HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 KRZYSZTOF ZIEMIA'NSKI Abstract.A homotopy (complex) representation of a compact connected Lie group L at prime p is a map from BL into the p-completion of the classif* *ying space of the unitary group BU(n)^p. In this paper we give a partial clas* *sifica- tion of homotopy representations of SO(7) and Spin(7) at prime 2. Motivi* *a- tion for considering this problem is twofold: first, one may hope that i* *t would help to understand maps between classifying spaces. Secondly, constructi* *on of a homotopy representation of Spin(7) is a crucial step in the constructi* *on of a faithful representation of the 2-compact group DI(4) [14]. Let p be a prime and let L be a compact connected Lie group. A group P is p-toral if it is an extension of a finite p-group by a torus; a p-toral subg* *roup P L is p-stubborn if NL(P )=P is finite and contains no non-trivial normal p- subgroups. Let Op(L) (resp. Rp(L)) be a category of L-orbits L=P for p-toral P (resp. p-stubborn P ) and L-equivariant maps. By [7], the map (hocolimL=P2Rp(L)(EL xL L=P )^p)^p-! BL^p is a weak homotopy equivalence. The category Rp(L) contains a maximal object G=N, where N is a p-normalizer of a maximal torus T . Let N1 be a p-discrete approximation of N, i.e. its dense subgroup such that T \ N1 be a subgroup of * *all elements of T having a p-power order. By a version of Dwyer-Zabrodsky theorem [8] the map Rep(N1 , U(n)) 3 ' 7! (B')^p! [(EL x L=N)^p, BU(n)^p], where Rep(N1 , U(n)) := Hom (N1 , U(n))= Inn(U(n)), is an isomorphism. A rep- resentation ' of N1 is Rp(L)-invariant iff (B')^pdefines a homotopy compatible family of maps in limL=P2Rp(L)[EL xL (.), BU(n)^p]. Obviously if (B')^pextends to a homotopy representation of L (i.e. a map BL^p! BU(n)^p), then ' is Rp(L)- invariant; the question is whether or not an Rp(L)-invariant representations ex- tends (after completion) to a homotopy representation of L. The main result of * *the present paper is the following Theorem. Let p = 2. If L = Spin(7) or L = SO(7), then every Rp(L)-invariant representation of N1 extends to a homotopy representation of L. Throughout the whole paper G = SO(7), "G= Spin(7), ss : "G! G is an obvious projection and u 2 "Gis a non-trivial lift of unity in G. For any subgroup H G let "H:= ss-1(H) "G. Organization of the paper. Section 1 contains generalization of some elemen- tary facts from representation theory of finite groups onto locally finite grou* *ps (in particular onto discrete approximations of p-toral groups). In Section 2 we des* *cribe a general method of construction homotopy representations of compact Lie groups. 1 2 KRZYSZTOF ZIEMIA'NSKI Section 3 contains a classification of 2-stubborn subgroups of orthogonal and s* *ym- metric groups, and a construction of a full inclusion R2( n) ! R2(O(n)). The la* *st result simplifies calculations of higher limits on R2(O(n)). In Section 4 some * *facts about representations of discrete approximations of 2-stubborn subgroups of SO(* *7) and Spin(7) are provided. Section 5 contains specific computations of higher li* *mits of atomic functors. Finally, the main theorem is proven in Section 6. 1. Representations of countable locally finite groups A group is countable locally finite if it satisfies the following equivalen* *t condi- tions: o is countable and every finitely generated subgroup of is finite. o There is an ascending sequence of finite groups (1.1) {1} = (0) (1) (2) . . . S such that s (s)= . By a complex representation of a group we mean a homomorphism ' : ! GL (V ), where V is a finite dimensional complex vector space. A representation* * is unitary if V is equipped with a hermitian scalar product which is preserved by * *'(g) for each G 2 . Throughout this section denotes anyScountable locally finite group and (s)* *is a fixed sequence (1.1). Let Rep( ) := n Rep( , GL(Cn)) and let IR( ) Rep( ) be the set of isomorphism classes of irreducible representations of . The foll* *owing proposition provides a main reason for similarity of representation theory of l* *ocally finite groups with the finite case. Proposition 1.2. If ' : ! GL (Va)nd _ : ! GL (W )are representations, then there is s < 1 such that Hom (V, W ) = Hom (s)(V, W ). Proof.A sequence Hom (0)(V, W ) Hom (1)(V, W ) Hom (2)(V, W ) . . .Hom (V, W ) eventually stabilizesTsince the dimensions of these spaces do. Moreover, we ha* *ve Hom (V, W ) = 1r=0Hom (r)(V, W ). Thus Hom (V, W ) = Hom (s)(V, W ) for large enough s. Here follow immediate corollaries of Proposition 1.2: Proposition 1.3. Fix representations ' : ! GL (V ) and _ : ! GL (W ). (1) If ' is irreducible, then for some s res (s)' is irreducible. (2) If for all s res (s)' and res (s)_ are isomorphic, then ' and _ are iso- morphic. (3) (Schur lemma) If ' is irreducible, then End (V ) contains only homotetie* *s. If additionally _ is irreducible and non-isomorphic to ', then Hom (V, * *W ) = 0. Proof.Fix s such that End (V ) = End (s)(V ) and Hom (V, W ) = Hom (s)(V, W ) (1) If res (s)' is reducible, then there is a projection f 2 End (s)(V ) ont* *o one of its irreducible summands. Since f 2 End (V ) it denies irreducibility* * of '. HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 3 (2) If Hom (s)(V, W ) contains an isomorphism, then Hom (V, W ) also does. (3) By (1) and (2) it follows from the Schur lemma for finite groups. Proposition 1.4. [12] Every representation of is semisimple, and its decompo- sition into irreducible summands is unique up to permutation of summands. Proof.Let ' : L ! GL (V ) be any representation and fix s such that End (s)V = End (V ). Let iWi be a decomposition of res (s)' onto irreducible summands. All Wi's are irreducible -subrepresentationsL(since any (s)-projection V ! Wi* * is also L-homomorphism).LThen iWi is a decomposition of '. If iWi and jXj are decompositions of ', then there is a (s)-isomorphism M M f : V = Wi- ! Xj = V i j permuting irreducible (s)-summands. But f is also -isomorphism. A decomposition into irreducible summands can be obtained functorially. Pick a representative oe : ! GL (Woe) of every element of IR( ). Proposition 1.5. For any representation ' : ! GL (V ) the evaluation M ev Woe Hom (Woe, V ) -! V oe2IR( ) is an isomorphism. Furthermore, the homomorphism M M End (Hom (Woe, V )) 3 foe7! oe2IR( ) oe iM j 7! ev O IdWoe foe O (ev)-1 2 End (V ) oe is an isomorphism. L Proof.By Proposition 1.4 we can assume that V = oe2IR(W)oloee. By the Schur lemma Hom (Woe, Woeloe) ~=Hom (Woe, V ). Then the evaluation is a composition M M M Woe Hom (Woe, V ) ~= Woe Hom (Woe, Woeloe) ~= Woe Cloe= V. oe oe oe Again by the Schur lemma M End (V ) ' End (Woe Hom (Woe, V )). oe2IR( ) Moreover, End (Woe Hom (Woe, V )) = Hom (Woe Hom (Woe, V ), Woe Hom (Woe, V )) = Hom (Hom (Woe, V ), Hom (Woe, Woe Hom (Woe, V ))) = Hom (Hom (Woe, V ), Hom (Woe, Woe) Hom (Woe, V )) = End(Hom (Woe, V )). Let Ch( ) C be a vector subspace spanned by all characters of representa- tions (it does not contain all class functions). 4 KRZYSZTOF ZIEMIA'NSKI Proposition 1.6. If O, O0 2 Ch( ), then the sequence (O|O0)s := (O| (s)|O0| (s)) stabilizes. In particular, (O|O0) := lims!1(O|O0)s is a hermitian product on Ch( ) and characters of irreducible representations f* *orm an orthonormal basis of Ch( ). Proof.By Proposition 1.4 it is enough to prove that the sequnce (O|O0)s stabili* *zes for O = O', O0 = O_, where ', _ 2 IR( ). By 1.3.(1) for large enough s both res (s)' and res (s)_ are irreducible. Then if ' and _ are isomorphic, then (O'|O_)s = 1; if not (O'|O_)s = 0 (cf. 1.3.(2)). From Propositions 1.4 and 1.6 we obtain Corollary 1.7. Two representations of a given locally finite group are isomorph* *ic iff their characters are equal. The next proposition states that representations of products behave similarly* * to the finite case: Proposition 1.8. Let and be countable locally finite groups. The map ~ : IR( ) x IR( ) 3 (', _) 7! res x ' res x _ 2 IR( x ) is a bijection. Proof.If ' 2 IR( ), _ 2 IR( ) then for some s both representations res (s)' and res (s)_ are irreducible. Hence res x(s)x (s)' ~_ is irreducible and so is * *' ~_. Then the map ~ is well-defined. Now let ! 2 IR( x ). There is r such that for s r res (s)! is irreducible and then it is isomorphic to a tensor product of irreducible representations of* * fac- tors. Then both res x(s)x{1}! and res{x1}x (s)! are sums of pairwise isomorphic irreducible representations. As a consequence of Proposition 1.3.(2) we have res xx{1}! ' ' dim_, res{x1}x! ' _ dim' for some ' 2 IR( ), _ 2 IR( ). For each s r the characters of ! and ' ~_ are equal on (s)x (s)and hence they are equal on x . Now Corollary 1.7 implies that ! is isomorphic to ' ~_. Up to this point we have considered linear representations when Dwyer-Zabrosky theorem requires unitary ones. The following proposition states that any complex representation admits a unique unitary structure. Proposition 1.9. Let V be a complex linear space with a hermitian scalar produc* *t. Then the map Rep( , U(V )) ! Rep( , GL(V )) is a bijection. Proof.Consider the following (obviously commutative) diagram (lim stands for an inverse limit) Rep( , U(V ))__________wRep ( , GL(V )) | | (1.10) || || |u |u lims!1Rep( (s), U(V_))__slim!1Rep(w(s), GL(V )) HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 5 where the vertical maps assign to any representation the collection of its rest* *rictions to subgroups (s). The lower horizontal arrow is bijective since groups (s)are finite. For L = U(V ) or L = GL (V ) let {'s} 2 lims!1 Hom ( (s), L). Define homomorphisms _s 2 Hom ( (s), L) by induction. For s > 0 let _s(g)S= h-1's(g)h where h is any element such that _s-1 = h-1's| (s-1)h. Then _ = s _s is a well- defined homomorphism ! L such that _| (s)is conjugate to 's. This implies the surjectivity of both vertical maps. Let ', _ 2 Hom ( , U(V )) be non-conjug* *ate homomorphisms and put Gs = {g 2 U(V ) : g-1'gT= _}. Gs is a non-increasing sequence of closed subsets of U(V ). Since rGr = ; and U(V ) is compact, then there is s such that Gs = ;. Then '| (s)is not conjugate to _| (s)and the map Rep( , U(V )) ! limsRep( (s), U(Vi))s injective. Injectivity of the right verti* *cal arrow follows from Proposition 1.7. Now the conclusion follows. We conclude with a corollary from Propositions 1.5 and 1.9: Corollary 1.11. For any unitary representation ' : ! U(V ) we have Y CU(V )('( )) = U (V ) ~= U(Hom (Woe, V )). oe2IR( ) 2. Homotopy representations of compact Lie groups The tool which appears the most appriopriate for calculating the set of homo- topy representations of L is a subgroup homotopy decomposition due to Jackowski, McClure and Oliver [7]. Recall that holds the following Theorem 2.1 ([7, 1.4]). The map (hocolimL=P2Rp(L)EL xL L=P )^p' BL^p induced by projection is a weak homotopy equivalence. Notice that EL xL L=P is homotopy equivalent to BP . The next theorem describes the mapping space map (BP, BU(n)^p) if P is p-toral. Theorem 2.2 ([8, Thm. 1.1]). Let P be a p-toral group and let H be a compact connected Lie group. Then: (1) The maps Rep(P 1, H) -B![BP 1, BH^p] - [BP, BH^p] are bijections. (2) For any ' : P 1 ! H the pairing BCH ('(P 1)) x BP 1 ! BH induces a homotopy equivalence BCH ('(P 1))^p-! map(BP, BH^p)B' . Hence limL=P[EL xL (.), BU(n)^p] ' limL=PRep(P 1, U(n)). Let N L be a p-normalizer of a maximal torus and let ' : N1 ! U(V ) be an Rp(L)-invariant representation. For i > 0 define contravariant functors i from Rp(L) to the category of groups (abelian if i > 1) 'i(L=P ) := ssimap ((EL=P )^p, BU(n)^p)B'^p|(EL=P)^p. By the theorem of Wojtkowiak [13], the obstructions to an existence of an exten* *sion of (B')^pto (BL)^plie in groups Hi+1(Rp(L); i) for i 1. Luckily functors i 6 KRZYSZTOF ZIEMIA'NSKI can be interpreted in terms of irreducible representations of restrictions resN* *1P1' for p-stubborn P . For any ! 