Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Subgroup Complexes and their Lefschetz Modules

Silvia Onofrei

Department of Mathematics Kansas State University ¡ AA ¡ A ¡ ¡ ¡ p-local structure ¡ ¡ ¡ ¡ complexes of p- ¡¡ AA ¡ A algebraic topology representation theory mod-p cohomology Lefschetz modules classifying spaces

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

nonabelian finite simple groups alternating groups (n 5) associated geometries groups of Lie type ≥ Tits buildings 26 sporadic groups sporadic geometries Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

nonabelian finite simple groups alternating groups (n 5) associated geometries groups of Lie type ≥ Tits buildings 26 sporadic groups sporadic geometries ¡ AA ¡ A ¡ ¡ group theory ¡ p-local structure ¡ ¡ ¡ ¡ complexes of p-subgroups ¡¡ AA ¡ A algebraic topology representation theory mod-p cohomology Lefschetz modules classifying spaces Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules Outline of the Talk

1 Terminology and Notation

2 Background, History and Context

3 An Example: GL3(2)

4 Distinguished Collections of p-Subgroups

5 Lefschetz Modules for Distinguished Complexes Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules Terminology and Notation: Groups

G is a finite group and p a prime dividing its H.K denotes an extension of H by K pn denotes an elementary of order pn

Op(G) is the largest normal p- in G

Q a nontrivial p-subgroup of G H G is p-local subgroup if H = N (Q) ≤ G Q is p-radical if Q = Op(NG(Q)) Q is p-centric if Z(Q) Syl (C (Q)) ∈ p G Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules Terminology and Notation: Collections

Collection family of subgroups of G C closed under G-conjugation partially ordered by inclusion

Subgroup complex = ∆( ) |C| C simplices: σ = (Q0 < Q1 < . . . < Qn), Qi n ∈ C isotropy group of σ: Gσ = N (Q ) ∩i=0 G i fixed point set of Q: ∆( )Q C Let k be a field of characteristic p. The reduced Lefschetz kG-module:

dim(∆) X Le (∆( ); k) := ( 1)i C (∆( ); k) G C − i C i=−1 Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules Standard Collections of p-Subgroups

Brown p(G) nontrivial p-subgroups S Quillen p(G) nontrivial elementary abelian p-subgroups A Bouc p(G) nontrivial p-radical subgroups B

Quillen, 1978 p(G) p(G) is homotopy equivalence A ⊆ S Le ( p(G) ; k) is virtual projective module G |S |

Thevenaz,´ Webb, 1991 p(G) p(G) p(G) areA equivariant⊆ S homotopy⊇ B equivalences Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules Webb’s Alternating Sum Formula

Webb, 1987

assumes: ∆ is a G - simplicial complex ∆Q is contractible, Q any subgroup of order p

proves: LeG(∆; Zp) is virtual projective module

n P dim(σ) n Hb (G; M)p = ( 1) Hb (Gσ; M)p σ∈∆/G − Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules Sporadic Geometries

first 2-local geometries constructed Ronan and Smith, 1980 Ronan and Stroth, 1984 geometries with projective reduced Lefschetz modules Ryba, Smith and Yoshiara, 1990 relate projectivity of the reduced Lefschetz module to p-local structure of the group Smith and Yoshiara, 1997 connections with standard complexes and mod-2 cohomology for the 26 sporadic simple groups Benson and Smith, 2004 Lefschetz characters for several 2-local geometries Grizzard, 2007 Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

An Example: GL3(2) The Tits building: the extrinsic approach

e h 1i Fano Plane 3 V = F2 = e1, e2, e3 e + e e + e h i h 1 2i h 1 3i p = e1 h i L = e1, e2 e + e + e h 1 2 3i h i pL = ( e1 e1, e2 ) h i ⊆ h i e e + e e h 2i h 2 3i h 3i

Stabilizers 1    1  ∗ ∗ ∗ ∗ ∗ ∗ ∗ Gp = 0  GL =   GpL = 0 1  0 ∗ ∗ 0∗ 0∗ 1∗ 0 0 1∗ ∗ ∗ Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

The Tits building for GL3(2): the intrinsic approach

The quotient of the action of G on The quotient of the action of G on its building: its Bouc complex: 2 2 p L 2a D8 2b

