalizer pair N(X, X0) = (N, * *N0). 4.1. Lemma. Suppose that (1)X0 is uniquely N-determined. (2)X is LHS. (3)H2 W=W0, ~Z(X0) ! H2 W=W0, Z(N~0) is injective. Then X is N-determined. Proof.Let X0 be another 2-compact group with maximal torus normalizer pair (N, * *N0). The as- sumption on the identity component X0means (2.14) that there exists an isomorph* *ism f0:X0 ! X00 under T . For any , 2 W=W0 = N=N0 = X=X0 = X0=X00, the isomorphism ,f0,-1 is a* *lso an isomorphism under T and thus ,f0 = f0, as X0 is uniquely N-determined. By the s* *econd assump- tion, the automorphism ff0 = AM (f0): N0 ! N0with ß*(Bff0) = 1 extends to an is* *omorphism ff: N ! Nwith ß*(Bff) = 1. Our aim is to find an isomorphism f :X ! X0 to fill in the based homotopy com* *mutative diagram Bf0 0 BX0 ___~=__//_BX0 | | | | fflffl| fflffl| BX ________//______________BX0 | | | | fflffl| fflffl| Bß0(X) __~=//_Bß0(X0) where the isomorphism between the base 2-compact groups is given by the isomorp* *hisms ß0(X) N=N0 ! ß0(X0). Since f0 is W=W0-equivariant up to homotopy, map(BX0, BX00)Bf0 i* *s a W=W0- 0 space. Composition with BX -Bj-BN Bj--!BX0 gives maps * map(BX0, BX00; Bf0)hW=W0 Bj--!map(BN0, BX00; B(j00ff))hW=W0 Bj0*-- hW=W0 ' map(BN0, BN0; Bff0) of homotopy fixed point spaces. The space to the right is non-empty for it con* *tains the iso- morphism Bff: BN ! BN . Using obstruction theory and the second assumption of t* *he lemma, we see that also the homotopy fixed point space to the~left is non-empty; it co* *ntains a morphism Bf :BX ! BX0 under Bf0:BX0 ! BX00 and over Bß0(X) =-!Bß0(X0) such that Bf O Bj* * and Bj O Bff are homotopic over B(N=N0) ! Bß0(X0). But since the fibre BX00of BX0! * *Bß0(X0) is simply connected this means that Bf OBj and BjOBff are based homotopic maps BN * *! BX0. 2-COMPACT GROUPS 11 4.2. Example. 1. Any 2-compact torus T is strongly N-determined for if j :T ! X* *is the max- imal torus normalizer for the connected 2-compact group X, then j is an isomorp* *hism. Indeed, H*(BT ; Q2) ~=H*(BX; Q2) [6, 9.7.(3)] and the connected space X=T has cohomolog* *ical dimen- sion cdF2(X=T ) = 0 [7, 4.5, 5.6] so is a point. 2. Any 2-compact toral group G is strongly N-determined: G clearly has N-determ* *ined auto- morphisms as G is its own maximal torus normalizer. If the 2-compact group X h* *as the same maximal torus normalizer pair (G, T ) as G, then X is a 2-compact toral group a* *nd j0:G ! X is an isomorphism. G is uniquely N-determined if and only if H1(ß0(G); ~T) = 0.* * In particular, GL(2, R) is uniquely and strongly N-determined. 4.3. Lemma. Let X be a connected 2-compact group and Z ! X its center. If X=Z * *is N- determined, so is X. Proof.Let j :N ! X be the maximal torus normalizer for X and j0:N ! X0 the maxi* *mal torus normalizer for some other connected 2-compact group X0. It suffices (2.14)to fi* *nd a morphism 0 f :X ! X0 under the maximal tori X- i T -i!X0. The 2-discrete center ~Zof X an* *d X0 is contained in the the 2-discrete maximal torus ~T[7, 7.5]. Factoring out [6, 8.* *3] these central monomorphisms we obtain the commutative diagram oo__Bi_____ _____Bi0___// BX~ BT~ BX~0 | | | | | | fflffl|B(i=Z) fflffl|B(i0=Z) fflffl| B(X=Z) ___________B(T=Z)oo___//_B(X0=Z) ___________________________________________________* *______________44______________________________________________________@ ____B(f=Z)_____________________________________* *______________________________________________________________________@ where the vertical maps are fibrations with fibre BZ~, the total spaces, such a* *s BX~, are the fibre- wise discrete approximations, and f=Z :X=Z ! X0=Z is the isomorphism under T=Z * *that exists because X?Z is N-determined. Construct the fibration map (BZ~, BZ~; B1) ! BZ~h(X=Z)! B(X=Z) whose sections are maps BX ! BX0 over B(f=Z) and under BZ~. There are two othe* *r such fibrations related to this one as shown in the commutative diagram map(BZ~, BZ~; B1)________map(BZ~, BZ~;_B1)______map(BZ~, BZ~; B1) | | | | | | fflffl| fflffl| * fflffl| BZ~h(X=Z)oo_____________BZ~h(T=Z)_____Bi_______//_BZ~h(T=Z) | | | | | | fflffl| fflffl| fflffl| B(X=Z) oo___B(i=Z)_____B(T=Z)___________________B(T=Z) where the middle fibration is the pull-back along B(i=Z) of the left fibration * *and the fibre over b 2 B(T=Z) of the right fibration consists of one component (remark about equiv* *ariance?) of the space of maps of the fibre BT~bover b into the fibre BX~0B(i0=Z)(b)over B(i* *0=Z)(b). The fibre equivalence Bi* is induced by Bi: BT~! BX~. The middle fibration has a section* * u0 such that Bi*O u0is the section Bi0:BT~! BX~0 of the right fibration. We now have fibre m* *aps u|X=T X=T ___________________//BZ~ | | | | fflffl| fflffl| BT~H________________//BZ~h(X=Z) HH rrr HHH rrr B(i=Z)H$$HH yyrrrr B(X=Z) 12 J.M. MØLLER where u is the composition of u0 and BZ~h(T=Z)! BZ~h(X=Z). The canonical map, * *given by constants, BZ~! map(X=T, BZ~) is a homotopy equivalence since X=T is simply con* *nected [21, 5.6] and hence a version [26, 6.6] of the Zabrodsky lemma implies that u = v O * *B(i=Z) for some section v :B(X=Z) ! BZ~h(X=Z)of the left fibration. The section v is, after fib* *re-wise completion, a fibre map BX ! BX0 under BT . Let X1 and X2 be two connected 2-compact groups with trivial centers and j1:N* *1 ! X1, j2:N2 ! X2 their maximal torus normalizers. The Splitting Theorem [8, 1.4] say* *s that if the monomorphism j :N1x N2 ! X is the maximal torus normalizer for some connected 2* *-compact group X then there exist an isomorphism s: X ! X1x X2 and an automorphism ff of* * N1 x N2 such that the diagram N1x N2 __ff~=//_N1x N2 j|| |j1xj2| fflffl|~= fflffl| X ___s___//X1x X2 commutes up to conjugacy. We record this in 4.4. Lemma. The product of two N-determined connected 2-compact groups is N-det* *ermined. The problem is now reduced to the connected and center-less case. Consider t* *herefore an extended 2-compact torus N and two connected, center-less 2-compact groups X an* *d X0 both having N as their maximal torus normalizer j 0 (4.5) X oo____N __j__//_X0 For each toral object (V, ) of A(X), let N :V ! N be the unique preferred l* *ift [27, 4.10] of (which factors through the identity component of N) and let (V, 0) be the ob* *ject defined by 0= j O N :V ! X0as in the commutative diagram V BB """""|NBB0B | BB ~~""" fflffl|__B X ojo_N_ _j0_//_X0 The functor A(X) t ! A(X0) tthat takes the object (V, ) to the object (V, 0) * *and is the identity on morphisms is an equivalence of toral Quillen categories [29, 2.8]. 