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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
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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
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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)
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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
Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 4/15
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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
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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.
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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.
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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.
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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.
Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 8/15
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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.
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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
Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 9/15
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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
Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 9/15
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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
Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 9/15
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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
Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 9/15
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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
Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 9/15
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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.
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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.
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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)|
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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).
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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
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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.
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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〉
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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〉
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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 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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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〉
Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 15/15
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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〉
Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 15/15