2 Rep ( , U(n)) let IR( , !) IR( ) be a subset of isomorphism classes of irreducible subrepresentations of !. Furthermore, let R( ) := Z[IR( * *)] and R( , !) := Z[IR( , !)]. Theorem 2.3. For each L=P 2 Rp(L) we have '1(L=P ) = '3(L=P ) = 0. Fur- thermore, there is a functorial isomorphism 1 '2(L=P ) ' Z^2 R(P 1, resNP1'). Proof.Let W be a1unitary space with a P 1-action which represents some element of Rep(P 1, resNP1'). Choose a monomorphism j : W ! V sand put 1 s (2.4) JP,': R(P 1, resNP1') 3 [_] 7! j*(lW ) 2 ss1(UP1 (V )), where lW 2 ss1(U(W )) is represented by a loop t 7! e2ssit. I. Maps JP,' does not depend on the choice of j since all inclusions W ! V sdetermine conjugate homomorphisms UP (W ) ! UP (V s). It also does0not depend on the choice of s since for s < s0the map UP (V s) ! UP (V s) induces an isomorphism on ss1. Let a : L=Q ! L=P be a morphism in Rp(L). Commutativity of a diagram 1 JP,' R(P 1, resNP1')_____wss1(UP1 (V )) | | a*| | | | |u 1 JQ,' |u R(Q, resNP1')______wss1(UQ1 (V )) implies that J' is a natural transformation. Moreover, both groups R(P 1, resN1* *P1') 1 and ss1(UP (V )) are free and abelian, the generator set of them both is IR(P 1* *, resNP1') and the generators are mapped to the corresponding generators. Therefore J' : R(-, ') ! ss1Aut (-)(V ) is a natural equivalence. Thus the composition Z^p R(-, ') Id-J'---!Z^p ss1(U(-)(V )) ~=Z^p ss1(CU(V )('(-))) ~= ^ ^ -! Zp ss2(BCU(V )('(-))) -! ss2(BCU(V )('(-)))p ~= ^ ^ ' -! ss2(map (EL=(-), BU(V )p)B'|P)p = 2(L=P ). (2.2) is a required natural equivalence. Furthermore, by 1.11 UP1 (V ) = CU(n)('(P 1)) is a product of unitary groups and hence ss0(Aut P1(V )) = ss2(Aut P1(V )) = 0. For any Rp(L)-invariant representation ' of N1 denote (') := Z^p R(-, '). The following is an immediate consequence of 2.3: Theorem 2.5. Let ' : N1 ! U(V ) be an Rp(L)-invariant representation. If H3(Rp(L); (')) = 0 and Hi(Rp(L); M) = 0 for i > 4 and any contraviariant functor M : Rp(L) ! Z^p-Mod , then B'^2extends to a map BL ! BU(m)^p . 3. Stubborn subgroups of symmetric and orthogonal groups In this section we recall results of Oliver [9], who gave the classification * *of 2- stubborn subgroups of orthogonal groups and the description of the categories HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 7 R2(O(n)). Furthermore, we recall the classification of 2-stubborn groups of sym- metric groups (given by Alperin and Fong [1]) and describe the categories R2( n* *). The section is concluded with a constuction of a full inclusion R2( n) ! R2(O(n* *)) which makes easier calculations of higher limits on R2(O(n)). 2-Stubborn subgroups of orthogonal groups. Each 2-stubborn subgroup of an orthogonal group is built up out of some number of "pieces" by taking products * *and wreath products. (Recall that the wreath product G o H is the semi-direct produ* *ct GH n H, where H acts on GH by shifting coordinates). Let In 2 GL n(R) denote the identity matrix. If M 2 GL m(R) and N 2 GL n(R), then M N 2 GL nm(R is the matrix with entries (M N)am+b-m,cm+d-m = Mb,dNa,c. for a, c = 1, . .,.n, b, d = 1, . .,.m. Let ` ' ` ' (3.1) A := 10 -01 , B := 01 10 , and for i < n define the matrices Ani, Bni2 O(2n) by (3.2) Ani:= I2i A I2n-i-1, Bni:= I2i B I2n-i-1. Let (3.3) 2n := <-I2n, An0, . .,.Ann-1, Bn0, . .,.Bnn-1> O(2n) (3.4) ~ 2n:= <{X I2n-1}X2SO(2), An0, . .,.Ann-1, Bn0, . .,.Bnn-1> O(2n) The group 2n is an extra-special group of order 22n+1 and ~ 2nis a 2-toral gro* *up with 1-dimensional torus. Note that 1 ~={ 1} = O(1) and ~ 2~=O(2). Definition 3.5. ([9, Def. 2]) Let T (k) Tirr(k) Tprod(k) be the sets of 2-t* *oral subgroups of O(k) defined as follows: o T (1) = { 1 = O(1)}, o T (2) = {~ 2= O(2)}, o T (2n) = { 2n, ~ 2n} for n > 1, o T (k) = ; for k 6= 2n, o Tirr(2n) is the set of those wreath products in O(2n) of the form o Ct12o . .o.Cts2, where 2 T (2l), n = l + t1 + . .+.ts and t1 > 1 if = 1, o Tprod(k) is the set of all products in O(k) of the form P = P1 x . .x.Ps O(k1) x O(k2) x O(ks) O(k), where Pi2 Tirr(ki) and k = k1 + . .+.ks. Remark. If O(2l), then the embedding o Ct2 O(2l+t) is chosen as follows: t l+t l+t l+t l+t o Ct2~=< 2 , Bl , Bl+1, . .,.Bl+t-1> O(2 ). By [9, Theorem 8] each 2-stubborn subgroup of O(k) is conjugate to a subgroup in Tprod(O(k)). Possibly not all elements 2 Tprod(O(k)) are 2-stubborn _ it depends on the Weyl group of . The following proposition follows immediately from [9, Theorem 6]: Proposition 3.6. Let WL(P ) := NL(P )=P . Then 8 KRZYSZTOF ZIEMIA'NSKI o WO(2n)( 2n) ' O+ (Xn), o WO(2n)(~ 2n) ' Sp(Vn), o WO(2n)( o Ct12o . .o.Cts2) ' WO(2m )( ) x GLt1(F2) x . .x.GLts(F2), 2 T (2, O(2m )). The following theorem determines which elements of Tprod(k) are in fact 2- stubborn subgroups of O(k): Theorem 3.7. ([9, Theorem 6]) Let P 2 O(k) be a 2-stubborn subgroup. Then P is conjugate to an element of Tprod(k). If P 2 Tprod(k) then P is 2-stubborn if* * and only if when written as a product P = P1 