On Fixed Point Sets and Lefschetz Modules for Sporadic Simple Groups

On Fixed Point Sets and Lefschetz Modules for Sporadic Simple Groups Silvia Onofrei in collaboration with John Maginnis Kansas State University Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 1/15

Terminology and Notation: Groups G is a finite group and p a prime dividing its order Q a nontrivial p-subgroup of G Q is p-radical if Q = O p (N G (Q)) Q is p-centric if Z (Q) Syl p (C G (Q)) G has characteristic p if C G (O p (G)) O p (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 Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 2/15

Terminology and Notation: Collections Collection C family of subgroups of G closed under G-conjugation partially ordered by inclusion Subgroup complex C = (C ) simplices: σ = (Q 0 < Q 1 <... < Q n ), Q i C isotropy group of σ: G σ = n i=0 N G(Q i ) fixed point set of Q: C Q = (C ) Q Standard collections all subgroups are nontrivial Brown S p (G) p-subgroups Quillen A p (G) elementary abelian p-subgroups Bouc B p (G) p-radical subgroups Bp cen (G) p-centric and p-radical subgroups Equivariant homotopy equivalences: A p (G) S p (G) B p (G) Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 3/15

Terminology and Notation: Lefschetz Modules k /G field of characteristic p subgroup complex the orbit complex of The reduced Lefschetz module alternating sum of chain groups element of Green ring of kg LG ( ;k) := i= 1 ( 1)i C i ( ;k) LG ( ;k) = σ /G ( 1) σ Ind G G σ k k for a Lie group in defining characteristic L G ( S p (G) ;k) is equal to the Steinberg module L G ( S p (G) ;k) is virtual projective module Thévenaz (1987): L G ( ;k) is X -relatively projective X is a collection of p-subgroups Q is contractible for every p-subgroup Q X Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 4/15

Background, History and Context if Q is contractible for Q any subgroup of order p then L G ( ;Z p ) is virtual projective module and Ĥn (G;M) p = σ /G ( 1) σ Ĥn (G σ ;M) p Webb, 1987 sporadic geometries with projective reduced Lefschetz modules Ryba, Smith and Yoshiara, 1990 relate projectivity of the reduced Lefschetz module for sporadic geometries to the p-local structure of the group Smith and Yoshiara, 1997 L( B p cen ; k) is projective relative to the collection of p-subgroups which are p-radical but not p-centric Sawabe, 2005 connections between 2-local geometries and standard complexes for the 26 sporadic simple groups Benson and Smith, 2008 Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 5/15

A 2-Local Geometry for Co 3 G - Conway s third sporadic simple group Co 3 - standard 2-local geometry with vertex stabilizers given below: P L M G p = 2.Sp 6 (2) G L = 2 2+6 3.(S 3 S 3 ) G M = 2 4.L 4 (2) Theorem [MO] The 2-local geometry for Co 3 is equivariant homotopy equivalent to the complex of distinguished 2-radical subgroups B 2 (Co 3 ) ; 2-radical subgroups containing 2-central involutions in their centers. Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 6/15

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 C p (G) be a collection of p-subgroups of G. Definition The distinguished collection C p (G) is the collection of subgroups in C p (G) which contain p-central elements in their centers. Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 7/15

A Homotopy Equivalence Proposition [MO] The inclusion A p (G) Ŝp(G) is a G-homotopy equivalence. Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 8/15

A Homotopy Equivalence Proposition [MO] The inclusion A p (G) Ŝp(G) is a G-homotopy equivalence. A poset C is conically contractible if there is a poset map f : C C and an element x 0 C such that x f (x) x 0 for all x C. Theorem [Thévenaz and Webb, 1991]: Let C D. Assume that for all y D the subposet C y = {x C x y} is G y -contractible. Then the inclusion is a G-homotopy equivalence. Proof. Let P Ŝp(G) and let Q A p (G) P. P is the subgroup generated by the p-central elements in Z (P). The subposet Ap (G) P is contractible via the double inequality: Q P Q P The poset map Q P Q is N G (P)-equivariant. Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 8/15

The Distinguished Bouc Collection If G has parabolic characteristic p, then B p (G) Ŝp(G) is a G-homotopy equivalence Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 9/15

The Distinguished Bouc Collection If G has parabolic characteristic p, then B p (G) Ŝp(G) is a G-homotopy equivalence Webb s alternating sum formula holds for Bp (G) H (G; L G ( B p ;k)) = 0 Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 9/15

The Distinguished Bouc Collection If G has parabolic characteristic p, then B p (G) Ŝp(G) is a G-homotopy equivalence Webb s alternating sum formula holds for Bp (G) B cen p H (G; L G ( B p ;k)) = 0 B p B p if G has parabolic characteristic p then B p = B cen p B p (G) is homotopy equivalent to the standard 2-local geometry for all but two (Fi 23 and O N) sporadic simple groups B p (G) preserves the geometric interpretation of the points of the geometry in cases where B cen p does not in Co 3, the 2-central involutions (the points of the geometry) are 2-radical but not 2-centric Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 9/15