4.6. Theorem. In the situation of ( 4.5), assume the following: (1)Centralizers of all toral rank 2 objects of A(X) have N-determined auto* *morphisms. (2)There exists a self-homotopy equivalence ff 2 H1(W ; ~T) Out(N) such th* *at for every object (L, ~) 2 Ob(A(X)) of rank 1 the diagram ff|CN(~N) CN (~N )________//_CN (~N ) j|CN(~N)|| j0|CN(~N)|| fflffl| fflffl| CX (~)_____f~___//CX0(~0) commutes for some isomorphism f~. (3)For any non-toral rank 2 object (V, ) of A(X) the composite monomorphism _(L) f(L, |L) 0L:V _______//_CX (L, _|L)~=__//CX0(L, ( |L)0)res//_X0 2-COMPACT GROUPS 13 and the induced isomorphism f ,L:CX (V, ) ! CX0(V,de0L)fined by the comm* *utative dia- gram Cf(L, |L) __ CCX(L, |L)(V, __(L))____//CCX0(L,( |L)0)(V, f(L, |L) O (L)) ~=|| |~=| fflffl| fflffl| CX (V, _)______f_,L________//CX0(V, 0(L)) do not depend on the choice of line L < V . (4)lim2(A(X) ; ß1(BZCX )) = 0 = lim3(A(X) ; ß2(BZCX )). Then there exists an isomorphism f :X ! X0 under T ( 2.14). Proof.The idea is that the isomorphisms f~: CX (~) ! CX0(~0)on the line central* *izers restrict to isomorphisms f :CX ( ) ! CX0( 0)for all centralizers in the F2-homology decompo* *sition hocolimA(X)BCX ( ) ! BX of BX. These locally defined isomorphisms combine to a globally defined isomorp* *hism BX ! BX0. First observe that the isomorphisms f~ on the line centralizers are uniquely * *determined by the cohomology class ff 2 H1(W ; ~T) (2.11.(1)). Let now (V, ) be a rank 2 object of A(X) and L a line in the plane V . If (V* *, ) is toral, define f :CX (V, ) ! CX0(V, t0)o be the isomorphism induced by f |L:CX (L, |L) ! CX0* *(L, (.|L)0 Since f is an isomorphism under ff|CN (V, N ) it does not depend on the choic* *e of L in V (2.11.(1)). If (V, ) is non-toral, define 0 to be 0Land define f :CX (V, ) ! CX0(V, t0)* *o be f ,L. By assumption 4.6.(3), the monomorphism 0and the isomorphism f ,Lare independent * *of the choice of L. This construction respects morphisms in A(X). Consider first, for instance, * *a morphism fi :(L1, ~1) ! (L2,b~2)etween two lines in X. Then ~1 = ~2fi and ~N1= ~N2fi. Th* *e commutative diagram of isomorphisms * CX (~1)|oo__________fi_____________CXe(~2)|eK99 | KKKK ssss | | KKK sss | | KK * sss | | N oofi_ N | | CN (~1 ) CN (~2 ) | | | f~1|| ff|CN(~N1)|| |ff|CN(~N2)||f~2| | fflffl| |fflffl | | CN (~N1)oo_*CN_(~N2) | | ss fi KKK | | sss KKK | | sss KKK | fflffl|yyss K%%fflffl| CX0(~01)oo___________fi*____________CX0(~02) shows that (fi*)-1 O f~1O fi* = f~2for they are both isomorphism under (fi*)-1 * *O ff|CN (~N1) O fi* = ff|CN (~N2). Second, by the very definition of f , the diagram CX (V, _)_f___//CX0(V, 0) | | | | fflffl| fflffl| CX (L, |L)f_|L//_CX0(L, ( |L)0) commutes whenever L < V and (V, ) is (toral or non-toral) rank 2 object of A(X* *). We have now defined natural isomorphisms f :CX (V, ) ! CX0(V, f0)or all obje* *cts (V, ) 2 Ob(A(X)) of rank 2. For any other object (E, ") of A(X), choose a line L < E * *and proceed as for toral rank 2 objects. That is, define "0:E ! X0 to be the monomorphism _"(L) f"|L E _______//CX (E, "|L)____//CX0(E, ("|L)0)res//_X0 14 J.M. MØLLER and define f":CX (E, ") ! CX0(E, "0)to be the isomorphism (f"|L)* _ CCX(E,"|L)(_"(L))_____//_CCX0(E,("|L)0)(f"|LO "(L)) ~=|| ~=|| fflffl| fflffl| CX (E, ")_______f"_______//_CX0(E, "0) induced by f"|L. If L1 and L2 are two distinct lines in E, let P = be t* *he plane generated by them. Then the commutative diagram f"|L1 CX (L1, "|L1)_~=___//_CX0(L1, ("|L1)0) _ nn77nOO OO QQQ "(L1)nnnn | | QQQresQQQ nnnn_ || || QQQ nnn "(P) ~= re0s QQ((Q P _________//PPPCX (P,_"|Pf)___//CX0(P, ("|P_)_)____//X066mm PPP | "|P | mmmmm _"(L2)PPPPPP| | mmmresmm P''Pfflffl| ~= fflffl|mmm CX (L2, "|L2)f"|L__//_CX0(L2, ("|L2)0) 2 shows that neither (E, "0) 2 Ob(A(X0)) nor the isomorphism f" depend on the cho* *ice of line in E. Thus we have constructed a collection of centric [4] maps (4.7) BCX (V, ) ! BX0, (V, ) 2 Ob(A(X)), that are homotopy invariant under A(X)-morphisms. The vanishing (4.6.(4)) of th* *e obstruction groups means [36] that these homotopy A(X)-invariant maps can be realized by a * *map Bf :BX -' hocolimBCX ! BX0 such that f O res= resO f for all (V, ) 2 Ob(A(X)). In particular, f is a map* * under T and an isomorphism (2.14). 4.8. Verification of condition 4.6.(2). Define ALHS(X) t to be the full subcate* *gory of the toral Quillen category A(X) t = A(W, t) [29, 2.2] generated by all objects wh* *ose centralizers CX ( ) are LHS and totally N-determined. For such an object, the solutions to t* *he isomorphism problem (4.9) CN ( N )_ff_//CN ( N ) | | | | fflffl| fflffl| CX ( )__f__//_CX0( 0) define a subset {ff } of H1(W ; ~T)( ) and (2.12.(1)) an element __ffof H1(W0; * *~T)W=W0 ( ). These elements respect the morphisms in ALHS(X) t (because the restriction of a solut* *ion is a solution) so they represent an element (__ff) of the limit group. If the two homomorphisms H1(W (X); ~T(X)) ! lim0(ALHS(X) t; H1(W ; ~T))! lim0(ALHS(X) t; H1(W0; ~T)W=* *W0 ) are surjective, this elements is the image og an element ff 2 H1(W (X); ~T(X)).* * This means that th isomorphism problems (4.9) have a coherent solution where ff = ff|CN ( N ) * *is the restriction of ff for all objects of ALHS(X) t. We can therefore replace 4.6.(1) and 4.6.(2) by o CX ( ) is LHS and totally N-determined for each toral elementary abelian * *2-subgroup (V, ) of X of rank 2 o lim1(ALHS(X) t; H1(W0; ~T)W=W0 ) = 0 The first property ensures that H1(W (X); ~T(X)) ! lim0(ALHS(X) t; H1(W ; ~T))~=lim0(A(X) t2; H1(W ; ~T)) ~=lim0(A(X) t; H1(W ; ~T)) 2-COMPACT GROUPS 15 is an isomorphism [10, 8.1] [32] and the second property that there is an exact* * sequence 0 ! lim0H1(W=W0; ~T W0) ! lim0H1(W ; ~T) ! lim0H1(W0; ~T)W=W0,! 0 where the limits are taken over over ALHS(X) t or A(X) t2. It is sometimes poss* *ible to compute the above lim1-term by means of Oliver's cochain complex [32]. 4.10. Verification of condition 4.6.(3). The following observation can sometime* *s be useful in the verification of condition 4.6.(3). 4.11. Lemma. Let (V, ) be a non-toral rank 2 object of A(X) and L < V a line i* *n V . Write C3 for the Sylow 3-subgroup of GL(V ). Suppose that (1)C3 A(X)(V, ) \ A(X0)(V, 0) (2)f ,L:CX (V, ) ! CX0(V,is0)C3-equivariant Then condition 4.6.