Equivalences between fusion systems of finite groups of ... · Groups appear in di erent contexts...

Equivalences between fusion systems of finitegroups of Lie type

Carles Broto

Universitat Autonoma de Barcelona

EMS-RSME Joint Mathematical Weekend.Bilbao, October 9, 2011

Carles Broto Equivalences between fusion systems of finite groups of Lie type

Groups and local methods in homotopy theory

Groups appear in different contexts in homotopy theory:

as objects of interest on their own, e.g. compact Lie groups,groups of symmetries, or

as algebraic invariants, like the fundamental group of atopological space.

Local methods, first applied to algebraic invariants, e.g. Serre:p-torsion in homotopy groups of spheres.

But then applied to topological spaces themselves: Sullivan,Bousfield-Kan localization and completion functors for topologicalspaces.


p-equivalences and p-completion

Fix a prime p.

Two spaces X and Y are p-equivalent if there is a 3rd space Z andmaps

X → Z ← Y

that induce isomorphisms in cohomology with coefficients in Fp.

p-completion is a coaugmented functor `X : X → X∧p that turnsp-equivalences into homotopy equivalences.

Found useful in order to study compact (connected) Lie groupsfrom a homotopy theoretic point of view, or its generalisations,finite loop spaces.


Classifying spaces

Given G, (compact Lie) group,there is a universal contractible G-space EG.

The space of orbits is BG = EG/G is determined by G, up tohomotopy.We call BG the classifying space of G

Examples

G Unitary group U(n), BU(n) Infinite grasmannians, egG = S1, BS1 ' CP∞ (ES1 ' S∞).)

G discrete, BG = K(G, 1), e.g. G = Z/2, EZ/2 ' S∞ andBZ/2 ' RP∞.

S1 ' BZ, (S1)∧p ' BZp.

Algebraic and geometric properties of G are reflected as homotopytheoretic properties of BG.


Homotopical group theory

Break BG into BG∧p for all p and the rationalization. Pieces canbe glued back again if G is connected, thus recovering BG.

∗ There is a way to introduce a twisting in the gluing process, soproducing many finite loop spaces that are not Lie groups.

Mod p finite loop spaces or p-compact groups [Dwyer-Wilkerson]are homotopy theoretic objects designed as generalizations ofBG∧p .

These has been classified as a list of simple objects that resemblesthe classification of compact Lie groups. eg. For p=2, there is onlyone simple mod 2 finite loop space which is not the 2-completionof a compact Lie group: DI(4) [Dwyer-Wilkerson]


Case of finite groups

Question: how much of G or BG is determined by H∗(G,Fp)?

Example:

Martino-Priddy: There are 2-equivalencesBSL2(q) '2 BSL2(q′) that are not induced by a grouphomomorphism or a sequence of group homomorphisms offinite groups.


More examples:

Martino-Priddy (BM11) '2 BPSL3(3).

Quillen: BGLn(q) and BGLn(q′) have same cohomology ringwith coefficients in Fp, p not dividing q and q′, and samecohomology operations, when q ≡ q′(p) andvp(q

s − 1) = vp(q′s − 1), s = ordp(q).

B-Levi: For an odd prime power q,H∗(X,F2) ∼= H∗(BPSL2(q),F2) as algebra+operators =⇒X and BPSL2(q) are 2-equivalent.(The cohomology ring only depends on q mod 8 andv2(q2 − 1).)


The Martino-Priddy conjecture



Fix S ∈ Sylp(G), then Fp(G) = FS(G) is a category with

objects: P ≤ S, the subgroups of S, and

morphisms:

HomFp(G)(P,Q) = {ϕ : P → Q|∃g ∈ G,ϕ(x) = gxg−1}∼= NG(P,Q)/CG(P ) .

If G′ is another group and S′ ∈ Sylp(G′), we say that

Fp(G) ' Fp(G′)if there is an isomorphism f : S −→ S′ that preserves fusion.

i.e.: ϕ ∈ HomFp(G)(P,Q) =⇒ (f |Q) ◦ ϕ ◦ (f |P )−1

∈ HomFp(G′)(f(P ), f(Q)).



Martino-Priddy: Given finite groups G and H,

BG∧p ' BH∧p =⇒ Fp(G) ' Fp(H)

M-P conjecture: Converse is also true.

Oliver: M-P conjecture holds. Depends on classification of finitesimple groups.

Chermak: Generalization to abstract fusion systems. Alsodepending on CFSG.


Finite groups of Lie type

What follows is joint work with Jesper Møller and Bob Oliver.

