Equivariant homology theories (Lecture IV)

Equivariant homology theories (Lecture IV)

Wolfgang Lück Münster Germany

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

Hangzhou, July 2007

Flashback

We have introduced theFarrell-Jones Conjectureand the Baum-Connes Conjecturefor torsionfree groups and discussed applications of these conjectures such as to theKaplansky Conjectureand theBorel Conjecture.

We have explained that the formulations for torsionfree groups cannot extend to arbitrary groups.

Our goal is to find a formulation which makes sense for all groups and all rings.

For this purpose we have introduced classifying spaces for families of subgroups of a groupGwhich we will recall next.

In the sequel group will mean discrete group.

Flashback

We have introduced theFarrell-Jones Conjectureand the Baum-Connes Conjecturefor torsionfree groups and discussed applications of these conjectures such as to theKaplansky Conjectureand theBorel Conjecture.

We have explained that the formulations for torsionfree groups cannot extend to arbitrary groups.

Our goal is to find a formulation which makes sense for all groups and all rings.

For this purpose we have introduced classifying spaces for families of subgroups of a groupGwhich we will recall next.

In the sequel group will mean discrete group.

Flashback

We have introduced theFarrell-Jones Conjectureand the Baum-Connes Conjecturefor torsionfree groups and discussed applications of these conjectures such as to theKaplansky Conjectureand theBorel Conjecture.

We have explained that the formulations for torsionfree groups cannot extend to arbitrary groups.

Our goal is to find a formulation which makes sense for all groups and all rings.

For this purpose we have introduced classifying spaces for families of subgroups of a groupGwhich we will recall next.

In the sequel group will mean discrete group.

Flashback

We have introduced theFarrell-Jones Conjectureand the Baum-Connes Conjecturefor torsionfree groups and discussed applications of these conjectures such as to theKaplansky Conjectureand theBorel Conjecture.

We have explained that the formulations for torsionfree groups cannot extend to arbitrary groups.

Our goal is to find a formulation which makes sense for all groups and all rings.

For this purpose we have introduced classifying spaces for families of subgroups of a groupGwhich we will recall next.

In the sequel group will mean discrete group.

Flashback

We have introduced theFarrell-Jones Conjectureand the Baum-Connes Conjecturefor torsionfree groups and discussed applications of these conjectures such as to theKaplansky Conjectureand theBorel Conjecture.

We have explained that the formulations for torsionfree groups cannot extend to arbitrary groups.

Our goal is to find a formulation which makes sense for all groups and all rings.

For this purpose we have introduced classifying spaces for families of subgroups of a groupGwhich we will recall next.

In the sequel group will mean discrete group.

Flashback

We have introduced theFarrell-Jones Conjectureand the Baum-Connes Conjecturefor torsionfree groups and discussed applications of these conjectures such as to theKaplansky Conjectureand theBorel Conjecture.

We have explained that the formulations for torsionfree groups cannot extend to arbitrary groups.

Our goal is to find a formulation which makes sense for all groups and all rings.

For this purpose we have introduced classifying spaces for families of subgroups of a groupGwhich we will recall next.

In the sequel group will mean discrete group.

Definition (Family of subgroups)

AfamilyF of subgroupsofGis a set of subgroups ofGwhich is closed under conjugation and finite intersections.

Examples forF are:

T R = {trivial subgroup};

F IN = {finite subgroups};

F CYC = {finite cyclic subgroups};

VCYC = {virtually cyclic subgroups};

ALL = {all subgroups}.

Definition (Family of subgroups)

AfamilyF of subgroupsofGis a set of subgroups ofGwhich is closed under conjugation and finite intersections.

Examples forF are:

T R = {trivial subgroup};

F IN = {finite subgroups};

F CYC = {finite cyclic subgroups};

VCYC = {virtually cyclic subgroups};

ALL = {all subgroups}.

Definition (Family of subgroups)

AfamilyF of subgroupsofGis a set of subgroups ofGwhich is closed under conjugation and finite intersections.

Examples forF are:

T R = {trivial subgroup};

F IN = {finite subgroups};

F CYC = {finite cyclic subgroups};

VCYC = {virtually cyclic subgroups};

ALL = {all subgroups}.

Definition (Family of subgroups)

AfamilyF of subgroupsofGis a set of subgroups ofGwhich is closed under conjugation and finite intersections.

Examples forF are:

T R = {trivial subgroup};

F IN = {finite subgroups};

F CYC = {finite cyclic subgroups};

VCYC = {virtually cyclic subgroups};

ALL = {all subgroups}.

