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 ¡ ¡ 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 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 order H:K denotes an extension of H by K pn denotes an elementary abelian group of order pn Op(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 (Q) ≤ G Q is p-radical if Q = Op(NG(Q)) Q is p-centric if Z(Q) Syl (C (Q)) 2 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 = ∆( ) jCj C simplices: σ = (Q0 < Q1 < : : : < Qn), Qi n 2 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 eL (∆( ); 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 jS j 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: eLG(∆; Zp) is virtual projective module n P dim(σ) n Hb (G; M)p = ( 1) Hb (Gσ; M)p σ2∆=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 01 1 0 1 01 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ Gp = @0 A GL = @ A GpL = @0 1 A 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) eL ( ) = H (∆) = St GL3(2) jB2j − 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) eL ( ) = H (∆) = St GL3(2) jB2j − 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 simple group 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 involutionsjB in theirj 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!C that x f (x) x for all x . 0 2 C ≤ ≥ 0 2 C THEOREM [Thevenaz´ and Webb,1991]: Let . Assume that for all y the subposet C ⊆ D 2 D ≤y = x x y C f 2 C j ≤ g 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 . 2 S 2 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. jB j 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. jB j 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 quotient group C = C=Op(C) has parabolic characteristic p. Then there is an NG(T )-equivariant homotopy equivalence T bp(G) bp(C) jB j ' jB j 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) jB j ' jS j ' jS >T j ' jS >T j ≤C ≤C ep(G) bp(G) S bp(C) bp(C) ' jS >OC j ' jS >OC j ' j j ' jS j ' jB j Some of the notations used: Sep(G) = p-subgroups of G which contain p-central elements , f g ≤H = Q P < Q H , C>P f 2 C j ≤ g OC = Op(C) and C = CG(t), ≤C S = P bp(G) Z(P) Z (S) = 1; f 2 S >OC \ 6 for S and S such that P S S , T ≤ T ≤ g S Sylp(C) which extends to S Sylp(G).