Equivariant Chern characters

Equivariant Chern characters

Wolfgang Lück

Mathematisches Institut Westfälische Wilhelms-Universität Münster

Einsteinstr. 62 D-48149 Münster

http://www.math.uni-muenster.de/u/lueck/

Göttingen, November 2006

Outline

Dolds rational computation of a generalized homology theory in terms of singular homology.

Equivariant homology theories and Chern characters Applications to the Farrell-Jones and the Baum-Connes Conjecture

Rational computation of the topologicalK-theory ofBGfor a groupG.

Theorem (Dold)

LetH_∗ be a generalized homology theory with values inΛ-modules for Q⊆Λ.

Then there exists for every n∈Zand every CW -complex X a natural isomorphism

M

p+q=n

H_p(X; Λ)⊗_ΛH_q(pt)−→ H^∼= _n(X).

This means that theAtiyah-Hirzebruch spectral sequence collapses in the strongest sense.

The assumptionQ⊆Λis necessary.

Dolds’ Chern character for aCW-complexX is given by the following composite

chn: M

p+q=n

Hp(X;H_q(∗))←−^α M

p+q=n

Hp(X;Z)⊗_ZH_q(∗)

L

p+q=nhur⊗id ∼=

←−−−−−−−−−−−− M

p+q=n

π^s_p(X₊,∗)⊗_ZH_q(∗)

L

p+q=nDp,q

−−−−−−−−→ H_n(X).

Definition (G-homology theory)

AG-homology theoryH_∗ is a covariant functor from the category of G-CW-pairs to the category ofZ-gradedΛ-modules together with natural transformations

∂n(X,A) :H_n(X,A)→ H_n−1(A) forn∈Zsatisfying the following axioms:

G-homotopy invariance;

Long exact sequence of a pair;

Excision;

Disjoint union axiom.

Definition (Equivariant homology theory)

Anequivariant homology theoryH^?_∗ assigns to every groupGa G-homology theoryH^G_∗. These are linked together with the following so calledinduction structure: given a group homomorphismα:H→G and aH-CW-pair(X,A)there are for alln∈Znatural homomorphisms

ind_α:H^H_n(X,A) → H^G_n(ind_α(X,A)) satisfying

Bijectivity

If ker(α)acts freely onX, then ind_α is a bijection;

Compatibility with the boundary homomorphisms Functoriality inα

Compatibility with conjugation

Examples for equivariant homology theories are Given aK_∗ non-equivariant homology theory, put

H^G_∗(X) := K_∗(X/G);

H^G_∗(X) := K_∗(EG×_GX) Borel homology.

Equivariant bordismΩ^?_∗(X);

Equivariant topologicalK-theoryK_∗^?(X);

Given a functorE:Groupoids→Spectrasending equivalences to weak equivalences, there exists an equivariant homology theoryH_∗^?(−;E)satisfying

H^H_n(pt)∼=H^G_n(G/H)∼=π_n(E(H)).

Theorem (L.)

LetH_∗^?be a proper equivariant homology theory with values in

Λ-modules forQ⊆Λ. Suppose thatH^?_∗ has a Mackey extension. Let I be the set of conjugacy classes(H)of finite subgroups H of G.

Then there is for every group G, every proper G-CW -complex X and every n∈Za natural isomorphism calledequivariant Chern character

ch^G_n: M

p+q=n

M

(H)∈I

H_p(C_GH\X^H; Λ)⊗_Λ[WGH]S_H

H^H_q(∗) _∼=

−→ H^G_n(X)

C_GH is the centralizer andN_GH the normalizer ofH ⊆G;

W_GH:=N_GH/H·C_GH(This is always a finite group);

S_H H_q^H(∗)

:=cok

L

K⊂H K6=H

ind^H_K :L

K⊂H K6=H

H^K_q(∗)→ H^H_q(∗)

. ch^?_∗ is an equivalence of equivariant homology theories.