( 3) is satisfied. Proof.Let L1 and L2 = L be lines in V . Choose an automorphism ff of (V, ) tha* *t takes L1 to L2. Then 0L2ff = 0- L1and f ,L1= CX (ff) O f ,L2O CX (ff)-1 (4.15). The following lemma assures that condition 4.11.(1) holds. 4.12. Lemma. Suppose that (1)There is (up to conjugacy) a unique monomorphism ~: Z=2 ! Xwith non-conne* *cted cent- ralizer (2)There is (up to conjugacy) a unique non-toral monomorphism :(Z=2)2 ! X Then the same holds for X0 and A(X)(V, ) = GL (V ) = A(X0)(V, 0) for the non-* *toral object (V, ) of A(X). Proof.Let 0:(Z=2)2 ! X0be a non-toral monomorphism from a rank elementary abel* *ian 2-group into X0and let i1:Z=2 ! (Z=2)2be the inclusion into the first summand. Then 0i* *1 corresponds to ~ under the bijection between rank 1 objects of A(X) and A(X0), i.e. 0i1 = * *~0. Moreover, the diagram (4.13) X oresoCX_(Z=2,g~)_____f~~=_____//_CX0(Z=2,g_0i1)res//_X0_________* *_________66________________ __________________________________ffLLL_077pp__________________* *_______________________________________________ ______________________________________________________________* *_____(Z=2)LLL(Z=2)ppp_________________________________________________@ ___________________________________________________________* *_________________LLppp________________________________________________@ _______________________________________________________* *______________________________________________________________________@ _(Z=2)2_______________________________________* *___________ is commutative. To see this, observe that = resOf-1~O__0(Z=2) by uniqueness o* *f , and __(Z=2) = f-1~O __0(Z=2) by uniqueness of canonical factorizations under Z=2 [28, 3.9]. * *We conclude that 0= resO __0(Z=2) = resO f~ O __(Z=2) is uniquely determined up to conjugacy. Note in connection with condition 4.11.(2) that by mapping (Z=2)2 into middle* * part of dia- gram (4.13) we see that f((Z=2)2, ) is a map under the canonical factorization* * in the sense that (4.14) (Z=2)2 _((Z=2)2)xxpp NNN_0((Z=2)2)N pppp NNNNN xxpppp ~= N''N CX ((Z=2)2,__)_f((Z=2)2,_)_//CX0((Z=2)2, 0) commutes where the canonical monomorphisms, __((Z=2)2)and __0((Z=2)2), are GL(E* *)-equivariant. Thus the restriction of f(V, ) to V is C3-equivariant. 16 J.M. MØLLER 4.15. Canonical factorizations. Let :V ! Xbe a monomorphism from an elementar* *y abelian p-group to the p-compact group X. The canonical factorization of through its * *centralizer is the central monomorphism __(V ): V ! CX (V,wh)ose adjoint is V x V +-!V -! X [6, * *8.2]. If ff: (V1, 1) ! (V2,is2)a morphism in A(X) then the canonical factorizations are* * related by a commutative diagram _1(V1) (4.16) V1 _______//_CXO(V1,__1)res//_XO ff|| CX(ff)|| |||| fflffl| | || V2 __2(V2)//_CX (V2,__2)res//_X and we shall write __2(V1): V2 ! CX (V1, 1) for CX (ff)O__2(V2) and call it th* *e canonical factorization of 2 through the centralizer of 1. The induced diagram (4.17) CCX(ff) __CCX(V1, 1)(ff) __ CCX(V2, 2)(V2, __2(V2))~=_//_CCX(V1, 1)(V2, 2(V1))__//_CCX(V1, 1)(V1, 1(V1* *)) ~=|| |~=| fflffl| fflffl| CX (V2, 2)_________________CX(ff)__________________//_CX (V1, 1) is a factorization of CX (ff). 5. The Quillen category of PGL (n + 1, C) For W is a finite group acting on a finite F2-vector space V , define A(W, V * *) to be the category whose objects are non-trivial subspaces of V and whose morphisms are group homo* *morphisms_in- duced by the W -action; the morphism set A(W, V )(V1, V2) is the set of orbits * *W (V1, V2)=W (V1, V1) for the action_of the point-wise stabilizer group W (V1, V1) = {w 2 W |wv = v f* *or allv 2 V } on the set W (V1, V2) = {w 2 W |wV1 V2}. 5.1. The toral subcategory of A(PGL (n+1, C)). An object (V, ) of the Quillen * *category of the 2-compact group X is toral if the monomorphism :V ! X is conjugate to a monom* *orphism that factors through the maximal torus T (X) of X. Let A(X) t denote the full subcat* *egory of A(X) generated by all toral objects. We shall determine this toral subcategory in ca* *se X = PGL (n+1, C). 5.2. Lemma. The monomorphism :V ! PGL (n + 1, C)is toral if and only if it li* *fts to a morph- ism V ! GL(n + 1, C). If n is even, all objects of A(PGL (n + 1, C)) are toral. Proof.All objects of A(GL (n + 1, C)) are toral by complex representation theor* *y. Any mono- morphism V ! (Cx)n+1=Cx lifts to (Cx)n+1 since Cx is a divisible abelian group.* * If n is even, PGL (n + 1, C) = SL(n + 1, C) as 2-compact groups and all monomorphisms V ! SL(* *n + 1, C) are toral by complex representation theory. 5.3. Proposition. [29, 2.8] The inclusion t(PGL (n + 1, C)) ! T (PGL (n + 1, C)* *) induces an equivalence of categories A( n+1, t(PGL (n + 1, C))) ! A(PGL (n + 1, C)) t. Proof.The functor is the identity on morphisms. Any morphism between two toral * *objects V1 ! V2 of A(PGL (n + 1, C)) is induced from the action by a Weyl group element. 5.4. Corollary. When n > 1, the limits limi(A(PGL (n + 1, C)) t; ßj(BZCPGL(n+1,* *C))) = 0 and limi(A(PGL (n + 1, C)); ßj(BZCPGL(n+1,C))) is isomorphic to limi(A(PGL (n + 1, C)) t; ßj(BZCPGL(n+1,C))~t)= limi(A(PGL (n + 1, C)); ßj(BZCPGL(n+1,C))* * t) for all i 0 and j = 1, 2. 2-COMPACT GROUPS 17 Proof.For any non-trivial toral subgroup V t(PGL (n + 1, C)) we have by (2.25* *)that ßj(BZCPGL(n+1,C)) = H2-j n+1(V ); L(PGL (n + 1, C)) , j = 1, 2, because the 2-discrete toral group ~ZCPGL(n+1,C)(V ) = ZCN~(PGL(n+1,C))(V ) = Z T~(PGL (n + 1, C)) o n+1(V ) = H0( n+1(V ); ~T(PGL (n + 1, C)* *)) and consequently the higher limits of these functors A(PGL (n + 1, C)) t ! Ab a* *re trivial while for i = 0 we get H2-j( n+1; L(PGL (n + 1, C))) which is trivial for n > 1. Appl* *y [29, 2.11] to get the isomorphisms. Let E be a non-trivial elementary abelian 2-group and Rep(E, GL(n + 1, C)) th* *e set of func- tionsPi: E_ ! N taking the dual E_ = Hom (E, Cx) of E into the natural numbers * *such that f2E_i(f) = n + 1. This set supports group actions E_ x Rep(E, GL(n + 1, C))_//Rep(E, GL(n + 1,oC))Repo(E,_GL(n + 1, C)) x GL(E) given by g.i = iOøg, g 2 E_, and i.A = iOA_, A 2 GL(E), where øg(f) = g+f and A* *_(f) = fOA-1 for all linear forms f 2 E_. The identity øA_(g)A_ = A_øg gives (g . i) . A = (* *(A-1)_g) . (i . A). We say that a subset S of linear forms on E has trivial equalizer, and write * *Eq(S) = 0, if S contains at least two elements and all the elements of S agree only on the triv* *ial