x . .x.Ps, (Pi2 Tirr(ki)) there is no factor Pi with WO(ki)(Pi) = 1 which occurs with multiplicity exactl* *y 2 or 4. Theorem 3.8. ([9, Proposition 9]) Let P, P 02 Tprod(k). If P 0is conjugate to a subgroup of P , then x-1P 0x P for some permutation matrix x which permutates irreducible factors of P 0. If P 0 P then the inclusion is a composite of prod* *ucts of the following types: (a)O(1) x O(1) O(2), (b)O(1) o C2t12o Ct22o . .o.Cts2 O(2) o Ct12o Ct22o . .o.Cts2, (c)( o Ct12o . .o.Cts2)ts+1 o Ct12o . .o.Cts2o Cts+12, (d) 2ko Ct12o . .o.Cts2 ~ 2ko Ct12o . .o.Cts2, (e) o . .o.Cti2o Cti+12o . .o.Cts2 o . .o.Cti+ti+12o . .o.Cts2, (f) 2k+t1o Ct22o . .o.Cts2 2ko Ct12o Ct22o . .o.Cts2, (g)~ 2k+t1o Ct22o . .o.Cts2 ~ 2ko Ct12o Ct22o . .o.Cts2, where stands for either 2k or ~ 2k. Corollary 3.9. Each morphism in R2(O(n)) is a composition of automorphisms and inclusions enlisted in 3.8. 2-Stubborn subgroups of symmetric groups. The following classification of 2- stubborn subgroups of symmetric groups is due to Alperin and Fong [1] (a subgro* *up of a finite group is 2-stubborn iff it is 2-radical). Note that if G m , H n, then the product G x H is a subgroup of m+n and the wreath product G o H is a subgroup of mn . Definition 3.10. For any sequence t1, . .,.ts of positive integers let B(t1, . .,.ts) := 1 o Ct12o . .o.Cts2 2t, where t = t1 + . .+.ts (we treat 1 as a subgroup of 1). The groups B(t1, . .,.* *ts) will be called basic subgroups of 2t. The set of all basic subgroups of 2twil* *l be denoted by Birr(2t). Definition 3.11. Let Bprod(n) denotes the family of all products of basic subgr* *oups in n, i.e. Bprod(n) = {P1 x . .x.Pr : Pi2 Birr(2ti), n = 2t1+ . .+.2tr}. Here follow two propositions which are consequences of [1, (2B)]: Proposition 3.12. If t = t1 + . .+.ts, then W 2t(B(t1, . .,.ts)) ' GL t1(F2) x . .x.GLts(F2). HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 9 Proposition 3.13. Let Pi 2ti, for i = 1, .P.,.r, be a collection of pairwise non-isomorphic basic subgroups, and let k = ri=12tili. Then _ r ! Y Yr W k Pili = W 2ti(Pi) o li. i=1 i=1 Theorem 3.14. Each 2-stubborn subgroup G k is, up to conjugacy, a product of basic subgroups (i.e. G 2 Bprod(k)). A group P 2 Bprod(k) is stubborn if and only if written as a product of basic subgroups P = P1 x . .x.Pr there is no fa* *ctor isomorphic to B(1, . .,.1) which occurs with multiplicity exactly 2 or 4. Proof.The first statement is a consequence of [1, (2A)]. Since GL n(F2) has no * *non- trivial normal 2-subgroups and n has a non-trivial normal 2-subgroup if and on* *ly if n = 2 or n = 4, then the second statement follows immediately from Propositi* *ons 3.12 and 3.13. Proposition 3.15. If groups P, Q 2 Bprod(k) are conjugate, then there exists a conjugacy between them which permutes its basic factors. Proof.Since each basic subgroup of a symmetric group acts transitively on the set of letters, basic factors of P (and, similarly, Q) are in bijection with th* *e set of P -orbits (Q-orbits). The conjugacy between P and Q permutes the orbits and therefore it also permutes its basic factors. Proposition 3.16. Fix collections of subgroups Pi ki for i = 1, . .,.r and Hj ljfor j = 1, . .,.s. Assume that for each i the group Gi acts transitivel* *y on the set of letters, and that n := k1 + . .+.kr = l1 + . .+.ls. If Q := Q1 x . .x.Qs P := P1 x . .x.Pr n, then Q = (Q \ P1) x . .x.(Q \ Pr). Proof.ForSeach i = 1, . .,.r let OPibe an orbit of Pi P . Note that {1, . .,.k* *} = r P Q i=1Oi is a decomposition onto G-orbits. Similarly define Q-sets Oj , for j = 1, . .,.s. Since Q P , then for each j there exists i such that OQj OPi. Hen* *ce Qj P \ OPi= Pi and the conclusion follows. Proposition 3.17. For any subgroup P k let ffi(P ) P denotes the subgroup generated by all elements g 2 P which have a fixed point. The following holds: o if P 2 Bprod(k) is a non-trivial product of basic subgroups, then ffi(P * *) = P . o ffi(B(t1, . .,.tr)) = B(t1, . .,.tr-1)2tr, Proof.The product Q x Q0is generated by Q x {1} [t{1} x Q0. It implies that the first statement holds, and that B(t1, . .,.tr-1)2 r ffi(B(t1, . .,.tr)). On th* *e other hand, for Ctr2 2trwe have ffi(Ctr2) = {1}. Then each element g 2 B(t1, . .,.t* *r) \ B(t1, . .,.tr-1)2tracts freely on the set {1, . .,.2t} (where t = t1 + . .+.tr)* *. As a consequence we obtain the second statement. Theorem 3.18. Let P, Q 2 Bprod(k). Assume that Q P . Then the inclusion Q P is a composite of products of inclusions of the following types: (a)B(t1, . .,.tr-1)2tr B(t1, . .,.tr-1, tr), (b)B(t1, . .,.tj+ tj+1, . .,.tr) B(t1, . .,.tj, tj+1, . .,.tr). 10 KRZYSZTOF ZIEMIA'NSKI Proof.If Q is reducible, then by Proposition 3.16 the inclusion is the product * *of in- clusions (Q\Pi) Pi, where Pi's are irreducible, so assume that P = B(t1, . .,* *.tr). If Q is reducible, then by Proposition 3.17 we obtain tr Q = ffi(Q) ffi(P ) = B(t1, . .,.tr-1)2 B(t1, . .,.tr-1, tr) = P. In this case the inclusion is the composition of an inclusion of type (a) with Q ffi(P ). Finally, if Q is irreducible then Q = B(t01, . .,.t0s). We have t0s 2tr ffi(Q) = B(t01, . .,.t0s-1)2 B(t1, . .,.tr-1) = ffi(P ). Since Q-orbits are contained in P -orbits, then t0s tr. If t0s= tr, then we a* *re reduced to the case of smaller inclusion of irreducible subgroups. If t0s> tr, * *then the inclusion t0-t B(t01, . .,.t0s-1)2 s r B(t1, . .,.tr-1) factors through B(t1, . .,.tr-2)2tr-1. Hence t0s- tr tr-1 and then t0s tr-1+* * tr. Finally, we obtain the factorization Q B(t1, . .,.tr-2, tr-1 + tr) B(t1, . .,.tr-2, tr-1, tr) = G. As a consequence we obtain Corollary 3.19. Each morphism in R2(O(n)) is a composition of automorphisms and inclusions enlisted in 3.18. A full inclusion R2( n) ! R2(O(n)). Definition 3.20. For any P 2 Bprod(n) let ~P2 Tprod(n) be given by P~= { 1} o Ct12o . .o.Ctr2 for P = B(t1, . .,.tr), t1 > 1 ~P= O(2) o Ct22o . .o.Ctr2 for P = B(1, t2, . .,.tr) P~= ~P1x . .x.