Contemporary Mathematics Minami-Webb type decompositions for compact Lie groups John Martino and Stewart Priddy 1. Introduction Let p be a fixed prime number. We extend to compact Lie groups some stable classifying space decompositions of Minami [M ], following Webb [W ]. One notable feature of [W ] is the use of a combinatorial Möbius function to encode p-local information about the cohomology of a finite group. We wish to show similar phenomena hold for com- pact Lie groups. However, for a compact Lie group G one is faced with the problem of an infinite number of conjugacy classes of p-toral sub- groups, that is, extensions of tori by finite p-groups. These groups are the analogs of p-groups for finite groups. We circumvent this problem by considering a certain finite G-complex which allows us to introduce combinatorial methods in the compact Lie group case. This complex is based on the notion of p-stubborn subgroups which arose earlier in modular representation theory of finite groups (where they were called p-radical groups) in connection with Alperin's conjecture [A ], [Bouc ], in group cohomology [W ], and in the study of homotopy classes of maps between classifying spaces of compact Lie groups [JMO ]. We also derive a decomposition based on the corresponding complex for el- ementary abelian p-subgroups. Several examples are given to illustrate the various decompositions. 2. Main Results We begin by defining the G-sets used to construct our stable de- compositions. Let = (G) be the poset of non-trivial p-toral sub- groups P G with finite Weyl groups WG (P ) = NG (P )=P . Let cO0000 (copyright holder) 1 2 JOHN MARTINO AND STEWART PRIDDY ( ) = |Nerve( )| be the geometric realization of the nerve of viewed as a category. Thus ( ) is a simplicial complex associated with . The n-simplices oe = (P0, P1, ..., Pn) of ( ) are sequences of inclusions P0 < P1 < ... < Pn of elements of . G acts on ( ) via conjugation. Our goal is to replace ( ) by a finite G-complex which can be used to study BG. We shall need a compact Lie group version of seminal results of Quillen [Q ] (following Brown [Bwn ]) regarding this complex and certain subcomplexes. Throughout sums of classifying spaces will be taken in the Grothendieck group of spectra under wedge sum. The complex (S) of p-stubborn subgroups: Let S = S(G) (G) be the poset of p-stubborn subgroups of G. Thus P 2 S if and only if 1)P is p-toral with WG (P ) finite, 2) P = Op(NG (P )), i.e., WG (P ) has no non-trivial normal p-subgroups. For finite groups homological properties of these subgroups were studied by Bouc [Bouc ]. In the compact Lie group case Jackowski-McClure- Oliver showed that up to conjugacy in G, S is finite [JMO ]. Let (S) = |Nerve(S)|. As we shall see (S) has desirable homotopical properties. Lemma 2.1. The complex (S)=G is finite. The proof is given in Section 5. If oe = (P0, P1, ..., Pn) is a simplex of (S) then the isotropy sub- group Gff= NG (P0) \ ... \ NG (Pn). Theorem 2.2. Stably X BG^p' (-1)dim(ff)(BGff)^p. _ff2 (S)=G Since (S)=G is a finite complex by Lemma 2.1, the summation is finite. The proof of Theorem 2.2, given in Section 6, is a topological variation of the proof of Webb's Theorem A [W2 ] for Mackey functors from the category of finite G-sets to a category of modules. MINAMI-WEBB TYPE DECOMPOSITIONS 3 The complex (A) of elementary abelian p-subgroups: By considering elementary abelian groups we obtain another de- composition. Let A = A(G) be the poset of non-trivial elementary