Barycentric subdivision of Tits building = Bouc complex 2 2 Gp = S4 = 2a.S3 = NG(2a) 2 2 GL = S4 = 2b.S3 = NG(2b) 1+2 GpL = D8 = 2 = NG(D8) 2 2 NG(2a < D8) = NG(2b < D8) = D8 Webb’s alternating formula for mod-2 cohomology:

∗ ∗ ∗ ∗ 0 H (GL3(2); F2) H (S4; F2) H (S4; F2) H (D8; F2) 0 → → ⊕ → → ∗ ∗ ∗ ∗ H (GL3(2); F2) = H (S4; F2) + H (S4; F2) H (D8; F2) −

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

The reduced Lefschetz module of the Bouc complex = Steinberg module for GL3(2)

Le ( ) = H (∆) = St GL3(2) |B2| − 1 − GL3(2) Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

The reduced Lefschetz module of the Bouc complex = Steinberg module for GL3(2)

Le ( ) = H (∆) = St GL3(2) |B2| − 1 − GL3(2)

Webb’s alternating formula for mod-2 cohomology:

∗ ∗ ∗ ∗ 0 H (GL3(2); F2) H (S4; F2) H (S4; F2) H (D8; F2) 0 → → ⊕ → → ∗ ∗ ∗ ∗ H (GL3(2); F2) = H (S4; F2) + H (S4; F2) H (D8; F2) − Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

A 2-Local Geometry for Co3

G - Conway’s third sporadic Co3 ∆ - subgroup complex with vertex stabilizers given below:

Gp = 2.S6(2) ◦P ◦ L ◦M G = 22+63.(S S ) L 3 × 3 4 GM = 2 .L4(2)

Theorem (Maginnis and Onofrei, 2004)

The 2-local geometry ∆ for Co3 is homotopy equivalent to the complex of distinguished 2-radical subgroups b2(Co3) ; 2-radical subgroups containing 2-central involutions|B in their| centers. Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules Distinguished Collections of p-Subgroups

An element of order p in G is p-central if it lies in the center of a Sylow p-subgroup of G.

Let p(G) be a collection of p-subgroups of G. C

Definition

The distinguished collection bp(G) is the collection of C subgroups in p(G) which contain p-central elements in their centers. C Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules Poset Homotopy

Two G-posets are G-homotopy equivalent if they are homotopy equivalent and the homotopies are G-equivariant.

A poset is conically contractible if thereC is a poset map f : and an element x suchC that →x C f (x) x for all x . 0 ∈ C ≤ ≥ 0 ∈ C THEOREM [Thevenaz´ and Webb,1991]: Let . Assume that for all y the subposet C ⊆ D ∈ D ≤y = x x y C { ∈ C | ≤ } is Gy -contractible. Then the inclusion is a G-homotopy equivalence. Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Proposition (Maginnis and Onofrei, 2005)

The inclusion bp(G) , bp(G) is a G-homotopy equivalence. A → S

Proof.

Let P bp(G) and let Q bp(G)≤P . ∈ S ∈ A Pb is the subgroup generated by the p-central elements in Z (P). The subposet bp(G)≤P is contractible via the double inequality: A Q Pb Q Pb ≤ · ≥ The poset map Q Pb Q is N (P)-equivariant. → · G Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules Groups of Parabolic Characteristic p

G has characteristic p if C (Op(G)) Op(G). G ≤ G has local characteristic p if all p-local subgroups of G have characteristic p.

G has parabolic characteristic p if all p-local subgroups which contain a Sylow p-subgroup of G have characteristic p.

Theorem (Maginnis and Onofrei, 2007) Let G be a finite group of parabolic characteristic p. Then the collections bp(G), bp(G) and bp(G) are G-homotopy equivalent. B A S Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules Fixed Point Sets

Proposition (Maginnis and Onofrei, 2007 ) Let G be a finite group of parabolic characteristic p. Let z be a p-central element in G and let Z = z . Z h i Then the fixed point set bp(G) is N (Z )-contractible. |B | G

Proposition (Maginnis and Onofrei, 2007 ) Let G be a finite group of parabolic characteristic p. Let t be a noncentral element of order p and let T = t . h i Assume that Op(CG(t)) contains a p-central element. T Then the fixed point set bp(G) is N (T )-contractible. |B | G Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Theorem (Maginnis and Onofrei, 2007 ) Assume G is a finite group of parabolic characteristic p. Let T = t with t an element of order p of noncentral type in G. h i Let C = CG(t). Suppose that the following hypotheses hold:

Op(C) does not contain any p-central elements;

The C = C/Op(C) has parabolic characteristic p.

