On Fixed Point Sets and Lefschetz Modules for Sporadic ... › onofrei.1 › slides › oberw-08.pdffor all but two (Fi23 and O0N) sporadic simple groups Bb p(G) preserves the geometric

On Fixed Point Sets and Lefschetz Modulesfor 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 orderQ a nontrivial p-subgroup of GQ is p-radical if Q = Op(NG(Q))

Q is p-centric if Z (Q) ∈ Sylp(CG(Q))

G has characteristic p if CG(Op(G))≤Op(G)

G has local characteristic p if all p-local subgroups of G havecharacteristic pG has parabolic characteristic p if all p-local subgroups whichcontain a Sylow p-subgroup of G have characteristic p


Terminology and Notation: Collections

Collection C family of subgroups of Gclosed under G-conjugationpartially ordered by inclusion

Subgroup complex |C |= ∆(C )simplices: σ = (Q0 < Q1 < .. . < Qn), Qi ∈ C

isotropy group of σ : Gσ = ∩ni=0NG(Qi )

fixed point set of Q: |C |Q = ∆(C )Q

Standard collections all subgroups are nontrivialBrown Sp(G) p-subgroupsQuillen Ap(G) elementary abelian p-subgroupsBouc Bp(G) p-radical subgroups

Bcenp (G) p-centric and p-radical subgroups

Equivariant homotopy equivalences: Ap(G)⊆Sp(G)⊇Bp(G)


Terminology and Notation: Lefschetz Modules

k field of characteristic p∆ subgroup complex∆/G the orbit complex of ∆

The reduced Lefschetz modulealternating sum of chain groups LG(∆;k) := ∑

|∆|i=−1(−1)iCi (∆;k)

element of Green ring of kG LG(∆;k) = ∑σ∈∆/G(−1)|σ |IndGGσ

k − k

• for a Lie group in defining characteristic LG(|Sp(G)|;k)is equal to the Steinberg module

• LG(|Sp(G)|;k) is virtual projective module• Thevenaz (1987): LG(∆;k) is X -relatively projective

X is a collection of p-subgroups∆Q is contractible for every p-subgroup Q 6∈X


Background, History and Context

if ∆Q is contractible for Q any subgroup of order pthen LG(∆;Zp) is virtual projective moduleand Hn(G;M)p = ∑σ∈∆/G(−1)|σ |Hn(Gσ ;M)p

Webb, 1987

sporadic geometries with projective reduced Lefschetz modulesRyba, Smith and Yoshiara, 1990

relate projectivity of the reduced Lefschetz module for sporadicgeometries to the p-local structure of the group

Smith and Yoshiara, 1997

L(|Bcenp |;k) is projective relative to the collection of p-subgroups

which are p-radical but not p-centricSawabe, 2005

connections between 2-local geometries and standardcomplexes for the 26 sporadic simple groups

Benson and Smith, 2008


A 2-Local Geometry for Co3

G - Conway’s third sporadic simple group Co3∆ - standard 2-local geometry with vertex stabilizers given below:

◦P Gp = 2.Sp6(2)

GL = 22+63.(S3×S3)

GM = 24.L4(2)

◦L

◦M

Theorem [MO]

The 2-local geometry ∆ for Co3 is equivariant homotopy equivalent tothe complex of distinguished 2-radical subgroups |B2(Co3)|;2-radical subgroups containing 2-central involutions in their centers.


Distinguished Collections of p-Subgroups

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

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

Definition

The distinguished collection Cp(G) is the collection of subgroups inCp(G) which contain p-central elements in their centers.


A Homotopy Equivalence

Proposition [MO]

The inclusion Ap(G) ↪→ Sp(G) is a G-homotopy equivalence.

A poset C is conically contractible if there is a poset map f : C → C and anelement x0 ∈ C such that x ≤ f (x)≥ x0 for all x ∈ C .

Theorem [Thevenaz and Webb, 1991]:Let C ⊆D . Assume that for all y ∈D the subposet C≤y = {x ∈ C |x ≤ y}is Gy -contractible. Then the inclusion is a G-homotopy equivalence.

Proof.

Let P ∈ Sp(G) and let Q ∈ Ap(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 ≥ PThe poset map Q→ P ·Q is NG(P)-equivariant.



Proposition [MO]




Proof.





Proposition [MO]




Proof.