abelian p-subgroups of G and (A) = |Nerve(A)| the associated sim- plicial complex. Lemma 2.3. The complex (A)=G is finite. The proof is given in Section 5. As in the case of finite groups we have Lemma 2.4. If P is a non-trivial p-toral subgroup of G then the fixed point complex (A)P is contractible. Proof. The proof given in [Q , 4.4] works equally well for compact Lie groups. With this lemma the proof of Theorem 2.2 applied to (A) proves Theorem 2.5. Stably X BG^p' (-1)dim(ff)(BGff)^p. _ff2 (A)=G A Möbius function and Minami-Webb type formula: Let Zp denote the p-adic integers. If G is a finite group, H G then uH denotes the permutation module Zp ZpH ZpG viewed as an element of the Green ring which is the Grothendieck group (over Q) of finitely generated indecomposable ZpG modules [W ]. We recall that a cyclic mod-p group is a extension of a finite p-group by a finite cyclic p0-group. Let C(G) be the collection of all cyclic mod-p subgroups of G. A Möbius function f : C(G) ! Z is defined recursively by X f(K) = 1. J K2C(G) Computation of f is facilitated by P. Hall's observation [Ha ] that f vanishes except on intersections of maximal subgroups. By Webb's formula [W , Theorem D'] X f(H) (1) uG = _________ uH H2C(G)*|WG (H)| where the sum is taken a set of representatives of conjugacy classes C(G)* of cyclic mod-p subgroups of G and WG (H) =: NG (H)=H, the 4 JOHN MARTINO AND STEWART PRIDDY Weyl group of H. From this Minami derives a corresponding formula for classifying spaces [M , Theorem 6.6] X f(H) (2) BG^p' _________ BH^p H2C(G)*|WG (H)| In this formula we may omit any p0-groups of C(G) since completed at p their classifying space contributes nothing to the sum. However as we shall see in Examples 2 and 3 of Section 3 it is necessary to leave these groups in formula (1). In the general compact Lie group case let C(S) be the set of all cyclic mod-p extensions of p-stubborn subgroups. Thus H 2 C(S) if and only if H contains a normal subgroup P 2 S such that H=P is a finite cyclic p0-group. Since the conjugacy classes of S are finite and each element has finite Weyl group it follows that C(S) has a finite number of conjugacy classes. Let cJK denote the number of conjugates of K which contain J. By Lemma 5.2, cJK is finite for J, K 2 C(S). Thus we can define a Möbius function f : C(S) - ! Z recursively by X (3) cJK f(K) = 1 J K2C(S)* for fixed J 2 C(S) with Op(J) 6= 1. Here C(S) *is a set of representa- tives for the conjugacy classes of C(S) . Theorem 2.6. Stably X f(H) BG^p' _________ BH^p. H2C(S)*|WG (H)| Here we have tensored the Grothendieck group of spectra with the rationals Q. The proof is given in Section 7. 3. Applications In these examples all spaces are stable and completed at p = 2. Example 3.1. G = SO(3). Then up to conjugacy S consists of {O(2), V } where V = O(1) x O(1) O(2) is an elementary abelian 2-group of rank 2. NG (O(2)) = O(2) and NG (V ) is the octahedral group isomorphic to 4, the symmetric group on four letters. This is easily checked from the information on the conjugacy classes of subgroups of G given in MINAMI-WEBB TYPE DECOMPOSITIONS 5 [TD2 ]. Let D8 be the dihedral group of order 8. Then up to conjugacy C(S) consists of {V, A4, D8, O(2)}: SO(3) | 6| I@ | @ | @ || @ | O(2) (1) | | | 6| | | | | | | | | (1) A4 | | | 6| | | | | | | | | | | D8 (0) | || ` | | | (-3) V where arrows represent inclusion and the values of the Möbius func- tion f are given in parentheses. These are computed inductively from equation (3). Since Q = O(2), A4 are maximal, f(Q) = 1. By Lemma 5.2, cD8,O(2)= 1 hence f(D8) + cD8,O(2)f(O(2)) = 1 implies f(D8) = 0. Similarly cV,O(2)= 3, cV,A4 = 1 hence B f(V ) + cV,A4f(A4) + cV,O(2)f(O(2)) = 1 implies f(V ) = -3. By Theorem 2.6 we have 1 (4) BSO(3) ' BO(2) + __(BA4 - BV ) 2 Similarly the elementary abelian 2-subgroups of G fall into two conjugacy classes {V, C} where C V has order two. Lemma 3.2. Let C0 V , V 0 O(2) be subgroups of SO(3) iso- morphic to C, V respectively. Then 1) C0 is N(V )-conjugate to C. 2) V 0is O(2)-conjugate to V . 6 JOHN MARTINO AND STEWART PRIDDY Proof. 1) N(V ) = V oGL2(F2). Thus any two non-trivial involutions of V are conjugate by an element which normalizes V . 2) V =< a, -I2 . a > where ~ ~ -1 0 a = 0 1 Let ~ ~ 0 1 ø = 1 0 Since SO(2) does not contain an elementary abelian 2-group of rank 2 we may assume V 0contains two generators a0, b0 2 O(2)-1, the non- identity component of O(2). Thus V 0= where a0= ø x, b0= ø y for some x, y 2 SO(2). However a0b0 = b0a0, hence xy-1 = yx-1. Thus x = -I2 . y and V 0= . Now since every non-trivial involution of O(2)-1 is O(2)-conjugate to a and -I2 is central, V 0is O(2)-conjugate to V . By Lemma 3.2(1), (A)=G has a single 1-simplex corresponding to C V and two 0-simplices corresponding to C, V . We have NG (C) = O(2), NG (V ) = 4 and NG (C) \ NG (V ) = D8. Thus by Theorem 2.5 (5) BSO(3) ' BO(2) + B 4 - BD8. Using S and Lemma 3.2(2), Theorem 2.2 yields the same result. These formulas are consistent with those of [Mitch-P ]. Example 3.3. G = SU(2) = S3. There are two conjugacy classes of 2-stubborn subgroups, H = NG (S1) = and K = Q8. It is easy to check that any pair of subgroups H0 K0 conjugate to H, K respectively is simultaneously conjugate. (This type of argument is illustrated in Lemma 3.2.) Thus (S)=G has a single one simplex corresponding to K H and two zero simplices corresponding to H, K. Furthermore NG (H) = H and_NG (K) is the binary octahedral group which is iso- morphic to 4, the two-fold cover of 4 and N(H) \ N(K) = Q8. Thus by the formula of Theorem 2.2 __ BS3 ' BNG (S1) + B 4- BQ8 Applying Theorem 2.6 we have the refinement 1 BS3 ' BNG (S1) + __B(Q8 o Z=3 - BQ8) 2 MINAMI-WEBB TYPE DECOMPOSITIONS 7 The interested reader can verify that this decomposition relates well to that of BQ8 given in [Mitch-P ] i.e., 1 -1BS3=BNG (S1) ' __B(Q8 o Z=3 - BQ8). 2 Example 3.4. G = U(2) with standard maximal torus T . The center C T of G, i.e., matrices of the form zI2, z 2 S1, is non-trivial so Theorem 2.5 does not yield a useful expression of BG. We could use Theorem 2.6. However, more simply G=C = SU(2)=< I2> = SO(3) and C is in every maximal subgroup. Therefore the Möbius functions for G and G=C correspond. Hence we may pull back formula (4) to obtain 1 (6) BU(2) ' B(T o Z=2) + __[B(Q o Z=3) - BQ] 2 where Q = ( S1 x Z=2) o Z=2, i.e., Q is generated by the elements ~ ~ z 0 1 0 z , z 2 S and the involutions a, ø defined in the proof of Lemma 3.2. The action of Z=3 is given by conjugation with the element ~ ~ 1 -1 - i -1 - i fi = __ . 