Subgroup Complexes and their Lefschetz Modules Silvia Onofrei - PowerPoint PPT Presentation

Outline Terminology History GL 3 ( 2 ) Distinguished Collections Lefschetz Modules Subgroup Complexes and their Lefschetz Modules Silvia Onofrei Department of Mathematics Kansas State University

Outline Terminology History GL 3 ( 2 ) Distinguished Collections Lefschetz Modules nonabelian finite simple groups associated geometries alternating groups ( n ≥ 5) Tits buildings groups of Lie type sporadic geometries 26 sporadic groups

Outline Terminology History GL 3 ( 2 ) Distinguished Collections Lefschetz Modules nonabelian finite simple groups associated geometries alternating groups ( n ≥ 5) Tits buildings groups of Lie type sporadic geometries 26 sporadic groups ✁ ❆ ✁ ❆ ✁ ❆ ✁ group theory ✁ ✁ p-local structure ✁ ✁ ✁ complexes of p-subgroups ✁ ❆ ✁ ❆ ✁ ❆ algebraic topology representation theory mod-p cohomology Lefschetz modules classifying spaces

Outline Terminology History GL 3 ( 2 ) Distinguished Collections Lefschetz Modules Outline of the Talk 1 Terminology and Notation Background, History and Context 2 An Example: GL 3 ( 2 ) 3 Distinguished Collections of p -Subgroups 4 Lefschetz Modules for Distinguished Complexes 5

Outline Terminology History GL 3 ( 2 ) Distinguished Collections Lefschetz Modules Terminology and Notation: Groups G is a finite group and p a prime dividing its order H . K denotes an extension of H by K p n denotes an elementary abelian group of order p n O p ( G ) is the largest normal p -subgroup in G Q a nontrivial p -subgroup of G H ≤ G is p-local subgroup if H = N G ( Q ) Q is p-radical if Q = O p ( N G ( Q )) Q is p-centric if Z ( Q ) ∈ Syl p ( C G ( Q ))

Outline Terminology History GL 3 ( 2 ) Distinguished Collections Lefschetz Modules 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 G σ = ∩ n isotropy group of σ : i = 0 N G ( Q i ) ∆( C ) Q fixed point set of Q : Let k be a field of characteristic p . The reduced Lefschetz kG-module : dim (∆) � � ( − 1 ) i C i (∆( C ); k ) L G (∆( C ); k ) := i = − 1

Outline Terminology History GL 3 ( 2 ) Distinguished Collections Lefschetz Modules Standard Collections of p-Subgroups Brown S p ( G ) nontrivial p -subgroups A p ( G ) Quillen nontrivial elementary abelian p -subgroups Bouc B p ( G ) nontrivial p -radical subgroups Quillen, 1978 A p ( G ) ⊆ S p ( G ) is homotopy equivalence � L G ( |S p ( G ) | ; k ) is virtual projective module Th´ evenaz, Webb, 1991 A p ( G ) ⊆ S p ( G ) ⊇ B p ( G ) are equivariant homotopy equivalences

Outline Terminology History GL 3 ( 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: L G (∆; Z p ) is virtual projective module H n ( G ; M ) p = � σ ∈ ∆ / G ( − 1 ) dim ( σ ) � � H n ( G σ ; M ) p

Outline Terminology History GL 3 ( 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 GL 3 ( 2 ) Distinguished Collections Lefschetz Modules An Example: GL 3 ( 2 ) The Tits building: the extrinsic approach � e 1 � Fano Plane V = F 3 2 = � e 1 , e 2 , e 3 � � e 1 + e 2 � � e 1 + e 3 � p = � e 1 � L = � e 1 , e 2 � � e 1 + e 2 + e 3 � pL = ( � e 1 � ⊆ � e 1 , e 2 � ) � e 2 � � e 2 + e 3 � � e 3 � Stabilizers       1 ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗       G p = 0 ∗ ∗ G L = ∗ ∗ ∗ G pL = 0 1 ∗ 0 ∗ ∗ 0 0 1 0 0 1