element of E. 5.5. Proposition. Let E be a non-trivial elementary abelian 2-group. (1)The set of conjugacy classes of toral monomorphisms :E ! PGL (n + 1, C)* *corresponds bijectively to the set E_\{i 2 Rep(E, GL(n + 1, C)) | Eq(S(i)) = 0} of E_-orbits. (2)A(PGL (n + 1, C))(E_i) = {A 2 GL(E) | (E_i) . A_ = E_i}. (3)ß0(CPGL(n+1,C)(E_i)) = {i 2 E_ | i . i = i}. (4)The set of isomorphism classes of dimFpE-dimensional toral objects of A(P* *GL (n + 1, C)) corresponds bijectively to the set E_\{i 2 Rep(E, GL(n + 1, C)) | Eq(S(i)) = 0}=GL (E) of E_ x GL(E)-orbits. Proof.1. Let :E ! PGL (n + 1, C)be a toral monomorphism and ~: E ! GL(n + 1, * *C)any lift of to GL(n + 1, C). The representation ~ is a sum of linear characters X ~ = i~(f)f f2E_ for some function i~ 2 Rep(E, GL(n + 1, C)). The condition that ~(E) intersects* * theScenter Cx trivially, translates to Eq(S(i~)) = 0 (or, equivalently, S(i~) spans V _and V * *= f2S(i~)kerf). Any other lift of has the form i~ for some i 2 E_. We have ii~= i . i~ for X X X (i~)(v) = i~(f)i(v)f(v) = (i~ O øi)(f)f(v) = (i . i~)(f)f(v) for all v 2 V . (Also, S(øii~) = øiS(i~) so the equalizer subspace does not cha* *nge.) 2. An automorphism A 2 GL (E) preserves the conjugacy class of :E ! PGL (n + * *1, C)if and only if ~A(v) = i(v)~(v) for some i 2 E_ (depending on A). Since X X X (~A)(v) = i~(f)(fA)(v) = (i~ O A_)(f)f(v) = (i~ . A)(f)f(v) for all v 2 V , this means that i~ . A = i . i~. Then (g . imu) . A = ((A-1)_g* *) . (imu . A) = ((A-1)_g) . (i . imu) = ((A-1)_g + i) . imu 2 E_imu for all g 2 E_. 3. The component group of CPGL(n+1,C)(E_i~) is [29, 5.11.(2)] isomorphic to the* * group of i 2 E_ for which ~ and i~ are conjugate in GL(n + 1, C). For the traces, this means th* *at i~ = i . i~. 18 J.M. MØLLER 5.6. Remark. 1. Since iøføi = iøf , iøi = i, the right hand side for the equati* *on in 5.5.(3) remains the same for all elements of the orbit E_i. 2. Let A 2 A(PGL (n + 1, C))(E_i) so that iA_ = iøi for some i 2 E_. Then iøA_(g)= i , iøA_(g)A_ = iA_ , iA_øg = iA_ , iøiøg = iøi , iøg = i for any g 2 E_, meaning that A_(g) 2 ß0(CPGL(n+1,C)(E_i)) , g 2 ß0(CPGL(n+1,C)(* *E_i)). Thus A(PGL (n + 1, C))(E_i)opacts on ß0(CPGL(n+1,C)(E_i)). 5.7. Example. (Toral lines and planes in PGL (m, C)) Let P (m, k) be the number* * of ways to write m = i0+ i1+ . .+.ik as a sum of k integers i0, i1, . .,.ik such that 1 i0 i* *1 . . .ik. There are P (m, 2) = [m=2] toral lines and P (m, 3) + P (m, 4) toral planes in PGL (m* *, C). The P (m, 2) toral lines of type i = (i0, i1) with i0, i1 > 0 and i0 + i1 = m have these Qui* *llen automorphism groups and centralizer component groups: (i0, i1):A(PGL (m, C))(L) = 1, ß0(CPGL(m,C)(L)) = 1 (i0, i0):A(PGL (m, C))(L) = 1, ß0(CPGL(m,C)(L)) = L_ The non-connected rank 1 centralizer is 2 7.7 CPGL(m,C)(L) = GL_(i0,_C)_GL(1,oC)L_, ZCPGL(m,C)(L) = L The P (m, 3) + P (m, 4) toral planes of type i = (i0, i1, i2, i3) with i0, i1, * *i2 > 0, i3 0, and i0+ i1+ i2+ i3 = m have these Quillen automorphism groups and centralizer compo* *nent groups: (i0, i1, i2,Ai3):(PGL (m, C))(V ) = 1, ß0(CPGL(m,C)(V )) = 1 (i0, i0, i2,Ai3):(PGL (m, C))(V ) = C2, ß0(CPGL(m,C)(V )) = 1 (i0, i0, i0,Ai3):(PGL (m, C))(V ) = GL(V ), ß0(CPGL(m,C)(V )) = 1 (i0, i0, i2,Ai2):(PGL (m, C))(V ) = C2, ß0(CPGL(m,C)(V )) = L_ for some lin* *e L < V . (i0, i0, i0,Ai0):(PGL (m, C))(V ) = GL(V ), ß0(CPGL(m,C)(V )) = V _ If V = F22is a plane, then V _and GL(V ) ~= 3together generate all permutations* * of the four letters (i0, i1, i2, i3). Thus there are P (m, 3) isomorphism classes of the form (i0, * *i1, i2, 0), i0, i1, i2 > 0, i0 + i1 + i2 = m and P (m, 4) isomorphism classes of the form (i0, i1, i2, i3),* * i0, i1, i2, i3 > 0, i0+i1+i2+i3 = m. If, for instance i = (i0, i0, i0, i3) with i0 6= i3, then V _i* * = iGL (V ) contains four elements, so ß0 ~=V _and Aut = GL(V ). In all cases, ß0CPGL(m,C)(V, ) = ß0ZCPG* *L(m,C)(V, ); this is clear in case ß0(CPGL(m,C)(V )) = 1 is trivial and in the remaining two* * cases it is a direct check. The character table for V =_C2x_C2_=_{e0,_e2,_e2,_e3 = e1+ e2} |____|e|0|e1_|e2_|e3_|_ |_æ0_|1||__1_|_1_|_1_| |_æ1_|1||__1_|-1_|-1_| |_æ2_|1||-1_|__1_|-1_| |_æ3_|1||-1_|-1_|__1_| contains four linear characters V _= {æ0, æ1, æ2, æ3}. In the list above, (i0, * *i1, i2, i3) means i0æ0+ i1æ1 + i2æ2 + i3æ3. Non-connected PGL (m, C)-centralizers only occur for induc* *ed GL (m, C)- representations: ( V (i æ + i æ ) i 6= i (i0, i0, i2, i2) = ind0V0 2 1 0 2 ind{0}(i0æ0) = i0regVi0 = i2 In the first case, the centralizer 2x GL(i1, C)2 7.7GL(1, C)* * x GL(1, C) CPGL(m,C)(V, æ) = GL_(i0,_C)__________GL(1,oC)L_, ZCPGL(m,C)(V, æ) = _________* *________GL(1,xC)L, is LHS and has ß*(N)-determined automorphisms (2.20). In the second case, we ha* *ve a pure rank 2 object, the only rank 1 sub-object is 2i0 times the regular representation of* * C2. Its centralizer 4 7.5 CPGL(m,C)(V, æ) = GL_(i0,_C)_GL(1,oC)V _, ZCPGL(m,C)(V, æ) = V, is LHS but does not does not have ß*(N)-determined automorphisms (2.20). 2-COMPACT GROUPS 19 5.8. The non-toral subcategory of A(PGL (n + 1, C)). For 2-compact group X, let* * A(X) t denote the full subcategory of A(X) on all non-toral objects and their sub-obje* *cts. We determine this non-toral subcategory A(PGL (n + 1, C)) t in case X = PGL (n + 1, C). For any non-trivial elementary abelian 2-group V in PGL (n + 1, C), let [ , ]* *: V x V ! F2be the symplectic bilinear form [16, II.9.1] given by [uCx, vCx] = [u, v] for all * *uCx, vCx 2 V . (The elements [u, v] and u2 lie in the center Cx of GL(n+1, C) so that E = [u2, v] =* * [u, v]u[u, v] = [u, v]2 and thus [u, v] 2 Cx has order 2. Therefore [u, v] = [u, v]-1 = [v, u].) 5.9. Lemma. V in PGL (n + 1, C) is toral, [V, V ] = 0 Proof.Let eiCx, 1 i d, be a basis for V . Since Cx is divisible, we can ass* *ume that each ei2 GL(n + 1, C) has order 2. If [V, V ] = 0, these eis commute and span a lift* * to GL(n + 1, C) of V PGL (n + 1, C). An extra special 2-group is of positive type if it is isomorphic to a central* * product of dihedral groups D8 of order 8. 