2 1 - i -1 + i Another decomposition can be obtained from S which (up to con- jugacy) consists of {N(T ), Q} with Q N(T ). This follows from Oliver's description of the p-stubborn subgroups of the classical groups [O ] where Q is denoted by U2. Lemma 3.5. If Q0 N(T ) is conjugate to Q then Q0is T -conjugate to Q. Proof. Since S1 = ZU(2), Q0= < (S1), a0, ø 0> where a0, ø 0cor- respond to a, ø under the given conjugation. Then a0, ø 0are non- commuting involutions. Since ø 02 N(T ) it has the form ø 0= ø ffl(z1, z2), ffl = 0, 1. Case 1: ffl = 1. ø20= [ø (z1, z2)]2 = (z1z2, z1z2) = 1 8 JOHN MARTINO AND STEWART PRIDDY Hence z2 = z1-1. Then ø 0= ø (z1, z1-1). However (z1, 1)ø (z1, z1-1)(z1-1, 1) = ø (1, 1) = ø Thus we may assume ø 0= ø . Now if a0 2 T , then a0 = a. If not a0= ø (z1, z2), then arguing as above we have z2 = z1-1. Hence -1 = ø a0ø a0= (z12, z1-2) Thus z1 = i and a0= (i, i)a. Thus either way Q0 is T -conjugate to Q. Case 2: ffl = 0. We have -1 = det(ø 0) = z1z2 and 1 = ø 02= (z1z2, z1z2). Hence z2 = -z1-1, z1 = 1. Thus ø 0= a. In this case a02= T since a0 and ø 0do not commute. Hence a0= ø (z1, z2) and so a0 is T -conjugate to ø . This implies Q0 is T -conjugate to Q. By Lemma 3.5, (S) has one 1-simplex and two 0-simplices. Com- puting normalizers we have N(N(T )) = N(T ) = T o Z=2, N(Q) = Q o 3 where the 3 action is generated by {ff, fi} with ~ ~ 1 -1 1 ff = ____p_ . 2 1 1 Then N(T ) \ N(Q) = Q o 2. Thus by Theorem 2.2 we have (7) BU(2) ' B(T o Z=2) + B(Q o 3) - B(Q o 2) We note that the same formula is obtained by pulling back equation (5). This gives another description of Q o 3. Finally we note that equation (7) transforms to equation (6) by simplifying B(Qo 3). This is done by pulling back Webb's formula (1) for 3 to obtain 1 B(Q o 3) ' B(Q o 2) + __[B(Q o Z=3 ) + BQ]. 2 Example 3.6. G = SU(3). There are three conjugacy classes of 2-stubborn sub- groups. By [O ] they are represented by the subgroups {T, NU(2)(T ), Q} of U(2) defined in Example 3.4. We consider these as subgroups of SU(3) by the usual monomorphism U(2) ! SU(3). Computing nor- malizers we find NSU(3)(T ) = T o 3, NSU(3)(NU(2)(T )) = NU(2)(T ) NSU(3)(Q) = Q o 3. MINAMI-WEBB TYPE DECOMPOSITIONS 9 From the inclusions T NU(2)(T ) Q we have NSU(3)(T ) \ NSU(3)(NU(2)(T )) = T o 3 NSU(3)(NU(2)(T )) \ NSU(3)(Q) = Q o 2 By Lemma 3.5, Q NU(2)(T ) is unique up to conjugation (even in U(2)), while T is the unique maximal torus of NU(2)(T ). Hence (S)=G has exactly two 1-simplices corresponding to T NU(2)(T ), Q NU(2)(T ) and three 0-simplices. Thus by Theorem 2.2 (8) BSU(3) ' B(T o 3) + B(Q o 3) + B(Q o 2). Theorem 2.6 yields 1 1 (9) BSU(3) ' B(T oZ=2)+ __[B(T oZ=3)-BT ]+ __[BQoZ=3-BQ]. 2 2 Alternatively, the conjugacy classes of elementary abelian 2-subgroups of G are represented by W of rank 2 generated by the diagonal matrices with entries (a, b, (ab)-1), a, b = 1 and by Z of rank one generated by (-1, -1, 1). Then NG (Z) = U(2), NG (W ) = T o 3, NG (Z) \ NG (W ) = T o Z=2. Hence Theorem 2.5 gives (10) BSU(3) ' BU(2) + B(T o 3) - B(T o Z=2). We can simplify B(T o 3) by pulling back Webb's formula (1) for 3, as in Example 3.4, to obtain 1 B(T o 3) ' B(T o Z=2) + __[B(T o Z=3) - BT ] 2 Thus 1 BSU(3) ' BU(2) + __[B(T o Z=3) - BT ] 2 which combined with (6) yields another derivation of (9). Similarly (10) combined with (7) gives (8). 4. A G-homotopy equivalence of complexes We shall need the following result in the proof of Theorem 2.2 Proposition 4.1. 1) Inclusion i : (S) ! ( ) is a G-homotopy equivalence. 2) Suppose P 2 is non-trivial p-toral subgroup then (S)P is contractible. 