~Pr for Pi2 Birr(ki) Remark. For each P 2 Bprod(n) holds (3.21) ~P\ { 1} o k = { 1} o P. Theorem 3.22. The formulae R2( n) 3 n=P 7! O(n)=P~2 R2(O(n)) Mor R2( n)(Q, P ) 3 gP 7! gP~2 MorR2(O(k))(Q~, ~P) define the functor I : R2( n) ! R2(O(n)) which is an inclusion onto the full subcategory. Proof.The functor I is well-defined. It is sufficient to check that for each generating morphism gP : Q ! P holds g-1Q~g P~. It is clear for automorphisms (cf. Propositions 3.12 and 3.13), * *so assume that g = 1 and Q ! P is a product of the inclusions enlisted in Theorem 3.18. If the inclusion Q P is a non-trivial product of inclusions Q1 P1 and Q2 P2, then ~Q ~Pif and only if ~Q1 ~P1and ~Q2 ~P2. Hence we are reduced to the case when the inclusion is of type (a) or type (b) (cf. 3.18). If tr Q = B(t1, . .,.tr-1)2 B(t1, . .,.tr-1, tr) = P, then for t1 > 1 we obtain ~Q= ({ 1} o Ct12o . .o.Ctr-12)2tr { 1} o Ct12o . .o.Ctr2= ~P HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 11 and for t1 = 1 ~Q= (O(2) o Ct22o . .o.Ctr-12)2tr O(2) o Ct22o . .o.Ctr2= ~P. If Q = B(t1, . .,.tj+ tj+1, . .,.tr) B(t1, . .,.tj, tj+1, . .,.tr) = P, then for j > 1 we obtain the inclusion ~Q= K o Ct22o . .o.Ctj+tj+12o . .o.Ctr2 K o Ct22o . .o.Ctj2o Ctj+12o . .o.Ct* *r2= ~P, where K = O(2) if t1 = 1 and K = { 1} o Ct12otherwise. Similarly if j = 1 and t1 > 1, then the inclusion Q~ P~is straightforward. The only non-trivial case appears when j = t1 = 1. Then Q~= { 1} o C1+t22o . .o.Ctr2 O(2) o Ct22o . .o.Ctr2= ~P, since { 1} o C1+t22 { 1} o C2 o Ct22 O(2) o Ct22. Hence I is well-defined. The functor I is faithful. By combining Propositions 3.6, 3.12 and 3.13 we see that for each subgroup P 2 Bprod(k) the homomorphism I : AutR2( k)(P ) ! AutR2(O(k))(P~) is actually an isomorphism. Now fix Q 6= P 2 Bprod(k) and choose morphisms g1P, g2P : Q ! P in the category of k-orbits. Let us consider the compositions Q giP--!g-1iQgi-1.P-!P for i = 1, 2. By Proposition 3.15 g-11Qg1 and g-12Qg2 differ by conjugation by * *an element h which permutes irreducible factors. Hence the conjugation by i(h), wh* *ere i : k ! O(k) is an obvious inclusion, sends the group g-11~Qg1 onto g-12~Qg2 a* *nd also permutes irreducible factors. By (3.21) h 2 P if and only if i(h) 2 ~P. It* * shows that g1P and g2P represent the same morphism Q ! P in R2( k) if and only if they represent the same morphism in R2( k). As a consequence we get that I is an isomorphism on sets of morphisms. 4. 2-Stubborn subgroups of G, "Gand their representations 2-Stubborn subgroups of O(7). By [9, Prop. 11], [9, Th. 12] and [7, Prop. 1.6.(i)] the functors (4.1) R2(O(7)) 3 O(7)=P 7! G=(P \ G) 2 R2(G) R2(G) 3 G=P 7! "G=P"2 R2(G") are natural equivalences. We have Tirr(1) = {{ 1}}, Tirr(2) = {O(2)}, Tirr(4) = {O(2) o C2, { 1} o C22, ~ 4, 4}. Therefore (4.2) Tprod(7) = {H x O(2) x { 1}, H x { 1}3}H2Tirr(4) [ {O(2)ix { 1}7-2i}i=0,...,3. All groups in Tprod(7) but O(2)2 x { 1}3 are 2-stubborn in O(7) (by 3.7). In- troduce the following notation for 2-stubborn subgroups of G and "G. Let N := 12 KRZYSZTOF ZIEMIA'NSKI O(2) o C2, K := { 1} o C22, J := ~ 4and M := 4, and for each H 2 {J, K, M, N} let (4.3) H1 := (H x O(2) x { 1}) \ G (O(4) x O(2) x O(1)) \ G G (4.4) H0 := (H x { 1}3) \ G (O(4) x O(3)) \ G G, and for i = 0, 1, 3 let (4.5) Li:= (O(2)ix { 1}7-2i) \ G G. Then Ob(R2(G)) = {G=P } and Ob(R2(G")) = {G"=P"}, where P is conjugate to one of Ji Ki, Mi, Ni (i = 0, 1) or Li (i = 0, 1, 3). Here follows the list of Weyl groups of 2-stubborn subgroups of G: (4.6) WG (N1) = 1, WG (N0) ~=WG (L3) ~=WG (K1) ~=WG (J1) ~= 3 WG (L1) ~= 5, WG (L0) ~= 7, WG (K0) ~=WG (J0) ~= 3 x 3 WG (M1) ~= 3 o 2, WG (M0) ~= 3 o 2 x 3 The set of morphisms of R2(O(7)) ~=R2(G) ~=R2(G") is generated by automor- phisms and by inclusions presented on the following diagram: N1u[^ A AACaeo|[ A A ae | [ A A ae | L3u N0 K1 J1 u [[AC^ [[^ u | aeoA|A Aaeo aeo | | aeA | ae[ ae[ | | aeAA | ae ae | L1u K0 [^ J0u M1 | aeo [ | aeo | ae [ | ae | ae | ae L0 M0 Representations. The remaining part of this section contains a partial classifi* *ca- tion of complex representations of discrete approximations of 2-stubborn subgro* *ups of G and "G. At the beginning let us introduce some notation. Let ` denote a tr* *ivial irreducible representation of any group and let ' be a non-trivial irreducible * *repre- sentation of an order 2 group. Every irreducible representation of a 2-discrete* * torus Z=21 ~=SO(2)1 is 1-dimensional and has the form ` ' (4.7) %k : Z=21 3 n_2t7! exp 2ssikn_2t2 U(1), k 2 Z^2 Finally, let 1 ^ (4.8) ffk:= indO(2)SO(2)1%k for k 2 1 + Z2 1 ^ (4.9) fik:= indO(2)SO(2)1%k for k 2 2Z2 \ {0}. We have IR(O(2)1 ) = {ffk}k21+Z^2[ {fik}k22Z^2\{0}[ {`, o := det}. HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 13 Odd representations of subgroups of "G. Let be any locally finite subgroup of G and let " := ss-1( ). An irreducible representation ' of " is o even if it is a restriction of some representation of (and O'(u) = dim* *') o odd if it is not (in this case O'(u) = - dim') The following lemma will be applied later to groups M"0and "L0: Proposition 4.10. Let 2 O(4) be a locally finite group and let ":= ss-1(( x { 1}3) \ G) "G. Then any odd representation of " is invariant under conjugation by an element l 2 "Gsuch that ss(l) = -(I5 B) (cf. 3.1). Proof.Let ' be an odd representation of " and let g 2 ". We have ss(g) = (h, t), where h 2 O(4), t 2 { 1}3. If t = (1, 1, 1) or t = (-1, -1, -1), then g and l commute and hence Ol*'(g) = O'(l-1gl) = O'(g). Otherwise l-1gl is conjugate to l-1glu and g is conjugate to gu. Since ' is odd, we have O'(g) = O'(gu) = -O'(g* *). Therefore O'(g) = 0 (and similarly O'(l-1gl) = 0). Then representations ' and l*' have equal characters and they are isomorphic. Representations of L0 and L11. Let r be an odd positive integer. For each sequence (~1, . .,.