Let G be a connected reductive group scheme over Z. Fix a primep, and finite fields Fq, Fq′ of char 6= p, then:

a) Fp(G(q)) ' Fp(G(q′)) if 〈q〉 = 〈q′〉 ≤ Z×p(i.e.,

ordp(q) = ordp(q′) = s and vp(q

s − 1) = vp(q′s − 1), if p is odd,

q ≡ q′ mod 8 and v2(q2 − 1) = v2(q′2 − 1), if p = 2.)

(e.g. Fp(SLn(q)) ' Fp(SLn(q′)) iff they have the same mod pcohomology).


b) For G = An, Dn, E6 and τ a graph automorphism,

Fp(τG(q)) ' Fp(τG(q′)) if 〈q〉 = 〈q′〉.

c) In case the Weyl group of G contains an element which actson the maximal torus by inverting all elements: ψ−1, then

Fp(G(q)) ' Fp(G(q′))

Fp(τG(q)) ' Fp(τG(q′)) if G and τ are as in (b)

provided 〈−1, q〉 = 〈−1, q′〉 ≤ Z×p .

d) For G of type An, Dn with n odd, or E6, and τ a graphautomorphism of order 2,

Fp(τG(q)) ' Fp(G(q′)) if 〈−q〉 = 〈q′〉 ≤ Z×p

(e.g. Fp(2E6(q)) ' Fp(E6(q′)) if −q and q′ generate thesame closed subgroup of Z∗p)


Some additional cases:If p ≡ 1 mod p,

p 6= 3, Fp(G2(q)) ' Fp(3D4(q))

p 6= 2, Fp(F4(q)) ' Fp(2E6(q))

If p is odd and q ≡ 1 mod p

Fp(SU2n(q)) ' Fp(Sp2n(q′))

Fp(SU2n+1(q)) ' Fp(Spin2n+1(q′))

Fp(Spin−2n(q)) ' Fp(Spin2n−1(q))


Ingridients of the proof

(i) Results by Friedlander and by Jacksowski-McClure-Oliver:

If G is a connected reductive group scheme over Z and (p, q) = 1,there is a homotopy pull-back square

BτG(q)∧p //

��

BG(C)∧p

∆��

BG(C)∧p1×τψq

// BG(C)∧p ×BG(C)∧p

where ψq is the unstable Adams map of exponent q and τ a graphautomorphism. (BG(C)∧p ' BG∧p , G the unitary form of G(C).)

The case G = GL was first considered by Quillen in order tocompute H∗(GLn(q),Fp). (Related to his work on K-theory offinite fields.)


(ii) Homotopy fixed points.

In a pulback diagram:

Xhα //

��

X

∆��

X1×α // X ×X

for α a self map of X,

we can interpret Xhα ={w : I → X

∣∣ w(1) = α(w(0))}

as aspace of homotopy fixed points. Then, for α a homeomorphism,Xhα = Γ(Xhα ↓ S1) is a space of sections of the fibre bundle:

X // Xhα// S1

where Xhα = X × I/ ∼, (0, x) ∼ (1, α(x)), is the mapping torus.

Similar considerations hold if α is a homotopy equivalence.


Key observation: Sometimes X // Xhα// S1(' BZ)

extends to a fibrationX∧p // (Xhα)∧p // (S1)∧p (' BZp)

thus, the action of Z generated by α extends to an action of Zp.

Theorem

Fix a prime p, X a connected and p-complete space satisfying

H∗(X,Fp) Noetherian

Out(X) detected H∗(X,Zp) := lim←−H∗(X,Z/pk).

Let α, β be self equivalences of X with 〈α〉 = 〈β〉 in Out(X), then

Xhα ' Xhβ.

Here, Out(X) is given the p-adic topology with basis of openneighborhoods of the identity:

Uk = {[f ] ∈ Out(X)|f∗ = id on H∗(X,Z/pk}


(iii) Classification of self equivalences of BG∧p , G compactconnected Lie group [Jackowski-McClure-Oliver]

i) Out(BG∧p ) detected on H∗Qp(BG∧p ) := H∗(BG∧p ;Zp)⊗Q.

ii) Detected on BT∧p : The action of the Weyl group W on themaximal torus T , gives W ≤ GLn(Zp) and a diagram

e.g. If G simple, W is irreducible and

CGLn(Zp)(W ) ∼= Z×p 3 q ←→ ψq unstable Adams map


Equivalences between fusion systems of finite groups of ... · Groups appear in di erent contexts...

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