10 JOHN MARTINO AND STEWART PRIDDY Lemma 4.2. If P is a non-trivial p-toral subgroup then ( ) P is contractible. Proof. We adapt Quillen's method. Let Q 2 P then P NG (Q). Then P Q NG (Q) is a compact Lie group which is a fi- nite extension of Q since WG (Q) is finite. Let S, T be the maximal torus of P , Q respectively. Then S E P , T E Q since P , Q are p-toral. Let T 0be a maximal torus of (P Q)0 which contains S. Since P Q is a finite extension of Q , T 0= T . Thus ST = T and so S T . It follows easily that ß =: P Q=T is a finite p-group generated by Q=T and P=S. Hence P Q is p-toral. We claim P Q 2 P . Since P NG (P Q) it remains to show WG (P Q) is finite. If not there is a torus T 00 NG (P Q) such that T 00 P Q Then T 00normalizes P Q but acts trivially on ß since Aut(ß) is finite. Thus T 00acts trivially on the quotient P Q=Q = ß=(Q=T ). Thus T 00normalizes Q and hence T 00 Q since WG (Q) is finite. This contradicts the existence of T 00proving the claim. Thus we have P P Q Q in . This proves ( )P is conically contractible by [Q , 1.5]. We recall a result of Th'evenaz-Webb and some generalizations. From here through the proof of Proposition 4.5 we will identify a poset with its geometric realization. If Y is a G-poset then Y y = {z 2 Y | z y } Y y = {z 2 Y | z y }. Theorem 4.3 (Th'evanaz-Webb). [TW , Th. 1] Let G be a group, let X, Y be G-posets and let OE : X ! Y be a map of G-posets. Suppose either 1) OE-1(Y y) is Gy-contractible for all y 2 Y 2) OE-1(Y y) is Gy-contractible for all y 2 Y Then OE is a G-homotopy equivalence. Lemma 4.4. Let P 2 . Then >P is NG (P )-contractible if and only if P =2S. Proof. This is [TW , Lemma 2.1]; the proof applies equally well to compact Lie groups. We shall also need the following generalization of another result of [TW ] extended to infinite groups. MINAMI-WEBB TYPE DECOMPOSITIONS 11 T Proposition 4.5. Let X Y be G-posets. Asssume X = Yi where Y0 = Y and Yi+1 is obtained by deleting from Yi the minimal elements of Yi- X. If y 2 Y - X implies Y>y is Gy-contractible then the inclusion X ! Y is a G-homotopy equivalence. Proof. Let OEi : Yi+1 ! Yi be the inclusion. If y 2 Yi+1 then OEi-1((Yi) y) = (Yi+1) y which has y as a minimal element. Thus (Yi+1) y is the cone on y which is Gy fixed. Hence (Yi+1) y is Gy contractible. If y 2 Yi- Yi+1 then OEi-1((Yi) y) = Y>y since y and no element above y is deleted when forming Yi+1. Y>y is Gy contractible by hypothesis, thus OEi is a G-homotopy equivalence by Theorem 4.3. Now for each closed subgroup H there is the usual Milnor exact sequence 0 ! lim1 ß*+1(YiH ) ! ß*(XH ) ! lim ß*(YiH ) ! 0 Since OEi*: ß*(Yi+1H ) ! ß*(YiH ) is an isomorphism, the lim1 term is zero and ß*(XH ) ! ß*(Y H) is an isomorphism and so X ! Y is a G-homotopy equivalence. Proof of Proposition 4.1. 1) Let S0 = S [ pos where pos denote the elements of of positive dimension. We will show the inclusions of posets S0 and S S0 induce G homotopy equivalences. (11) (S0) ! ( ) (12) (S) ! (S0) We wish to apply Prop. 4.5. Let X = S0, Y = . Then the elements ß 2 Y - X are finite p-groups By consideringTtheir order |ß|, one sees that ß 2 Yi then |ß| pi. Thus X = Yi. Hence Lemma 4.4 and Prop. 4.5 imply (11) is a G-homotopy equivalence. For (12) we induct on dimension. Since this induction is finite the full strength of Prop. 4.5 is not needed. Part 2) follows from Part 1) and Lemma 4.2 5. Proof of Lemmas 2.1 and 2.3 The following result of Bredon [Brd , Cor. II 5.7] will be useful. Lemma 5.1. Let K H G be compact Lie groups. Then the orbit space (G=H)K =WG (K) of the right translation action of WG (K) is finite. 