Outline Terminology History GL 3 ( 2 ) Distinguished Collections Lefschetz Modules The Tits building for GL 3 ( 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 2 2 p L D 8 a b � � � � � Barycentric subdivision of Tits building = Bouc complex G p = S 4 = 2 2 a . S 3 = N G ( 2 2 a ) G L = S 4 = 2 2 b . S 3 = N G ( 2 2 b ) G pL = D 8 = 2 1 + 2 = N G ( D 8 ) N G ( 2 2 a < D 8 ) = N G ( 2 2 b < D 8 ) = D 8

Outline Terminology History GL 3 ( 2 ) Distinguished Collections Lefschetz Modules The reduced Lefschetz module of the Bouc complex = Steinberg module for GL 3 ( 2 ) � L GL 3 ( 2 ) ( |B 2 | ) = − H 1 (∆) = − St GL 3 ( 2 )

Outline Terminology History GL 3 ( 2 ) Distinguished Collections Lefschetz Modules The reduced Lefschetz module of the Bouc complex = Steinberg module for GL 3 ( 2 ) � L GL 3 ( 2 ) ( |B 2 | ) = − H 1 (∆) = − St GL 3 ( 2 ) Webb’s alternating formula for mod-2 cohomology: 0 → H ∗ ( GL 3 ( 2 ); F 2 ) → H ∗ ( S 4 ; F 2 ) ⊕ H ∗ ( S 4 ; F 2 ) → H ∗ ( D 8 ; F 2 ) → 0 H ∗ ( GL 3 ( 2 ); F 2 ) = H ∗ ( S 4 ; F 2 ) + H ∗ ( S 4 ; F 2 ) − H ∗ ( D 8 ; F 2 )

Outline Terminology History GL 3 ( 2 ) Distinguished Collections Lefschetz Modules A 2-Local Geometry for Co 3 G - Conway’s third sporadic simple group Co 3 ∆ - subgroup complex with vertex stabilizers given below: ◦ ◦ G p = 2 . S 6 ( 2 ) ◦ P L M G L = 2 2 + 6 3 . ( S 3 × S 3 ) G M = 2 4 . L 4 ( 2 ) Theorem (Maginnis and Onofrei, 2004) The 2 -local geometry ∆ for Co 3 is homotopy equivalent to the complex of distinguished 2 -radical subgroups | � B 2 ( Co 3 ) | ; 2 -radical subgroups containing 2 -central involutions in their centers.

Outline Terminology History GL 3 ( 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 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.

Outline Terminology History GL 3 ( 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 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 . T HEOREM [Th´ evenaz 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.

Outline Terminology History GL 3 ( 2 ) Distinguished Collections Lefschetz Modules Proposition (Maginnis and Onofrei, 2005) The inclusion � → � A p ( G ) ֒ S p ( G ) is a G-homotopy equivalence. Proof. Let P ∈ � S p ( G ) and let Q ∈ � A p ( G ) ≤ P . � P is the subgroup generated by the p -central elements in Z ( P ) . The subposet � A p ( G ) ≤ P is contractible via the double inequality: Q ≤ � P · Q ≥ � P The poset map Q → � P · Q is N G ( P ) -equivariant.

Outline Terminology History GL 3 ( 2 ) Distinguished Collections Lefschetz Modules Groups of Parabolic Characteristic p 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 . Theorem (Maginnis and Onofrei, 2007) Let G be a finite group of parabolic characteristic p. Then the collections � B p ( G ) , � A p ( G ) and � S p ( G ) are G-homotopy equivalent.

Outline Terminology History GL 3 ( 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 � . B p ( G ) | Z is N G ( Z ) -contractible. Then the fixed point set | � 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 � . Assume that O p ( C G ( t )) contains a p-central element. B p ( G ) | T is N G ( T ) -contractible. Then the fixed point set | �

Outline Terminology History GL 3 ( 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. 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 ) |