~r) 2 IR({ 1})r define r (4.11) o(~1,...,~r):= res{{1} 1}r\SO(r)~1~ . .~.~r. L Let jri:= o(~1,...,~r), where the sum is taken over all sequences in which ' * *appears exactly i times. Obviously jri~ jrr-i. Then each r-invariant representation * *of { 1}r \ SO(r) (where r acts by permutations) is isomorphic to a direct sum of jri's for 0 i < r_2. In particular, holds Corollary 4.12. Any WG (L0) ' 7-invariant representation of L0 is isomorphic to a direct sum of j7i's for i = 0, 1, 2, 3. An isomorphism O(2)1 x ({ 1} \ SO(5)) 3 (g, h) 7! (g, h . detg) 2 L11 provides a bijection IR(L11) ' IR(O(2)1 ) x IR({ 1}5 \ SO(5)); moreover, the action of 5 ' WG (L1) on IR(L11) coincides with an action on IR({ 1}5\SO(5))by permutations. Thus every 5-invariant representation of L11 is isomorphic to a product of representations of the form ~ ~j5i, where ~ 2 IR(O(2)1 ), i = 0, 1, * *2. L Next we calculate restrictions of representations of L11 to L0. Let j7i(j):= o(~1,...,~7), where the sum is taken over all sequences such that exactly j of the representations ~1, ~2 are isomorphic to ' and exactly i of the representat* *ions ~1, . .,.~7 are isomorphic to '. Elementary calculations of characters provide * *the following Corollary 4.13. 5-invariant representations of L11 restrict to L0 as follows: ffk~ j507! j71(1)` ~j507! j70 o ~j507! j72(0) ffk~ j517! j72(1)` ~j517! j73(2)o ~j517! j71(0) ffk~ j527! j73(1)` ~j527! j72(0)o ~j527! j73(0) 1 L1 Moreover, resL1L0fi2k~j5i' res1L0(` o) ~j5i. 14 KRZYSZTOF ZIEMIA'NSKI Representations of M0 and M11. Isomorphisms M x O(2)1 3 (g, h)7! (g, h, deth) 2 M11 G \ (O(4) x O(2) x O(1)) M x { 1}2 3 (g, h1, h2)7! (g, h1, h2, h1h2) 2 M0 G \ (O(4) x O(1)3) provide an identifications IR(M11) ' IR(M x O(2)1 ) ' IR(M) x IR(O(2)1 ) and IR(M0) ' IR(M x { 1}2) ' IR(M) x IR({ 1}2). Furthermore, the restriction from M11 ' M x O(2)1 to M0 ' M x { 1}2 is product-wise. Representations of K0. All irreducibleNrepresentations of K ' { 1}oC22are sub- representations of indK{ 1}4~ a2C22~a for ~a 2 IR({ 1}) = {`, '}. By Mackey's criterion representations fl1 := indK{ 1}4` ~` ~` ~' fl3 := indK{ 1}4` ~' ~' ~' are irreducible. Furthermore, For ~, 2 IR(C2) = {`, '} define 2 ~, C2 fl~,0:= resC2K(~ ~ ), fl4 := res2K(~ ~ ) det. There are decompositions 2 M ~, ind{{1}oC2`1}4~` ~`'~` fl0 ~, 2IR(C2) 2 M ~, ind{{1}oC2'1}4~' ~''~' fl4 . ~, 2IR(C2) For any a 2 C22\ {(0, 0)} let ia := ` ~` ~' ~' 2 IR({ 1}4) (where a is the difference between coordinates with the same isomorphism4class of representatio* *n). Following [11, Section 8.2] we see that ind{{1} n1}4ia splits onto the sum o* *f non- isomorphic one-dimensional representations i+aand i-a. Moreover, for ffl 2 {+, * *-} representations flffl2,a:= indK{ 1}4nifflaare irreducible. Obviously indK{ * *1}4ia ' fl+2,a fl-2,a. As a consequence we obtain Corollary 4.14. IR(K) = {fl1, fl3} [ {fl~,0, fl~,4}~, 2IR(C2)[ {flffl2,a}ffl2{+,-}a2C2* *2\{(0,0)}. Since the action of WO(4)(K) ~=GL 2(F2) ' 3 on K is natural, we have Corollary 4.15. Here follow the orbits of an action of WO(4)(K) on IR(K): {fl`,`0}, {fl`,'0, fl',`0, fl','0}, {fl`,`4}, {fl`,'4, fl',`4, fl','4}* *, {fl1}, {fl3} {fl+2,(0,1), fl+2,(1,0), fl+2,(1,1)}, {fl-2,(0,1), fl-2,(1,0), fl* *-2,(1,1)} Finally we show how WG (K0)-invariant representations of K0 restrict to L0. An isomorphism K x { 1}2 3 (g, h1, h2) 7! (g, h1, h2, h1h2) 2 K0 G \ (O(4) x O(1)3) identifies IR(K0) and IR(K) x IR({ 1}2) (and obviously an action of WG (K0) ' 3 x 3 is product-wise).LDenote oe := ` ~' ' ~` ' ~' 2 Rep({ 1}2). Finally, let j7i[j]:= o(~1,...,~7), where the sum is taken over all sequences such tha* *t ex- actly j of the representations ~1, ~2, ~3, ~4 are isomorphic to ' and exactly i* * of the representations ~1, . .,.~7 are isomorphic to '. By comparing characters we obt* *ain HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 15 Corollary 4.16. WG (K0)-invariant representations of K0 restrict to L0 as follo* *ws: fl~,0~`7! j70 fl~,0~oe7! j72[0]fl~,4~`7! j73[0]fl~,4~oe7! j71[0] fl1~ `7! j71[1] fl1~ oe7! j73[1]fl3~ `7! j73[3] fl3~ oe7! j72[1] flffl2,a~`7! j72[2]flffl2,a~oe7! j73[2] for any ~, 2 IR({ 1}), a 2 C22\ {0, 0}, ffl 2 {+, -}. 5. Cohomology of Z^2[R2(L)]-modules By an R[C]-module, where R is a commutative ring R and C is a small category, we mean a contravariant functor M : C ! R - Mod and Hn(C; M) stands for a higher limit limnCM. Let A be a ring of p-adic integers and let be a finite group. For any A[ ]-module M let FM be an atomic Op( )-module with value M concentrated on =1. Following [7, 5.3] define n( ; M) := Hn(Op( ); FM ). An importance of groups * comes from the following Theorem 5.1. ([7, 5.4]) Let L be a compact Lie group and M an atomic A[Rp(L)]- module concentrated on an object L=Q. Then H*(Rp(L); M) ~= *(Aut Rp(L)(L=Q); M(L=Q)) = *(WL(Q); M(L=Q)). Jackowski, McClure and Oliver [7] provided the following inductive method of calculation of groups *( , M). Proposition 5.2. Let M be an A[ ]-module. Then (1) [7, 6.1.(i)] If p divides | |, then 0( ; M) = 0. Otherwise 0( ; M) = M and i( ; M) = 0 for i > 0, (2) [7, 6.2.