12 JOHN MARTINO AND STEWART PRIDDY Lemma 5.2. For subgroups J, K G the number, cJK , of conju- gates of K which contain J is finite if WG (J) is finite. Moreover if WG (K) is also finite then cJK = |(G=K)J|=|WG (K)| Proof. Suppose WG (J) is finite. By definition cJK = |{g 2 G : gKg-1 J}=N(K)|. On the other hand (G=K)J = {gK : gKg-1 J} = {g : gKg-1 J}=K is finite since it has a finite number of WG (J) orbits by Lemma 5.1. Thus cJK is finite. Since WG (K) acts on (G=K)J with orbit space {g 2 G : gKg-1 J}=N(K) it follows that cJK = |(G=K)J|=|WG (K)| if WG (K) is also finite. Proof of Lemma 2.1: Consider a simplex oe = (Q1, Q2, ..., Qn), Q1 < Q2 < ... < Qn. Since there are only finitely many conjugacy classes of subgroups in S there are only a finite number of choices of Q1 up to conjugacy. Since the number of G-conjugacy classes of S is finite, Lemma 5.2 implies there are only a finite number of choices of Q2 which contain Q1. This process is finite and terminates after the conjugacy classes of S have been used. Proof of Lemma 2.3: Since G is compact there is a bound d for the rank of all maximal elementary abelian subgroups of G. Then the dimension of (A) < d. Let (E0, E1, ..., E(d-1)) be a simplex of highest dimension. Since G has only finitely many conjugacy classes of elementary abelian subgroups [Q2 , Lemma 6.3], it follows that in (A)=G the subgroup E(d-1)ranges over a finite set. Hence (A)=G is a finite complex. 6. Proof of Theorem 2.2 Let {Spectra} be the Grothendieck group of spectra completed at p. Let {G-space} be the category of G-spaces with a continuous left G-action and define a functor F : {G-space} ! {Spectra} by F (X) = 1 EG+ ^G X+ . In what follows, as elsewhere in the paper, we work stably and omit the symbol 1 for suspension spectrum. Then F is a Mackey functor [W2 ] with restriction and induction given by res#HK = F (i) : F (K) ! F (H) ind"HK = transfer : F (H) ! F (K) MINAMI-WEBB TYPE DECOMPOSITIONS 13 where i : K ! H is an inclusion of closed subgroups of G. At this point we formulate a special case of Webb's theorems the proof of which is applicable to the case of compact Lie groups. Theorem 6.1. [Webb] Let G be a finite group, M a Mackey func- tor, X and Y collections of subgroups of G closed under conjugation and taking subgroups, and a finite G-complex. Suppose 1) For every simplex oe of the vertices of lie in distinct G-orbits. 2) For every subgroup H 2 X - Y , H is contractible. 3) A Sylow p-subgroup Gp 2 X and M*(pr) : M(Gp x T ) ! M(T ) is a split surjection natural in T . 4) For every Y 2 Y, M(Y ) = 0. Then M M(G) _ (-1)dim(ff)M(Gff) ff2 =G Proof. [W2 ] Theorem A. Proof of Theorem 2.2: The homotopy category Ho{Spectra} is an additive category and Theorem 6.1 applies even though it is stated for Mackey functors with R-modules as target. Addition of stable maps gives the hom sets the structure of abelian groups and direct sum is given by the wedge product. In order to define terms let = (S) , X equal the set of all p-toral subgroups of G, and Y = {1}. Hypothesis (1) was observed in [W2 ]. For (2) we note H 2 X implies H is contractible by Lemma 4.2. For (3) we recall Ø(G=Gp) is prime to p. Thus the transfer for the (space level) fibration pr G=Gp ! EGx G(Gp x T ) -! EGx GT implies pr* + + F (Gp x T ) = EG+ ^G (Gp x T )+ - ! EG ^G T = F (T ) is a natural split surjection as required. We note X is closed under conjugation and under taking subgroups. Thus the proof of Theorem 6.1 applies. 