5.10. Lemma. [12, 3.1] [29, 5.4] Let :V ! PGL (n, C)be a non-toral monomorphi* *sm of a non- trivial elementary abelian 2-group V into PGL (n + 1, C). Then there exists a m* *orphism of short exact sequences of groups 1____//_Z(P_)______//_"P`E__________//_"V`______//_"1` | | | | | | fflffl| fflffl| fflffl| 1_____//_Cx____//_GL(n + 1,_C)_//_PGL(n + 1,_C)_//_1 where P E is the direct product of an extra special 2-group P GL(n + 1, C) of* * positive type and an elementary abelian 2-group E GL(n + 1, C) with P \ E = {1} = [P, E]. Write Cn+1 = C2d Cm for some d > 0 and some m 0. Let the extra-special 2-g* *roup 21+2d+ act faithfully on the first factor of the tensor product and let the (possibly * *trivial) elementary abelian 2-group E act faithfully on the second factor such that no non-trivial * *element of E acts as scalar multiplication. This makes Cn+1 a C[21+2d+x E]-module. The image of* * the group 21+2d+x E GL(n + 1, C) in PGL (n + 1, C) is a non-toral elementary abelian 2-* *group (5.9) and any non-toral elementary abelian 2-group in PGL (n + 1, C) has this form (5.10). Let G = i= P O C4x E be the group generated by E and the central produ* *ct P O C4 of P and the cyclic group C4 = Cx with Z=2 amalgamated. The image of G in PGL* * (n + 1, C) is V and q(vCx) = v2, v 2 G, is a quadratic form on V such that q(uCx + vCx) = * *q(uCx) + q(vCx) + [uCx, vCx] for all uCx, vCx 2 V . 5.11. Lemma. A(GL (n + 1, C))(G, G) ! A(PGL (n + 1, C))(V, V ) is surjective. Proof.Suppose that B 2 GL (n + 1, C) is such that V BCx = V . Then GB G . Cx:* * For any g 2 G there exist h 2 G and z 2 Cx such that gB = hz. But since G has exponent * *4, z4 = 1 so z 2 C4 and gB 2 G. A monomorphic conjugacy class :V ! PGL (n + 1, C)is said to be a (2d + r, r* *) object of A(PGL (n+1, C)) if the underlying symplectic vector space of (V, ) is isomorph* *ic to V = HdxV ? where H denotes the symplectic plane over F2 and dimFpV ?= r [16, II.9.6] (so t* *hat dimFpV = r + 2d). An (r, r) object is the same thing as an r-dimensional toral object. W* *e write Sp(V ) or Sp(2d + r, r) (abbreviated to Sp(2d) if r = 0) for the group of linear automorp* *hisms of V that preserve the symplectic form. 5.12. Corollary. Suppose that n + 1 = 2dm for some natural numbers d 1 and m * * 1. (1)There is up to isomorphism a unique (2d, 0) object Hd of A(PGL (n + 1, C)* *), and A(PGL (n + 1, C))(Hd) = Sp(2d), CPGL(n+1,C)(Hd) = Hd x PGL(m, C) for this object. 20 J.M. MØLLER (2)Isomorphism classes of (2d + r, r), r > 0, objects V of A(PGL (2dm, C)) c* *orrespond biject- ively to isomorphism classes of (r, r) objects V ? of A(PGL (m, C)), and ` ' A(PGL (2dm, C))(V ) = Sp(2d)*A(PGL (m0, C))(V ?) CPGL(2dm,C)(V ) = V=V ?x CPGL(m,C)(V ?) for these objects. Proof.1. The group 21+2d+O4 has [17, 7.5] 21+2dcharacters of degree 1 and 2 irr* *educible characters of degree 2d (interchanged by the action of Out(21+2d+O 4) ~=Sp(2d) x Aut(C4) [* *11, pp. 403-404]) given by ( d~(g) g 2 C4 Ø~(g) = 2 0 g 62 C4 where ~: C4 ! Cxis an injective group homomorphism (~(i) = i). The linear char* *acters vanish on the derived group 2 = [21+2d+O 4, 21+2d+O 4] but the irreducible characters * *of degree 2d do not. Thus the only faithful representations of 21+2d+O 4 with central centers are mu* *ltiples mØ~ of Ø~ for a fixed ~. Phrased slightly differently, GL (m2d, C) contains up to conjugacy a* * unique subgroup with central center isomorphic to 21+2d+O 4. For this group and its image Hd in* * PGL (2dm, C) we have A(GL (m2d, C))(21+2d+O 4, 21+2d+O 4) ~=Sp(2d) ~=A(PGL (m2d, C))(Hd, Hd) CGL(m2d,C)(21+2d+O 4) ~=GL(m, C), CPGL(m2d,C)(Hd) ~=Hd x PGL(m, C) where the last isomorphism is a consequence of [29, 5.9]. 2. The (2d+r, r) object (V, ) of A(PGL (2dm, C)) and the (r, 0) object (V ?, * *? ) of A(PGL (m, C)) correspond to each other iff there is an m-dimensional representation ~: V ?! G* *L(m, C)such that C2d ~ is a lift of |V ? and ~ a lift of ? . According to 5.10 any lift of |* *V ? has this form for some ~ uniquely determined up to the action of (V ?)_. We use 5.11 to calculate the Quillen automorphism group of a (2d + r, r) obje* *ct Hd x V ? of A(PGL (2dm, C)). Let Hd x V ? be covered by the group P O C4 x V ? as in 5.10. * *Let ff be an automorphism of P OC4, let fi be any homomorphism of the form P OC4 ! Hd ! V ?,* * and let fl be any Quillen automorphism of (V ?, ? ). Choose a homomorphism i1:P OC4 ! HdxC4=* *C2 ! C4 such that ~(i1(x)ff(x)) = ~(x) for all x 2 C4 and a homomorphism i2:V ?! C4 su* *ch that ~(i2(v))~(fl(v)) = ~(v) for all v 2 V ?. Then the automorphism of P O C4 that * *takes (x, v) to (i1(x)i2(v)ff(x), fi(x) + fl(v)) preserves the trace of Ø~#~ and therefore the * *automorphism in- duced on the quotient is a Quillen automorphism of Hd x V ?. Conversely, any a* *utomorphism of P O C4 x V ? takes the center C4 x V ? isomorphically to itself and hence it* * is of the form (x, v) ! (i(x, v)ff(x), fi(x) + fl(v)) for some automorphism ff of P O C4, some* * homomorphism fi :P O C4 ! Vv?anishing on C4, and some homomorphism i :P O C4x V ?! C4. Such * *an auto- morphism preserves the trace of Ø~#~ iff ~(i(x, v)ff(x)) = ~(fl(v)) for all (x,* * v) 2 Z(P OC4xV ?) = C4x V ?. But this means that the induced automorphism of Hd x V ?is of the stat* *ed form. 5.13. Example. (Oliver's cochain complex [32]) The non-toral objects of A(PGL (* *2m, C)) of rank 4 are o One (2, 0) object H, A(PGL (2m, C))(H) = Sp(2) = GL(2, F2), ß0 = H. o P (m, 2) (3, 1) objects V , A(PGL (2m, C))(V ) = Sp(3, 1),`ß0 = V=V ?or V* * . ' o P (m, 3)+P (m, 4) (4, 2) objects E, A(PGL (2m, C))(E) = Sp(2)*A(PGL (m0,* * C))(E? ), A(PGL (m, C))(E? ) = 1, C2, GL(E), ß0 = E=E? , E=E? or E=L, E=E? or E. o One (4, 0) object if m is even. The (2, 0) object H contributes Hom Sp(2)(St(H), H) ~=F2 The (3, 1) objects V contribute HomSp(3,1)(St(V ), V ) ~=HomSp(3,1)(St(V ), V=V ?) ~=F2 2-COMPACT GROUPS 21 The (4, 2) objects E with A(PGL (m, C))(E? ) = 1 contribute Hom0 1(St(E), E=E? ) ~=F22 @ Sp(2) 0A * 1 and the (4, 2) objects E with A(PGL (m, C))(E? ) = C2 contribute Hom 0 1(St(E), E=L) ~=Hom0 1(St(E), E=E? ) ~=F2 @Sp(2) 0 A @ Sp(2) 0 A * C2 * C2 The (4, 0) object (if it exists) and the (4, 2) objects with A(PGL (m, C))(E? )* * = GL (E) do not contribute to the cochain complex for the corresponding Hom -groups are trivial* *. Thus the cochain complex for computing higher limits of the functor ß1(BZCPGL(2m,C)) will have t* *he form 1Y ffi2 (5.14)0 ! Hom Sp(2)(St(H), H) ffi-! HomSp(3,1)(St(V ), V=V ?) -! [m=2] Y Y Hom 0 1(St(E), E=E? ) x Hom 0 1(St(E), E=E? ) ! .* * . . @Sp(2) 0A @Sp(2) 0 A * 1 * C2 To show vanishing of the relevant higher limits it suffices to show that ffi1 i* *s injective and that the rank of ffi2 is P (m, 2) - 1. 