Theorem (Artin’s Theorem) Let G be finite. Then the map

M

C⊂G

ind^G_C : M

C⊂G

RepC(C)→Rep

C(G)

is surjective after inverting|G|, where C⊂G runs through the cyclic subgroups of G.

LetCbe a finite cyclic group. TheArtin defectis the cokernel of the map

M

D⊂C,D6=C

ind^C_D : M

D⊂C,D6=C

Rep_C(D)→Rep_C(C).

For an appropriate idempotentθ_C ∈Rep_Q(C)⊗_ZZ h 1

|C|

i

the Artin defect is after inverting the order of|C|canonically isomorphic to

θ_C·Rep_C(C)⊗_ZZ 1

|C|

.

LetK_∗^G be equivariant topologicalK-theory. We get for a finite subgroupH ⊆G

K_n^G(G/H) =K_n^H(pt) =

Rep_C(H) ifnis even;

{0} ifnis odd.

Example

Let G be finite, X ={∗}andH^?_∗ =K_∗^?. Then we get an improvement of Artin’s theorem, namely, the equivariant Chern character induces to an isomorphism

ch^G₀(pt) : M

(C)

Z⊗_Z[W

GC]θ_C·Rep_C(C)⊗_ZZ 1

|G|

∼=

−→Rep_C(G)⊗_ZZ 1

|G|

Theorem (Davis-L)

Let R be a ring (with involution). There exist covariant functors K_R:Groupoids → Spectra;

L^hji_R :Groupoids → Spectra;

K^top:Groupoids^inj → Spectra with the following properties:

They send equivalences of groupoids to weak equivalences of spectra;

For every group G and all n∈Zwe have π_n(K_R(G)) ∼= K_n(RG);

πn(L^hji_R (G)) ∼= L^hji_n (RG);

π_n(K^top(G)) ∼= K_n(C_r^∗(G)).

Definition (Family of subgroups)

AfamilyF of subgroupsof the groupGis a set of subgroups ofG which is closed under conjugation and taking subgroups.

Examples for families are {1} trivial subgroup F IN finite subgroups

VCYC virtually cyclic subgroups ALL all subgroups

Definition (Classifying space of a family)

LetF be a family of subgroups ofG. A model for theclassifying space of the familyF is aG-CW-complexEF(G)such thatEF(G)^H is

contractible ifH∈ F and is empty ifH 6∈ F.

SometimesE G:=EF IN(G)is called theclassifying space for proper G-actions.

Theorem (tom Dieck)

The G-CW -complex EF(G)is characterized uniquely up to

G-homotopy by the property that for every G-CW -complex X whose isotropy groups belong toF there is up to G-homotopy precisely one G-map X →EF(G).

ObviouslyE_{1}(G) =EGandEALL(G) =G/G.

The spacesE G are interesting in their own right and have often very nice geometric models which are rather small. For instance

Rips complexfor word hyperbolic groups;

Teichmüller spacefor mapping class groups;

Outer spacefor the group of outer automorphisms of free groups;

L/K for a connected Lie groupL, a maximal compact subgroup K ⊆LandG⊆La discrete subgroup;

CAT(0)-spaceswith proper isometricG-actions, e.g., Riemannian manifolds with non-positive sectional curvature or trees.

Conjecture (Farrell-Jones)

TheFarrell-Jones Conjecture for algebraic K -theorywith coefficients in R for the group G predicts that the assembly map

H_n^G(EVCYC(G),K_R)→H_n^G(pt,K_R) =K_n(RG) is bijective for all n∈Z.

The Farrell-Jones Conjecture gives a way to computeKn(RGH)in terms ofK_m(RV)for all virtually cyclic subgroupsV ⊆Gand allm≤n.

Theorem (Bartels-L.-Reich)

The (Fibered) Farrell-Jones Conjecture for algebraic K -theory with (G-twisted) coefficients in any ring R is true for word-hyperbolic groups G.

It is analogous to the Baum-Connes Conjecture which is the version for topologicalK-theory of (reduced) groupC^∗-algebras.