14 JOHN MARTINO AND STEWART PRIDDY 7. Proof of Theorem 2.6 Let G be a compact Lie group_which_we initially assume has a normal maximal torus_T_. Let C(G ) be the set of cyclic mod-p subgroups of the finite group G =: G=T . Let WG (H) =: NG (H)=H, the Weyl group of H. Theorem 7.1. Suppose G has a normal maximal torus T . Then stably X f(H) BG^p' _________ BH^p __H2C(__G)*|WG (H)| ___ where H = H=T runs over_a set of representatives of the_finite set of conjugacy_classes C(G )* of cyclic mod-p_subgroups of G and f(H) =: f(H ) is the Möbius function f : C(G ) ! Z satisfying X f(H) = 1. __J __H2C(__G) __ __ for fixed J 2 C(G ). Proof. By a slight variation on Feshbach's construction [F ] one can construct a nested sequence of finite subgroups_Gk Gk+1 G with normal subgroups Tk T such that Gk=Tk = G and colim H*(BGk; Z=p) H*(BG; Z=p) __ ___ Let ßk : Gk ! G be projection and set Hk = ßk-1(H ). Let uH denote the permutation module Zp ZpH ZpG. Then from [W , Theorem D' and Lemma 7.1] X f(H) uGk = _________ uHk __H2C(__G)*|WG (H)| From this and (2) it follows directly that X f(H) (BGk)^p' _________ (BHk)^p __H2C(__G)*|WG (H)| Passing to the colimit over k gives the desired result since BH^p' colim(BHk)^p. Proof of Theorem 2.6: Let oe = (P0, P1, ..., Pn) 2 (S). Let T0 denote the normal and hence unique maximal torus of P0. Since WG (P0) is finite T0 is a maximal torus of NG (P0) which is also normal in NG (P0). Since P0 \NG (Pi) NG (P0), T0 is a normal maximal torus of Gff= MINAMI-WEBB TYPE DECOMPOSITIONS 15 __ \NG (Pi). Let G ff= Gff=T0. As indicated above we can apply Theorem 7.1 to decompose BGff. Thus Theorem 2.2 yields X X X fff(K) BG^p' (-1)dim(ff)(BGff)^p' (-1)dim(ff) __________BK^p _ff2 (S)=G _ff2 (S)=G __K2C(___Goe)*|WGoe(K)| Collecting terms in this last expression we have X X fff(K) X f(H) (-1)dim(ff) __________ BK^p= ________ BH^p _ff2 (S)=G __K2C(___Goe)*|WGoe(K)| H2C(R)* |W (H)| Thus we obtain a formula for f(H) X X |WG (K)| f(H) = (-1)dim(ff) fff(K)__________ _ff2 (S)=G __K2C(___G)* |WGoe(K)| H Goe K~GH oe Lemma 7.2. Let J 2 C. If K H G then a g (G=K)J = (H=K)J g2(G=H)J . Proof. This follows from the bundle H=K ! G=K ! G=H using the usual homeomorphism _ : G xH H=K ! G=K of G-spaces given by OE(g, hK) = ghK. It remains to show f(H) satisfies the requisite formula of the the- orem. Computing we have X cJH f(H) = J H2C(R)* X X X |WG (K)| |(G=H)J|=|WG (H)| (-1)dim(ff) fff(K)__________ J H2C(R)* _ff2 (S)=G __K2C(___Goe)*|WGoe(K)| H Goe K~GH X X X |(Gff=K)Jg| = (-1)dim(ff) fff(K)____________ _ff2 (S)=G g2(G=Goe)JJg K2C(___G)* |WGoe(K)| J Goe oe by Lemma 7.2. Summing over __oe2 (S)=G, J G ff 16 JOHN MARTINO AND STEWART PRIDDY is equivalent to summing over oe 2 (S)J and dividing by |(G=Gff)J| which is finite. We note for use below that (S)J is thus also finite by Lemma 5.1. Therefore the last expression becomes 0 1 X (-1)dim(ff) X X |(Gff=K)Jg| = __________J @ fff(K)____________A ff2 (S)J|(G=Gff) |g2(G=Goe)J Jg K2C(___Goe)* |WGoe(K)| For each g the expression in parentheses is 1 by the defining property of fff. Thus the entire expression simplifies to X = (-1)dim(ff)= Ø( (S)J) = 1 ff2 (S)J by Proposition 4.1(1). MINAMI-WEBB TYPE DECOMPOSITIONS 17 References [A] J. Alperin, Weights for finite groups Proceedings of Symposia in Pure Mathe- matics, 47, I, (1987), 369-380. [Bouc]S. Bouc, Homologie de certain ensembles ordonn'e, C.R. Acad. Sc. Paris S'erie I, 299 (1984), 49-52. [Brd]G. Bredon, Introduction to Compact Lie Groups, Academic Press, (1972). [Bwn] K. 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Department of Mathematics and Statistics, Western Michigan Uni- versity, Kalamazoo, MI 49008 E-mail address: martino@math-stat.wmich.edu Department of Mathematics, Northwestern University, Evanston, IL 60208 E-mail address: priddy@math.nwu.edu