6.N-determinism of the A-family By inductively applying 3.3 and 4.6 we show that the 2-compact groups PGL (n * *+ 1, C), n 1, are uniquely N-determined. 6.1. Lemma. Suppose that n + 1 = 2m 2 is even. (1)There is a unique monomorphism conjugacy class ~: Z=2 ! PGL (n + 1, C)wit* *h discon- nected centralizer. The centralizer of this monomorphism is GL(m, C)2=Cx * *o Z=2 (2)There is a unique monomorphism conjugacy class :H ! PGL (n + 1, C), H =* * (Z=2)2, such that is non-toral. The centralizer of this monomorphism is H x PGL* * (m, C) and the Quillen automorphism group is GL(H). Proof.Use that any monomorphism of Z=2 into PGL (n + 1, C) lifts to ~: Z=2 ! GL* *(n + 1, C). The only possibility is that ~ = m . regis a direct sum of regular representati* *ons. The result for non-toral rank 2 objects in A(PGL (n + 1, C)) is a special case of 5.10. 6.2. Lemma. Suppose that PGL (r + 1, C) is uniquely N-determined for all 0 r * *< n. Then PGL (n + 1, C), n 1, satisfies conditions 4.6.( 1), 4.6.( 2), and 4.6.( 3). Proof.We shall verify 4.6.(1) and 4.6.(2) by establishing the alternative two c* *onditions from 4.8. Let (V, ) be a toral elementary abelian 2-subgroup of PGL (n + 1, C) of rank* * 2 and C( ) = CPGL(n+1,C)( ) its centralizer. We have seen that C( ) is LHS (2.20) and that * *Z~(C( )0) = ~Z(N0(C( ))) as C( )0 does not contain a direct factor isomorphic to GL(2, C)=G* *L (1, C) = SO(3) (2.24, 5.7). The identity component C( )0 has ß*(N)-determined automorphisms ac* *cording to 3.2 and 3.4, and C( ) has N-determined automorphisms by 3.1. The identity componen* *t C( )0 is N-determined according to 4.3 and 4.4, and C( ) is N-determined by 4.1. Thus C(* * ) is LHS and totally N-determined. The functor H1(W=W0; ~T0W) is zero on A(PGL (n + 1, C)) t2except on the objec* *t (V, ) = (i0, i0, i0, i0), when n + 1 = 4i0, where it has value Z=2. However, this objec* *t has Quillen auto- morphism group GL(V ) and since the only GL(V )-equivariant homomorphism St(V )* * = V ! Z=2 is the trivial homomorphism, lim1(A(PGL (n+1, C)) t2; H1(W=W0; ~T0W)) = 0 follo* *ws from Oliver's cochain complex [32]. When n + 1 = 2m is even, we verify condition 4.6.(3) by applying 4.11. Let X0* * be a connected 2-compact group with maximal torus normalizer j0:N(PGL (n + 1, C) !.XSince the * *first item in 4.11 is satisfied by 4.12 and 6.1, it suffices to show that the isomorphism * *(from 4.6.(3)) f ,L:CPGL(2m,C)(H) = H x PGL(m, C) ! CX0(H, 0) 22 J.M. MØLLER defined by choosing one of the three lines L in H, is C3-equivariant. Now [24] Aut(H x PGL(m, C)) = GL(H) x Aut(PGL (m, C)) so that f ,Lis C3-equivariant if ß0f ,Land the restriction of f ,Lto the identi* *ty components are C3-equivariant. Here, Aut(PGL (m, C)) = Zx2(or Zx2={ 1} if m = 2) since PGL* * (m, C) has ß*(N)-determined automorphisms by induction hypothesis so C3 must act trivially* * on the identity components for purely group theoretic reasons. The commutative triangle (4.14) ß0(H)N ~=mmmmmm NNN~=NN mmm NNNN vvmmmm &&N ß0(CPGL(2m,C)(H, ))___ß_________//ß0(CX0(H, 0)) 0(f ,L) in which the slanted arrows, representing the canonical factorizations, are C3-* *equivariant (even GL(H)-equivariant) shows that ß0(f ,L) is C3-equivariant. We shall next compute the higher limits from 3.3.(2) and 4.6.(4) by means of * *5.4 and the cochain complex 5.14 from [32]. As 5.4 is not valid for PGL (2, C) we first consider th* *is case separately. 6.3. Proposition. The 2-compact group PGL (2, C) is uniquely N-determined. Proof.The functor CPGL(2,C)takes the Quillen category of PGL (2, C), consisting* * (5.7, 5.13, 6.1) of one toral line, L, and one non-toral plane, H, __________________________________* *_____________________________ (6.4) L ___________//_HbbGL(H)__________________________* *________________________________________________________ to the diagram _____________________________* *__________________________________ (6.5) GL (1, C)2=GL (1, C) ooC2o_______H bb_GL(H)op___________________* *______________________________________________________________ of uniquely N-determined 2-compact groups. The 2-compact toral group to the lef* *t is uniquely N-determined (4.2) because H1(C2; Z=21 ) = 0 for the non-trivial action. The c* *enter functor takes this diagram back to the starting point (6.4) for which the higher limits* * vanish [29, 12.7.4]. PGL (2, C) is thus uniquely N-determined by 3.3 and 4.6. 6.5. Lemma. The low degree higher limits of the functors ßj(BZCPGL(n+1,C)), j =* * 1, 2, are: (1)limi(A(PGL (n + 1, C)), ß1(BZCPGL(n+1,C))) = 0 for i = 1, 2, (2)limi(A(PGL (n + 1, C)), ß2(BZCPGL(n+1,C))) = 0 for i = 2, 3, for all n 1. Let V = F2e1+ F2e2+ F2e3 be a 3-dimensional vector space over F2 with basis {* *e1, e2, e3} and (degenerate) symplectic inner product matrix 0 1 0 1 0 @1 0 0A 0 0 0 Let F2[1] be the 21-dimensional F2-vector space on all length one flags [P > L]* * and F2[0] the 14- dimensional F2-vector space on all length zero flags, [P ] or [L], of non-trivi* *al and proper subspaces of V . The Steinberg module St(V ) over F2 for V is the 23 = 8-dimensional_kern* *el of the linear map d: F2[1] ! F2[0]given by d[P > L] = [P_] + [L]. Define f1 = f1| St(V ): St(* *V )a!sVthe restriction to St(V ) of the linear map f1:F2[1] ! Vwith values ( __ L P \ P ?= 0 f1[P > L] = 0 otherwise on the basis vectors. Let E = F2e1 + F2e2 + F2e3 + F2e4 be a 4-dimensional vector space over F2 wit* *h basis {e1, e2, e3, e4} and (degenerate) symplectic inner product matrix 0 1 0 1 0 0 BB1 0 0 0CC @ 0 0 0 0A 0 0 0 0 2-COMPACT GROUPS 23 Let F2[2] be the 315-dimensional F2-vector space on all length two flags [V > P* * > L] and F2[1] the also 315-dimensional F2-vector space on all length one flags, [P > L] or [V* * > L] or [V > P ], of non-trivial, proper subspaces of E. The Steinberg module St(E) over F2 for E* * is the 26 = 64- dimensional kernel of the linear_map d: F2[2] ! F2[1]given by d[V > P > L] = [P* * > L] + [V > L]_+ [V > P ]. Define F1 = F1| St(E): St(E) !aEs the restriction to St(E) of th* *e linear map F1:F2[2] ! Ewith values ( __ L P \ P ?= 0, V \ V ?= F2e3 (6.6) F1[V > P > L] = 0 otherwise __ on the basis elements. Define F2 = F2| St(E): St(E) !sEimilarly but`replace'th* *e condition V \ V ? = F2e3 by V \ V ? = F2e4. The linear maps F1 and F2 are Sp(2)* 01-equ* *ivariant because this group preserves the symplectic inner product on E and preserves V * *? = F2 pointwise. 