Then there is an NG(T )-equivariant homotopy equivalence

T bp(G) bp(C) |B | ' |B | Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Sketch of Proof: The proof requires a combination of equivariant homotopy equivalences:

T T ≤C ≤C bp(G) bp(G) bp(G) ep(G) |B | ' |S | ' |S >T | ' |S >T | ≤C ≤C ep(G) bp(G) S bp(C) bp(C) ' |S >OC | ' |S >OC | ' | | ' |S | ' |B | Some of the notations used:

Sep(G) = p-subgroups of G which contain p-central elements , { } ≤H = Q P < Q H , C>P { ∈ C | ≤ } OC = Op(C) and C = CG(t),

≤C S = P bp(G) Z(P) Z (S) = 1, { ∈ S >OC ∩ 6 for S and S such that P S S , T ≤ T ≤ } S Sylp(C) which extends to S Sylp(G). T ∈ ∈ Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules Terminology from Representation Theory

k, a field of characteristic p, splitting field for G and all its subgroups; kG, the group algebra of G over the field of coefficients k; G IndH (N) = kG kH N, the induced module, ⊗ for kH-module N and H G; ≤ Ind G (k) k[X], permutation module, Gx ' for X a G-transitive set and x X. ∈ A kG-module M is relatively H-projective if M is a direct summand of an module induced from H. Let M be an indecomposable kG-module; V is a vertex of M if M is relatively V -projective, but is not relatively U-projective, for any proper subgroup U of V . Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

A block B of kG is an indecomposable two-sided ideal of kG. The defect group of a block B, is a subgroup D G with the property that δD = (g, g); g D G is a vertex≤ of the k(G G)-module B{. ∈ ≤ } × The Green ring is a free abelian group generated by the isomorphism classes [M] of finitely generated indecomposable kG-modules. The ring structure is given by direct sums and k-tensor products. The reduced Lefschetz module, an element of the Green ring: X Le (∆; k) = ( 1)|σ|Ind G (k) k G − Gσ − σ∈∆/G

THEOREM [Robinson, 1988 ]: The number of indecomposable summands of LeG(∆; k) with vertex Q is equal to the number of indecomposable summands (∆Q; ) of LeNG(Q) k with the same vertex Q. Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules The Reduced Lefschetz Module for the Distinguished p-Radical Complex

Webb’s alternating sum formula holds for the distinguished Bouc complex:

n X dim(σ) n Hb (G; k) = ( 1) Hb (Gσ; k) − σ∈|Bbp(G)|/G

Assume that G has parabolic characteristic p. The vertices of the reduced Lefschetz module associated to bp(G) are p-subgroups of pure noncentral type. B Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

A 2-Local Geometry for Fi22

G = Fi22 has parabolic characteristic 2. G has three conjugacy classes of involutions:

CFi22 (2A) = 2.U6(2), C (2B) = (2 21+8 : U (2)) : 2, Fi22 × + 4 C (2C) = 25+8 :(S 32 : 4). Fi22 3 × The class 2B is 2-central. ∆ is the 2-local geometry with vertex stabilizers: 1+8 5+8 H1 = (2 2+ : U4(2)) : 2 H2 = 2 :(S3 A6) 6 × 10 × H3 = 2 : Sp6(2) H4 = 2 : M22

∆ is G-homotopy equivalent to b2(Fi22). B Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Theorem (Maginnis and Onofrei, 2007)

Let ∆ be the 2-local geometry for the Fi22. a The fixed point sets ∆2B and ∆2C are contractible. b The fixed point set ∆2A is equivariantly homotopy equivalent to the building for the U6(2). c There is precisely one nonprojective summand of the reduced Lefschetz module, it has vertex 2A and lies in a block with the same group as defect group. h i d As an element of the Green ring:

Le (∆) = P (ϕ12) P (ϕ13) 6ϕ15 12P (ϕ16) ϕ16. Fi22 − Fi22 − Fi22 − − Fi22 − Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules Summary of the Talk

New collections of subgroups were introduced: emphasize the role of p-central elements; have homotopy properties similar to the standard collections of p-subgroups; are suited for cohomology computations; are related to the p-local geometries for the sporadic simple groups. Further objectives: determine the vertices of the reduced Lefschetz modules for other classes of groups; obtain a general description of the indecomposable summands of the reduced Lefschetz modules and their distribution into the blocks of the group ring.