The Distinguished Bouc Collection

If G has parabolic characteristic p, then Bp(G) ↪→ Sp(G) is aG-homotopy equivalence

Webb’s alternating sum formula holds for Bp(G)

H∗(G; LG(|Bp|;k)) = 0

Bcenp ⊆ Bp ⊆Bp

if G has parabolic characteristic p then Bp = Bcenp

|Bp(G)| is homotopy equivalent to the standard 2-local geometryfor all but two (Fi23 and O′N) sporadic simple groups

Bp(G) preserves the geometric interpretation of the points of thegeometry in cases where Bcen

p does notin Co3, the 2-central involutions (the points of the geometry) are2-radical but not 2-centric





H∗(G; LG(|Bp|;k)) = 0

Bcenp ⊆ Bp ⊆Bp









H∗(G; LG(|Bp|;k)) = 0

Bcenp ⊆ Bp ⊆Bp









H∗(G; LG(|Bp|;k)) = 0

Bcenp ⊆ Bp ⊆Bp









H∗(G; LG(|Bp|;k)) = 0

Bcenp ⊆ Bp ⊆Bp






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 |Bp(G)|Z is NG(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 Op(CG(t)) contains a p-central element.Then the fixed point set |Bp(G)|T is NG(T )-contractible.


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 |Bp(G)|Z is NG(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 Op(CG(t)) contains a p-central element.Then the fixed point set |Bp(G)|T is NG(T )-contractible.


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. LetC = 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

|Bp(G)|T ' |Bp(C)|


Fixed Point Sets: Sketch of the Proof of Theorem 3

The proof requires a combination of equivariant homotopyequivalences:

|Bp(G)|T ' |Sp(G)|T ' |Sp(G)≤C>T | ' |Sp(G)≤C

>T |

' |Sp(G)≤C>OC| ' |Sp(G)≤C

>OC| ' |S| ' |Sp(C)| ' |Bp(C)|

Some of the notations used:Sp(G) = {p-subgroups of G which contain p-central elements},

C≤H>P = {Q ∈ C | P < Q ≤ H},

OC = Op(C) and C = CG(t),

S = {P ∈ Sp(G)≤C>OC

∣∣∣ Z (P)∩Z (S) 6= 1,

for ST and S such that P ≤ ST ≤ S},ST ∈ Sylp(C) which extends to S ∈ Sylp(G).


A 2-Local Geometry for Fi22

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

CFi22 (2A) = 2.U6(2)

CFi22 (2B) = (2×21+8+ : U4(2)) : 2, are 2-central

CFi22 (2C) = 25+8 : (S3×32 : 4)

∆ is the standard 2-local geometry for G, it is G-homotopyequivalent to B2(G) and has vertex stabilizers:

◦6

•1

JJJJJ

t

•

•

5

10

H1 = (2×21+8+ : U4(2)) : 2

H5 = 25+8 : (S3×A6)

H6 = 26 : Sp6(2)

H10 = 210 : M22


A 2-Local Geometry for Fi22

Proposition 4 [MO]

Let ∆ be the 2-local geometry for the Fischer group 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 Lie group U6(2).c. There is precisely one nonprojective summand of the reduced

Lefschetz module, it has vertex 〈2A〉 and lies in a block with thesame group as defect group.

d. As an element of the Green ring:

LFi22 (∆) =−PFi22 (ϕ12)−PFi22 (ϕ13)−6ϕ15−12PFi22 (ϕ16)−ϕ16.


Proof of the Proposition 4

Theorem [Robinson]: The number of indecomposable summands of LG(∆)

with vertex Q is equal to the number of indecomposable summands ofLNG(Q)(∆Q) with the same vertex Q.

the involutions 2B are central, Proposition 1 implies ∆2B iscontractibleO2(CG(2C)) contains 2-central elements, Proposition 2 impliesthat ∆2C is contractible∆Q is mod-2 acyclic for any 2-group Q containing an involution oftype 2B or 2C (Smith theory) thus LNG(Q)(∆Q) = 0

no vertex of an indecomposable summand of LG(∆) contains aninvolution of type 2B or 2CCG(2A)/O2(CG(2A)) = U6(2), Theorem 3 implies that ∆2A ishomotopy equivalent to the building for U6(2)

∆Q is contractible for any Q > 〈2A〉there is one nonprojective summand, it has vertex 〈2A〉













no vertex of an indecomposable summand of LG(∆) contains aninvolution of type 2B or 2C

CG(2A)/O2(CG(2A)) = U6(2), Theorem 3 implies that ∆2A ishomotopy equivalent to the building for U6(2)








∆Q is contractible for any Q > 〈2A〉

there is one nonprojective summand, it has vertex 〈2A〉









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On Fixed Point Sets and Lefschetz Modules for Sporadic ... › onofrei.1 › slides › oberw-08.pdffor all but two (Fi23 and O0N) sporadic simple groups Bb p(G) preserves the geometric