Fixed Point Sets Proposition 1 [MO] Let G be a finite group of parabolic characteristic p. Let z be a p-central element in G and let Z = z. Then the fixed point set B p (G) Z is N G (Z )-contractible. Proposition 2 [MO] Let G be a finite group of parabolic characteristic p. Let t be a noncentral element of order p and let T = t. Assume that O p (C G (t)) contains a p-central element. Then the fixed point set B p (G) T is N G (T )-contractible. Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 10/15

Fixed Point Sets Theorem 3 [MO] 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. Let C = C G (t). Suppose that the following hypotheses hold: O p (C) does not contain any p-central elements; The quotient group C = C/O p (C) has parabolic characteristic p. Then there is an N G (T )-equivariant homotopy equivalence B p (G) T B p (C) Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 11/15

Fixed Point Sets: Sketch of the Proof of Theorem 3 The proof requires a combination of equivariant homotopy equivalences: B p (G) T Ŝp(G) T Ŝp(G) C >T S p (G) C >T S p (G) C >O C Ŝp(G) C >O C S Ŝp(C) B p (C) Some of the notations used: S p (G) = {p-subgroups of G which contain p-central elements}, C H >P = {Q C P < Q H}, O C = O p (C) and C = C G (t), C S = {P Ŝp(G) >O Z (P) Z (S) 1, C for S T and S such that P S T S}, S T Syl p (C) which extends to S Syl p (G). Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 12/15

A 2-Local Geometry for Fi 22 G = Fi 22 has parabolic characteristic 2 G has three conjugacy classes of involutions: C Fi22 (2A) = 2.U 6 (2) C Fi22 (2B) = (2 2 1+8 + : U 4(2)) : 2, are 2-central C Fi22 (2C) = 2 5+8 : (S 3 3 2 : 4) is the standard 2-local geometry for G, it is G-homotopy equivalent to B 2 (G) and has vertex stabilizers: 6 1 t 5 10 H 1 = (2 2 1+8 + : U 4(2)) : 2 H 5 = 2 5+8 : (S 3 A 6 ) H 6 = 2 6 : Sp 6 (2) H 10 = 2 10 : M 22 Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 13/15

A 2-Local Geometry for Fi 22 Proposition 4 [MO] Let be the 2-local geometry for the Fischer group Fi 22. 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 Lie group U 6 (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. d. As an element of the Green ring: L Fi22 ( ) = P Fi22 (ϕ 12 ) P Fi22 (ϕ 13 ) 6ϕ 15 12P Fi22 (ϕ 16 ) ϕ 16. Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 14/15

Proof of the Proposition 4 Theorem [Robinson]: The number of indecomposable summands of L G ( ) with vertex Q is equal to the number of indecomposable summands of L NG (Q)( Q ) with the same vertex Q. the involutions 2B are central, Proposition 1 implies 2B is contractible O 2 (C G (2C)) contains 2-central elements, Proposition 2 implies that 2C is contractible Q is mod-2 acyclic for any 2-group Q containing an involution of type 2B or 2C (Smith theory) thus L NG (Q)( Q ) = 0 no vertex of an indecomposable summand of L G ( ) contains an involution of type 2B or 2C Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 15/15

Proof of the Proposition 4 Theorem [Robinson]: The number of indecomposable summands of L G ( ) with vertex Q is equal to the number of indecomposable summands of L NG (Q)( Q ) with the same vertex Q. the involutions 2B are central, Proposition 1 implies 2B is contractible O 2 (C G (2C)) contains 2-central elements, Proposition 2 implies that 2C is contractible Q is mod-2 acyclic for any 2-group Q containing an involution of type 2B or 2C (Smith theory) thus L NG (Q)( Q ) = 0 no vertex of an indecomposable summand of L G ( ) contains an involution of type 2B or 2C C G (2A)/O 2 (C G (2A)) = U 6 (2), Theorem 3 implies that 2A is homotopy equivalent to the building for U 6 (2) Q is contractible for any Q > 2A Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 15/15

Proof of the Proposition 4 Theorem [Robinson]: The number of indecomposable summands of L G ( ) with vertex Q is equal to the number of indecomposable summands of L NG (Q)( Q ) with the same vertex Q. the involutions 2B are central, Proposition 1 implies 2B is contractible O 2 (C G (2C)) contains 2-central elements, Proposition 2 implies that 2C is contractible Q is mod-2 acyclic for any 2-group Q containing an involution of type 2B or 2C (Smith theory) thus L NG (Q)( Q ) = 0 no vertex of an indecomposable summand of L G ( ) contains an involution of type 2B or 2C C G (2A)/O 2 (C G (2A)) = U 6 (2), Theorem 3 implies that 2A is homotopy equivalent to the building for U 6 (2) Q is contractible for any Q > 2A there is one nonprojective summand, it has vertex 2A Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 15/15