(ii)] Let p be a Sylow p-subgroup of and let ~ be the equival* *ence relation among p-Sylow subgroups generated by nontrivial intersection. S* *et := {g 2 : g-1 pg ~ p}. Then 1( ; M) ~=M =M . (3) [7, 5.2.(ii)] Define A[ ]-modules F0M( =P ) := MP and F00M( =P ) := F0M=* *FM . Then Hi(Rp( ); F0M) = 0 for i > 0 and H0(Rp( ); F0M) = M . As a con- sequence, i( ; M) = Hi(Rp( ); FM ) ' Hi-1(Rp( ); F00M) ' Hi-1(Rp( ) \ { =1}; F00M) for i > 1 (Rp( ) \ { =1} stands for a full subcategory with =1 omitted). If an A[Rp(L)]-module M is not atomic, then there is a spectral sequence converging to H*(Rp(L); M). Choose a strictly decreasing map of posets ht : Ob(Rp(L)) ! Z. Here follows reformulation of [6, 1.3]. Theorem 5.3. There is a spectral sequence E*,**:= E(Rp(L), M)*,**with the first term M Es,t1:= s+t(WL(P ); M(L=P )) ht(L=P)=s converging to H*(Rp(L); M). A differential ds,t1: Es,t1! Es+1,t1is a sum M s,t ds,t1= d1 (P, Q), ht(L=P)=s, ht(L=Q)=s+1 16 KRZYSZTOF ZIEMIA'NSKI where ds,t1(P, Q) : s+t(WL(P ); M(L=P )) ! s+t+1(WL(Q); M(L=Q)) is a differ- ential from a long exact sequence associated to a short exact sequence 0 -! M|{Q} -! M|{P,Q}-! M|{Q} -! 0, The module M|X , for X Ob(Rp(L)) is ( (5.4) M|X (L=P ) = M(L=P ) if L=P 2 X 0 if L=P 62 X and M|X (L=P ! L=Q) = M(L=P ! L=Q) if L=P, L=Q 2 X (of course, this definition is valid only for subsets X which are convex, i.e. for any sequence * *L=P ! L=Q ! L=R such that L=P, L=R 2 X also L=Q 2 X). Proposition 5.5. If ht(L=P )+1 = ht(L=1) = n, then dn-1,t1(P, 1) is a compositi* *on n-1+t(W (P ); M( =P )) -! n-1+t(W (P ); M( =1)P ) -! Hn-1+t(Op( ); F00M( =1)) ' n+t( ; M( =1)) Proof.A homomorphism M ! F0M( =1)which is an identity on =1 induces a commutative diagram ds,t1(P,1) n-1+t(W (P ); M( =P ))______wn+t( ; M( =1)) ||||||||| | | |||||||||||||||| | ||||||||||||| | | |||||||||||||||| |u |||||||| n-1+t(W (P ); M( =1)P )____w n+t( ; M( =1)) The conclusion follows. For any group let dimp( ) be the greatest integer n such that n( ; M) 6= 0 for some A[ ]-module M. By [7, p. 229] dim p( ) is less or equal to the rank of p-Sylow-subgroup of . An application of a spectral sequence 5.3 to an A[Rp( )]- module F00Mshows that (5.6) dimp( ) 1 + max dim p(W (P )). =P2Rp( ) The remaining part of the section contains calculations of groups *( ; M) for some groups and modules M for p = 2. Cohomology of R2( 3). Note that J := R2( 3) has two objects, namely 3=1 and 3=C2. By 5.2 and 5.6 we have ( 2=M 3 for n = 1 (5.7) n( 3; M) = M 0 for n 6= 1. Define a height function of J by ht( 3=1) = 1, ht( 3=C2) = 0. For any A[J ]- module M, the spectral sequence 5.3 degenerates to an exact sequence 0 -! H0(J ; M) -! M( 3= 2) -d!M( 2 3 1 3=1) =M( 3=1) -! H (J ; M) -! 0, where (by 5.5) d is the composition (5.8) M( 3= 2) M(1-2)----!M( 3=1) 2 -i M( 3=1) 2=M( 3=1) 3. HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 17 Cohomology of R2( 5). By (3.14) there are, up to conjugacy, four stubborn subgroups of 5, namely 1, C2, 1 o C22and C2 o C2. Its Weyl groups (in 5) are isomorphic to 5, 3, 3 and 1 respectively. Proposition 5.9. Let M be an A[ 5]-module. Then ( 2x 2x1=(M 2x 3 + M 4x1) for n = 2 n( 5; M) = M 0 for n 6=.2 Proof.By 5.2 and 5.6 we have n( 5; M) = 0 for n 6= 2 and 2( 5; M) ~= H1(R2( 5); F00M). Introduce a height function on R2( 5) by ht( 5=(C2o C2)) = 0, ht( 5=(1 o C22)) = ht( 5=C2) = 1 and ht( 5=1) = 2. The spectral sequence (5.3) calculating H*(R; F00M) degenerates to an exact sequence d0,01(1,C2) d0,01(1,1oC22) 0 -! H0(R; F00M) -! MC2oC2---------------! MC2xC2=MC2x 3 MC2oC2=M 4 -! H1(R; F00M) -! 0. Full subcategories with object sets respectively { 5=(C2 o C2), 5=(1 o C22)} a* *nd { 5=(C2o C2), 5=C2} are both isomorphic to R2( 3). Thus by 5.5 d0,01(1, C2) is* * a composition MC2oC2,! MC2xC2 i MC2xC2=MC2x 3 and d0,01(1, 1 o C22) is a natural projection. Thus 2( 5; M) ~= H1(R; F00M) ' M 2x 2x1=(M 2x 3 + M 4x1). The groups *( 7; -). *-groups of 7 are more difficult to calculate. Denote R := R2( 7), and let a = (12), b = (34), c = (56), s = (13)(24). By 3.14 there are 7 conjugacy classes of stubborn subgroups of 7, namely 1, C2, C* *32, K := 1 o C22(the Klein group), K x C2, D8 ' C2 o C2, D8 x C2. Its automorphism groups are respectively 1 (for D8 x C2), 3 (for D8, C22and K x C2), 5 (for C2* *), 3 x 3 (for K) and 7 (for 1). Proposition 5.10. If M is an A[ 7]-module, then i( 7; M) = 0 for i 6= 2, 3. Proof.For n = 0, 1 the conclusion is clear. Since dim2(Aut R( 7=P )) 2 for a* *ll orbits 7=P 2 R except 7=1, then for each n > 3 n( 7; M) ~=Hn-1(R; F00M) ~=Hn-1(R \ { 7=1}; F00M) = 0. Proposition 5.11. i( 7; M(7, 1)) = 0 for each i 0. Proof.It is enough to prove that for each 7=P 2 R and each n > 1 holds nP:= n(Aut R( 7=P ); FM00(7,1)( 7=P )) = 0. For P = L0 = 1 we have FM00(7,1)( 7=1) = 0 and for P = N1, N0, K1, L3 we have dim2(P ) < 2 and therefore nP= 0 for n 2. Moreover F00M(7,1)( 7=C2)= M(7, 1)C2 ~=A{x1 + x2} M(5, 1){xi}7i=3 F00M(7,1)( 7=K)~=A ~A{x1 + x2 + x3 + x4} A ~M(3, 1){xi}7i=5 Then, for i > 1, we have nL1= 0 by 5.9 and iK0= 0 by 5.7 and [7, 6.1.v]. Proposition 5.12. 3( 7; M(7, 2)) = 0. 18 KRZYSZTOF ZIEMIA'NSKI Proof.Introduce a height function on R := R2( 7) by 8 >>>0for P = D8 x C2 <1 for P = D , C2, K x C ht( 7=P ) = > 8 2 2 >>:2for P = C2, K 3 for P = 1 Let E*,**:= E(R2( 7), F00M(7,2))*,**. Since dim 2(W 7(P )) ht( 7=P ) for al* *l 2- stubborn P it is sufficient to prove that E2,01= 2( 5; M(7, 2)C2) 2( 23; M(7, 2)K ) is killed by the differential d1,01. We have M(7, 2)C2~=A{x12} M(5, 1){x1i+ x2i}7i=3 M(5, 2){xij}3 i and the conclusion follows. The groups *( 3 o C2; -). There are (up to conjugacy) three 2-stubborn sub- groups of 3oC2, namely 1, 2 and 2oC2. Its Weyl groups are respectively 3oC2, 3 and 1. The full subcategory of R := R2( 3 o C2) containing 2 and 2 o C2 is isomorphic to J = R2( 3). Proposition 5.13. Let M be an A[ 3 o C2]-module. Then ( 2x 2=M 2x 3 + M 2oC2 for n = 2 n( 3 o C2; M) = M 0 for n 6=.2 Proof.Since the rank of 3oC2 is even, and the relation ~ from 5.2.