6.7. Lemma. Let f1 and F1, F2 be the linear maps defined above. (1)The vector f1 is a basis vector for Hom0 1(St(V ), V ) ~=Hom0 1(St(V ), V=V ?) ~=F2 @ Sp(2) 0A @ Sp(2) 0A * 1 * 1 (2)The set {F1, F2} is a basis for Hom0 1(St(E), E) ~=Hom0 1(St(E), E=E? ) ~=F22 @ Sp(2) 0A @ Sp(2) 0A * 1 * 1 The sum F1+ F2 is the linear map defined as in ( 6.6) but with condition * *V \ V ?= F2e3 replaced by V \ V ?= F2(e3+ e4). Proof.This can be directly verified by machine computation. 6.8. Proposition. The differentials in the cochain complex 5.14 are given as fo* *llows: (1)Let H be the (2, 0) object and V a (3, 1) object of A(PGL (2m, C)). The V* * -component of the coboundary map ffi1V:Hom Sp(2)(St(H), H) ! Hom 0 1(St(V ), V ) @Sp(2) 0A * 1 is an isomorphism of 1-dimensional F2-vector spaces. (2)Let V be the (4, 2) object of A(PGL (2m, C)) corresponding ( 5.12, 5.7) t* *o the two dimen- sional toral object (1, i - 1, m - i, 0) of A(PGL (m, C)), 1 < i m=2, m* * 4. Then ffi2E(xi) = (x1+ xi)F1+ (x1+ xi-1)F2 where Q ffi2E: 1 i m=2Hom0 1(St(Vi), Vi) ! Hom 0 1(St(E), E) @ Sp(2) 0A @Sp(2) 0A * 1 * 1 Q is the E-component of the coboundary map and (xi) 2 1 i m=2Hom Sp(3,1)(S* *t(V ), V ). Proof.1. The non-zero vector in Hom Sp(2)(St(H), H) is the restriction to St(H)* * F32of the linear map F2[0] = F32! H that takes a basis vector [L] in F32to L 2 H. In the composi* *tion M M + St(V ) ! St(P ) ! P -! V V >P V >P the middle maps St(P ) ! P equal the map just described if P < V is non-toral, * *P \ P ?= 0, and are trivial if P < V is toral, P \ P ?= P . This is precisely the map f1. 2. For any non-toral three dimensional subspace V of E we have either o V \ V ?= F2e3, and then V = Vi, or, o V \ V ?= F2e4, and then V = Vi-1, or, o V \ V ?= F2(e3+ e4), and then V = V1, 24 J.M. MØLLER and thus the composite linear map M L xi M + St(E) ! St(V ) ---! V -! E E>V equals xiF1+ xi-1F2+ x1(F1+ F2) = (x1+ xi)F1+ (xi-1+ x1)F2. Proof of Lemma 6.5.Since we already know that these higher limits vanish when n* *+1 is odd (5.4) we can assume that n + 1 = 2m is even. 1. In Oliver's cochain complex 5.14, the coboundary map ffi1 is injective and k* *erffi2 is 1-dimensional by 6.8 when m 4. See 6.3 for the case m = 1. For m = 2 and m = 3, the cochain* * complexes 5.14 reduce to 1 0 ! Hom Sp(2)(St(H),-H)ffi!HomSp(3,1)(St(V ),!V=V0?) 1 0 ! Hom Sp(2)(St(H),-H)ffi!HomSp(3,1)(St(V ),!V=VH?)om0 1(St(E), E=E? * *) = 0 @Sp(2) 0 A * GL (E? ) with two non-trivial groups, both 1-dimensional F2-vector spaces, and with just* * one differential ffi1 which is an isomorphism (6.8). Thus the higher limits vanish in these cases as * *well. 2. Oliver's cochain complex for computing these higher limits over A(PGL (2m, C* *)) involve the Z2-modules Hom0 1 (St(E), ß2(BZCPGL(2m,C)(E))), dimF2 E = 3, 4, @ Sp(2) 0 A * A(PGL (m, C))(E? that are submodules of finite products of Z2-modules of the form Hom 0 1(St(E), Z2), dimF2 E = 3, 4, @Sp(2) 0A * 1 where the action on Z2 is trivial. According to the computer program magma, the* *se latter modules are trivial. Proof of Theorem 1.1.By induction over n using 3.3 and 4.6. The start of the in* *duction is provided by 6.3. Use (2.7) to compute the automorphism group. Proof of Corollary 1.2.The connected 2-compact group GL (n, C) is uniquely N-de* *termined be- cause (3.2, 4.3) its adjoint form PGL (n, C) is (1.1). Since the maximal torus* * normalizer for GL(n, C) is a split extension, we get (2.7) that Aut(GL (n, C)) is isomorphic t* *o Z( n)\ AutZ2 n(Zn2). This finishes the discussion of the 2-compact groups in the A-family. The rel* *evance of these are that they occur as centralizers of elementary abelian subgroups of many other 2* *-compact groups. Here is a result illustrating this. 6.9. Theorem. [34, 1.3] The simple 2-compact group G2 is uniquely N-determined * *and its auto- morphism group Aut(G2) equals Zx\Zx2x C2. Proof.The Quillen category A(G2) is equivalent to the category A(GL (V ), V ) o* *f all non-trivial subspaces of V = F32[12, 6.1] [10, 1.6] [9, 5.3] and the value of centralizer f* *unctor BCG2 on the three isomorphism classes of objects L, P , V is SL(4, R), T o Z=2, V . The ran* *k one centralizer, SL(4, R) = SL(2, C)OSL (2, C), is uniquely N-determined (6.3, 3.2, 3.4, 4.3, 4.* *4). Condition 4.6.(2) is satisfied because H1(W (X); ~T(X)) = 0 for X = G2, SL(4, R) [13], 4.6.(1) an* *d 4.6.(3) because the only rank two object in G2 is toral and its centralizer is a 2-compact tora* *l group. The functor ß1(BZCG2) is the identity functor and ß2(BZCG2) the zero functor so the obstruc* *tion groups vanish. Now 3.3 and 4.6 show that G2 is uniquely N-determined. The short exact * *sequence (2.7) can be used to calculate the automorphism group. We have Aut(G2) = W (G2)\NGL(2* *,Z2)(W (G2)) as the extension class e(G2) = 0 [3]. Using the description of the root system * *from [2, VI.4.13] 2-COMPACT GROUPS 25 with short root ff1 = "1- "2 and long root ff2 = 2" - "2- "3 generating the int* *egral lattice in Z32 one finds that ` ' x ff p___ 0 3 NGL(2,Z2)(W (G2)) = Z2, A, W (G2), A = -3 1 0 and therefore Aut(G2) = Zx2=Zx x C2 where the cyclic group of order two is gene* *rated by the exotic automorphism A interchanging the two roots. 7.Miscellaneous This section contains auxiliary results that are used at various places in th* *e main argument of this paper. 7.1. The 2-compact toral groups O(2) and Pin(2). Let H = {a + bj|a, b 2 C}, whe* *re j2 = -1 and ja = _aj for a 2 C, be the quaternion algebra. The normalizer of Cx in Hx * *is the Lie group NHx (Cx) = generated by the multiplicative Lie group Cx and j. The* * short exact sequence 1 ! Cx ! NHx (Cx) ! = <-1>! 1 does not split for all elements of jCx have order 4. Its discrete approximation* * P~in(2) = ~NHx(Cx) = j Hx, the non-split extension 1 ! Z=21 ! ~NHx(Cx) ! Z=2 ! 1 of Z=2 by Z=21 , is the discrete approximation to 2-compact toral group Pin(2).* * The semi-direct product ~O(2) = Z=21 o Z=2 is the discrete approximation to the 2-compact toral* * group O(2) or to GL(2, R). 