Conjecture (Baum-Connes)

TheBaum-Connes Conjecturepredicts that the assembly map K_n^G(EG) =H_n^G(E_Fin(G),K^top)→H_n^G(pt,K^top) =K_n(C_r^∗(G)) is bijective for all n∈Z.

Theorem (L.)

Let G be a group. Let T be the set of conjugacy classes(g)of elements g ∈G of finite order. There is a commutative diagram

L

p+q=n

L

(g)∈TH_p(BC_Ghgi;C)⊗_ZK_q(C) //

Kn(CG)⊗_ZC

L

p+q=n

L

(g)∈THp(BC_Ghgi;C)⊗_ZK_q^top(C) ^//K_n^top(C_r^∗(G))⊗_ZC The vertical arrows come from the obvious change of rings and of K-theory maps.

The horizontal arrows can be identified with the assembly maps occurring in the Farrell-Jones Conjecture and the Baum-Connes Conjecture by the equivariant Chern character.

Splitting principle.

One can spell out the Farrell-Jones Conjecture also for other theories liketopological Hochschild homologyandtopological cyclic homologyand compute the source of assembly map rationally using equivariant Chern characters.

Injectivity and Bijectivity results have been obtained for such theories byL.-Rognes-Reich-Varisco.

In particularL.-Rognes-Reich-Variscoextend the result of Bökstedt-Hsiang-Madsenfor{1}toF IN thus proving rational injectivity of theK-theoretic Farrell-Jones assembly map for coefficients inZunder mild homological assumptions.

Theorem (Atiyah-Segal) Let G be a finite group.

Then there are isomorphisms of abelian groups K⁰(BG) ∼= Rep

C(G)IbG

∼= Z× Y

p prime

Ip(G)⊗_ZZbp ∼= Z× Y

p prime

(Zbp)^r(p);

K¹(BG) ∼= 0.

For a primepdenote byr(p) =|con_p(G)|the number of

conjugacy classes(g)of elementsg6=1 inGofp-power order.

IG is the augmentation ideal of Rep_C(G).

LetIp(G)be the image of the restriction homomorphism I(G)→I(G_p).

Theorem (L.)

Let G be a discrete group. Denote by K^∗(BG)the topological

(complex) K-theory of its classifying space BG. Suppose that there is a cocompact G-CW -model for the classifying space E G for proper G-actions.

Then there is aQ-isomorphism chⁿ_G:Kⁿ(BG)⊗_ZQ

∼=

−→

Y

i∈Z

H²ⁱ⁺ⁿ(BG;Q)

!

× Y

p prime

Y

(g)∈conp(G)

Y

i∈Z

H²ⁱ⁺ⁿ(BC_Ghgi;Qbp)

! ,

The multiplicative structure can also be determined.

Theorem (L.)

Let X be a proper G-CW -complex. LetZ⊆Λ^G ⊂Qbe the subring of Qobtained by inverting the orders of all the finite subgroups of G.

Then there is a natural isomorphism ch^G: M

(C)

Kn(C_GC\X^C)⊗_Z[W

GC]θ_C·Rep_C(C)⊗_ZΛ^G

∼=

−→K_n^G(X)⊗_ZΛ^G, where(C)runs through the conjugacy classes of finite cyclic

subgroups.

Here is a problem concerning the theorem above.

TakeX =E G. Elements inK₀(E G)are given by elliptic G-operatorsP over cocompact properG-manifolds with Riemannian metrics.

What is the concrete preimage of its class under ch^G₀?

One term could be the index ofP^C onM^Cgiving an element in K₀(C_GC\E^C)which isK₀(BC_GC)after tensoring withΛ^G.

Another term could come from the normal data ofM^C inMwhich yields an element inθ_C·Rep_C(C).

Strategy: Use the pairing

K₀^G(X)⊗K_G⁰(X)→Z

given by twisting aG-operator with aG-vector bundle and then taking its index and the cohomological Chern character forK_G⁰ which hasK_G⁰as source and which is compatible with the obvious pairing on the “easy´´ sides of the two Chern characters.