(2) is trans* *itive, then n( 3 o C2; M) = 0 for n = 0, 1. For n > 1 we have n( 3 o C2; M) ~=Hn-1(R2( 3 o C2); F00M) ~=Hn-1(J ; resRJFM00). By 5.8 we have ( 2x 2=M 2x 3 + M 2oC2 for n = 2 Hn-1(J ; resRJFM00) ~= M 0 for n 6=.2 6. Homotopy representations of G and "G Let R := R2(G) ~=R2(G"). In this section we prove that for every R-invariant representation of N (resp. N") the map B'^2extends to a homotopy representation of G (resp. G"). Denote A := Z^2. Let ' be an R-invariant representation of N and let := (') be a module introduced in Section 2. By Theorem 2.5.(a) we need to prove that H3(R; ) vanishes, and that cohomology of R with coefficient in any A[R]-module vanish above dimension 4. Proposition 6.1. For each G=P 2 R we have dim2(WG (P )) 3. Proof.Each group WG (P ) is isomorphic to one of the following groups: 1, 3, 3x 3, 5, 3oC2, 3oC2x 3, 7 (cf. 4.6). We have dim2(1) = 0, dim2( 3) = 1 (by 5.7), dim2( 5) = 2 (by 5.9), dim2( 3oC2) 2 (by 5.13) and dim2( 7) 3 (by 5.10). Moreover, dim2( 3 x 3) = 2 and dim2( 3 o C2 x 3) = dim2( 3 o C2) + 1 3. As a consequence we obtain Corollary 6.2. cdim2(R) 3. In order to calculate H3(R; ) we use a spectral sequence (5.3). Define a hei* *ght function on R by putting 8 >>>0for P = N1 <1 for P = N , L , K , J (6.3) ht(G=P ) = > 0 3 1 1 >>:2for P = L3, K0, J0, M1 3 for P = L0, M0 20 KRZYSZTOF ZIEMIA'NSKI Let E*,**:= E(R; )*,**. Proposition 6.4. H3(R; ) = coker(d2,01: E2,01! E3,01). Proof.By 6.1 we have i(WG (P )) = 0 for i > ht(P ). Hence Es,t1vanishes if t >* * 0 or if s > 3 (since 3 is a maximal value of the height function). Therefore E3,0* *1is the only rank 3 entry at the first table which possibly does not vanish and d2,* *01the only possibly non-trivial differential hitting E3,01. Proposition 6.5. A homomorphism d2,01(M1, M0) : 2( 3 o C2; (G=M1)) -! 3( 3 o C2 x 3; (G=M0)) (cf. 5.3) is an epimorphism. 1 Proof.Let {Xi}i2Ibe orbits of an action of 3oC2x 3 on IR(M0, resN1M0'). Each Xihas the form YixZi, where Yiis an 3oC2-orbit in IR(M) and Ziis an 3-orbit in IR({ 1}2) (see p.14). By [7, 6.1.v] M 3( 3 o C2 x 3; (G=M0)) ~= 3( 3 o C2 x 3; A[Xi]) i2I M ' 2( 3 o C2; A[Yi]) 1( 3; A[Zi]) i2I Each Zi is equal either to {` ~`}, or to {` ~', ' ~`, ' ~'}. In the first1case * *we have 1( 3; A[Zi]) =.0In the second case, there is an orbit X0iin IR(M11, resN1M11')* * ' IR( 14) x IR(O(2)1 ) such that X0i= Yix {ff2k+1} for some k 2 A. Furthermore, the restriction of d2,01to 2( 3 o C2; A[X0i]) is a composition 2( 3 o C2; A[X0i]) -'! 2( 3 o C2; A[Yi0]) A{ff2k+1} 1xd--! 2( 3 o C2; A[Yi]) 1( 3; A{` ~', ' ~`, ' ~'}) 3( 3 o C2 x 3; (G=M0)). The differential d : A[{ff2k+1}] ! 1( 3; A[{` ~', ' ~`, ' ~'}]) is an epimorph* *ism by 5.8. Hence d2,01is an epimorphism. Proposition 6.6. We have F0M(7,3)(L1)' M(5, 1){x12k}k 3 M(5, 2){x1kl+ x2kl}k,l 3 M(5, 2){xklm}k,l,m 3 F0M(7,3)(K0)' (A ~A){x123+ x124+ x134+ x234} (M(3, 1) ~M(3, 1)){x12k+ x34k, x13+ x24k, x14+ x23k}k 5 (A ~M(3, 1)){x1kl+ x2kl+ x3kl+ x4kl}k,l 5(A ~A){x567} Proposition 6.7. A homomorphism d2,01(L1, L0) d2,01(K0, L0) : 2( 5; (G=L1)) 2( 23; (G=K0)) -! 3( 7; (G=L0* *)) is an epimorphism. Proof.Let I : R2( 7) ! R2(O(7)) ~=R be a full inclusion from Theorem 3.22 and let F := F0 (G=L0). Let ~E*,**:= E(R2( 7), F)*,**. There is a homomorphism of A[R2( 7)]-modules I* ! F0 (G=L0)which is an isomorphism on L0 and it HOMOTOPY REPRESENTATIONS OF SO(7) AND Spin(7) AT PRIME 2 21 induces a transformation of spectral sequences E*,**! ~E*,**. In particular, th* *ere is a commutative diagram 2( 5; (G=L1)) 2( 23; (G=K0))___w 2( 5; (L0)C2) 2( 23; (G=L0)K ) | | | | |d2,0 |~d2,0 | 1 | 1 | | |u |uu 3( 7; (G=L0)) _______________________ 3( 7; (G=L0)) (where K 4 7 is a Klein group). Since 1 resN1L0' ' (j70) l0 (j71) l1 (j72) l2 (j73) l3, (cf. 4.12) then (G=L0) is a direct sum of some of the modules M(7, i), i =1 0, . .,.3, where M(7, i) appears as a summand of (G=L0) if and only if resN1L0' contains a subrepresentation1isomorphic to j7i. From 5.11 and 5.12 follows that 3( 7; (G=L0)) = 0 if resN1L0' does not contain a subrepresentation isomorphic to j73, so assume otherwise. Let xijk, 1 i < j < k 7 be the generators of t* *he sub- module of1 (G=L0) corresponding to j73. By 4.13 there exists a subrepresentation of resN1L11' which is isomorphic either to o ~j52or to fi2k~j52for some k 2 A, * *and another subrepresentation isomorphic to ff2k0+1~j52, where k02 A. Then (G=L1) contains the direct sum of two A[ 5]-submodules isomorphic to M(5, 2): the one generated by irreducible subrepresentations of o ~j52(or fi2k~j52) maps onto the summand M(5, 2){xklm}3 k 0. Theorem 6.10. Any R-invariant representation of N"11extends to a homotopy representation of "G. Proof.It follows from 6.1, 6.9 and 2.5. References [1]J. L. Alperin and P. Fong, Weights for Symmetric and General Linear Group* *s, J. Alg. 131 (1990), 2-22 [2]A. K. Bousfield, The localization of spaces with respect to homology, Top* *ology 14 (1975), 133-150 [3]A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions and Localizat* *ions, Lec- ture Notes in Mathematics Vol. 304. Springer, New York (1972) [4]W. G. Dwyer and C. W. Wilkerson, A new finite loop space at the prime 2, * *J. 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