7.2. Type An, n 1. (Cf. [20, 19, 13]) The discrete maximal torus normalizer f* *or the center-less 2-compact group PGL (n + 1, C) = GL(n + 1, C)=Cx is the extended 2-discrete tor* *al group N~(PGL (n + 1, C)) = ~U(1)n+1=~U(1) o n+1 = ~To n+1 where ~U(1) = Z=21 is a discrete 2-torus of rank 1. In the coefficient sequen* *ce for ~U(1) ! ~U(1)n+1 ! ~Twe have H*( n+1; ~U(1)n+1) ~=H*( n; ~U(1)) by Shapiro so that i+1 1 res i+1 1 Hi(W ; ~T) ~=kerH ( n+1; Z=2 ) --!H ( n; Z=2 ) is trivial for n + 1 > 2(i + 1) by [30, 5.8, 6.7]. For small values of i we have ( ( 0 n 6= 1 1 0 n 6= 3 H0 W ; ~T= and H W ; ~T= Z=2 n = 1 Z=2 n = 3 as can be seen by using that the Schur multiplier H2( n; Z) is of order 2 for n* * 4 and trivial for 1 n 3 [16, V.25.12]. Thus the center ZN(PGL (n + 1, C)) of the maximal toru* *s normalizer is trivial for n > 1 but cyclic of order 2 for n = 1. For n = 3, the crossed h* *omomorphism 4 ! ~U(1)4=~U(1) whose values on the three generators (12), (23), (34) 2 4 [1* *6, I.19.7] are the columns of the matrix 0 1 -1 +1 +1 BB-1 -1 -1CC @+1 -1 -1A -1 -1 -1 is not principal. 7.3. Type Bn, n 2. (Cf. [20, 19, 13]) The discrete maximal torus normalizer f* *or the center-less 2-compact group SL(2n + 1, R) is the extended 2-discrete torus N~(SL(2n + 1, R)) = ~O(2) o n = (Z=21 o Z=2) o n = (Z=21 )n o (Z=2 o * * n) where Z=2 acts on Z=21 by sign. There is an isomorphism H1(Z=2 o n; (Z=21 )n) ~=Hom( n-1, Z=2) Hom( n, Z=2) ~=Z=2 Z=2 that to the pair (v, Ø) 2 Z=2 Hom( n, Z=2) associates the derivation D(v, Ø) * *given by D(v, Ø)("i, oe) = (v + Ø(oe), . .,.v + Ø(oe), Ø(oe), v + Ø(oe), . .,* *.v + Ø(oe)) 26 J.M. MØLLER where "iis the ith canonical basis vector for (Z=2)n and Ø(oe) is in the ith co* *ordinate. To see this, use the exact sequence from the Lyndon-Hochschild-Serre spectral sequence 0 ! H1( n; (Z=2)n) ! H1(Z=2 o n; (Z=21 )n) ! H1((Z=2)n; (Z=21 )n) n where H1( n; (Z=2)n) ~=Z=2 (n 3) and also the third term is of order 2 as, in* * general, H*(Gn; Mn) = H*(Gn; M)n = H*(Gn-1; H*(G; M)) = . .=.H*(G; . .;.H*(G; M) . .)* *.n for a group G and a G-module M. This gives ( H0(W ; ~T) = Z=2, H1(W ; ~T) = Z=2 n = 2 (Z=2)2 n 3 in this case. The computation of H0(W ; ~T) uses that the center of the maximal* * torus normalizer Z(N~(SL(2n + 1, R))) = Z(O~(2) o n) = ZO~(2) = Z=2 is cyclic of order two for all n 2 (whereas ~ZSL(2n + 1, R) = 0). 7.4. The center of a semi-direct product. Let Go be the semi-direct product fo* *r the action ! Aut(G) of the group on the group G. Let G = {g 2 G| g = g} and G = {oe * *2 |oe(g) = g for all g 2}G. 7.5. Lemma. The center Z(G o ) = G xAut(G)Z( ) of G o is the pull-back Z(G o )______//Z( ) | | | | fflffl| fflffl| G _______//Aut(G) of the action map restricted to the center of along the map G ! Aut(G) given* * by inner automorphisms. Proof.Suppose that (g, oe) 2 G x is in the center of G o . Since (g, oe) . (1, ø) = (g, oeø) = (1, ø) . (g, oe) = (ø(g), øoe) for all ø 2 , g is fixed by and oe is central in . Moreover, from (g, oe) . (h, 1) = (g . oe(h), oe) = (h, 1) . (g, oe) = (hg, * *oe) we see that oe(h) = hg for all h 2 G. 7.6. Corollary. If G is abelian, Z(G o ) = G x Z( )G is a direct product. Proof.The bottom horizontal homomorphism G ! Aut(G) is trivial. 7.7. Corollary. Let G be a group and Z 6= G a central subgroup. Let the cyclic * *group Cp of prime order p act on Gp=Z by cyclic permutation. Then Z(G)=Z x {z 2 Z|zp = 1} ~=Z(Gp=Z o Cp) via the isomorphism that takes the element z 2 Z of order p to (1, z, . .,.zp-1* *)Z 2 Gp=Z and is the diagonal on Z(G)=Z. Proof.Observe that ~ G=Z x {z 2 Z|zp = 1} =-!Gp=Z Cp via the isomorphism that takes (gZ, z) to g(1, z, . .,.zp-1)Z. To see this, co* *nsider an element (g1, . .,.gp)Z which is fixed by Cp. Then (g1, g2, . .,.gp)Z = (gp, g1, . .,.gp* *-1)Z so there exists an element z 2 Z so that g2 = g1z, g3 = g2z = g1z2, . .,.gp = g1zp-1, g1 = g1zp. T* *herefore, zp = 1 and (g1, g2, . .,.gp) = g1(1, z, . .,.zp-1). Thus Z(Gp=Z o Cp) is the pull back of the group homomorphisms G=Z x {z 2 Z|zp = 1} '-!Aut(Gp=Z) Cp where '(gZ, z)((g1, . .,.gp)Z) = (gg1, . .,.ggp)Z. Let ((gZ, z), oe) be an ele* *ment of the pull back. Assume that oe is non-trivial. Since p is a prime number, oe has no fixed point* *s. The equation 8g1, . .,.gp 2 G: (gg1, . .,.ggp)Z = (goe(1), . .,.goe(p))Z 2-COMPACT GROUPS 27 shows that gg1Z = goe(1)Z. This is impossible unless oe is the identity since o* *therwise we can find a g1 2 Z and a goe(1)62 Z. Thus the permutation oe must be the identity. The re* *quirement for ((gZ, z), 1) to be in the pull back is that 8(g1, . .,.gp) 2 Gp9u 2 Z :(gg1, gg2, . .,.ggp) = (g1u, g2u, . .,* *.gpu) which implies that [g1, g] = u = [g2, g] for all g1, g2 2 G. If we take g1 = 1 * *to be the identity, we see that g must be central. 7.8. Action in Lie case. Let :V ! Gbe a monomorphism of a non-trivial element* *ary abelian p-group to a compact Lie group G. There is a canonical map BCG( (V )) ! map (B* *V, BG)B from the classifying space of the Lie theoretic centralizer of (V ) to the map* *ping space component containing B . Write cg for conjugation with g 2 G. 7.9. Lemma. Suppose that ff = cg for some element g 2 G and some automorphism* * ff 2 GL(V ). Then conjugation by g takes CG( (V )) to CG(cg (V )) = CG( ff(V )) = CG( (V )) * *and the diagram BCG(O(VO))____//_map(BV, BG)B Bcg|~=| ~=(Bff)*|| | fflffl| BCG( (V ))____//_map(BV, BG)B is homotopy commutative. Proof.The commutative diagram of Lie group morphisms V x CG( (V ))_x1//_ (V ) x CG( (Vm))ult//_G ffxcg|| |||| fflffl| || V x CG( (V ))_x1//_ (V ) x CG( (Vm))ult//_G induces a commutative diagram B(multO( x1)) BV x BCG( (V ))_________//_BG BffxBcg|| |||| fflffl|B(multO( x1||)) BV x BCG( (V ))_________//_BG of classifying spaces. Taking adjoints, we obtain the homotopy commutative diag* *ram BCG(O(VO))____//_map(BV, BG)B Bcg|| (Bff)*|| | fflffl| BCG( (V ))____//_map(BV, BG)B as claimed. 7.10. Corollary. Suppose that ~: V ! N(G) is a monomorphism and that ~ff = cn~ * *for some ff 2 GL(V ) and n 2 N(G). Then w-1 = ß2((Bff)*): ß2(BT (G))ß0(~)(V!)ß2(BT (G))ß0(~)(V ) where w 2 W (G) is the image of n 2 N(G). Proof.There is a commutative diagram ~= ß2(BTO)______ß2(BN(G))oo_?ß2(BCN(G)(V,`~))___//ß2(mapO(BV,OBN),OB~)OO w|| |ß2(Bcn)| |ß2(Bcn)| |ß2((Bff)*)| | | | ~= fflffl| ß2(BT )______ß2(BN(G))oo_?ß2(BCN(G)(V,`~))___//ß2(map (BV, BN), B~) 28 J.M. MØLLER where ß2(BCN(G)(V, ~)) = ß2(BT (G))ß0(~)(Vd)enotes the fixed point group for th* *e group action ß0(~): V ! W (G) Aut(ß2(BT (G))). Since Bcn: BN ! BN is freely homotopic to * *the identity along the loop w 2 ß1(BN) its effect on the Zp[ß1(BN)]-module ß2(BN) is multipl* *ication by w. References [1]K. Andersen, J. Grodal, J. Mfiller, and A. 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