Definition (Family of subgroups)

AfamilyF of subgroupsofGis a set of subgroups ofGwhich is closed under conjugation and finite intersections.

Examples forF are:

T R = {trivial subgroup};

F IN = {finite subgroups};

F CYC = {finite cyclic subgroups};

VCYC = {virtually cyclic subgroups};

ALL = {all subgroups}.

Definition (Family of subgroups)

AfamilyF of subgroupsofGis a set of subgroups ofGwhich is closed under conjugation and finite intersections.

Examples forF are:

T R = {trivial subgroup};

F IN = {finite subgroups};

F CYC = {finite cyclic subgroups};

VCYC = {virtually cyclic subgroups};

ALL = {all subgroups}.

Definition (Family of subgroups)

AfamilyF of subgroupsofGis a set of subgroups ofGwhich is closed under conjugation and finite intersections.

Examples forF are:

T R = {trivial subgroup};

F IN = {finite subgroups};

F CYC = {finite cyclic subgroups};

VCYC = {virtually cyclic subgroups};

ALL = {all subgroups}.

Definition (Family of subgroups)

AfamilyF of subgroupsofGis a set of subgroups ofGwhich is closed under conjugation and finite intersections.

Examples forF are:

T R = {trivial subgroup};

F IN = {finite subgroups};

F CYC = {finite cyclic subgroups};

VCYC = {virtually cyclic subgroups};

ALL = {all subgroups}.

Definition (ClassifyingG-CW-complex for a family of subgroups) LetF be a family of subgroups ofG. A model for theclassifying G-CW -complex for the familyF is aG-CW-complexEF(G)which has the following properties:

All isotropy groups ofE_F(G)belong toF;

For anyG-CW-complexY, whose isotropy groups belong toF, there is up toG-homotopy precisely oneG-mapY →X.

We abbreviateE G:=EF IN(G)and call it theuniversal G-CW -complex for proper G-actions.

We also writeEG=ET R(G).

A model forEF(G)exists and is unique up toG-homotopy.

Definition (ClassifyingG-CW-complex for a family of subgroups) LetF be a family of subgroups ofG.A model for theclassifying G-CW -complex for the familyF is aG-CW-complexEF(G)which has the following properties:

All isotropy groups ofE_F(G)belong toF;

For anyG-CW-complexY, whose isotropy groups belong toF, there is up toG-homotopy precisely oneG-mapY →X.

We abbreviateE G:=EF IN(G)and call it theuniversal G-CW -complex for proper G-actions.

We also writeEG=ET R(G).

A model forEF(G)exists and is unique up toG-homotopy.

Definition (ClassifyingG-CW-complex for a family of subgroups) LetF be a family of subgroups ofG. A model for theclassifying G-CW -complex for the familyF is aG-CW-complexEF(G)which has the following properties:

All isotropy groups ofE_F(G)belong toF;

For anyG-CW-complexY, whose isotropy groups belong toF, there is up toG-homotopy precisely oneG-mapY →X.

We abbreviateE G:=EF IN(G)and call it theuniversal G-CW -complex for proper G-actions.

We also writeEG=ET R(G).

A model forEF(G)exists and is unique up toG-homotopy.

Definition (ClassifyingG-CW-complex for a family of subgroups) LetF be a family of subgroups ofG. A model for theclassifying G-CW -complex for the familyF is aG-CW-complexEF(G)which has the following properties:

All isotropy groups ofE_F(G)belong toF;

For anyG-CW-complexY, whose isotropy groups belong toF, there is up toG-homotopy precisely oneG-mapY →X.

We abbreviateE G:=EF IN(G)and call it theuniversal G-CW -complex for proper G-actions.

We also writeEG=ET R(G).

A model forEF(G)exists and is unique up toG-homotopy.

Definition (ClassifyingG-CW-complex for a family of subgroups) LetF be a family of subgroups ofG. A model for theclassifying G-CW -complex for the familyF is aG-CW-complexEF(G)which has the following properties:

All isotropy groups ofE_F(G)belong toF;

For anyG-CW-complexY, whose isotropy groups belong toF, there is up toG-homotopy precisely oneG-mapY →X.

We abbreviateE G:=EF IN(G)and call it theuniversal G-CW -complex for proper G-actions.

We also writeEG=ET R(G).

A model forEF(G)exists and is unique up toG-homotopy.

Definition (ClassifyingG-CW-complex for a family of subgroups) LetF be a family of subgroups ofG. A model for theclassifying G-CW -complex for the familyF is aG-CW-complexEF(G)which has the following properties:

All isotropy groups ofE_F(G)belong toF;

For anyG-CW-complexY, whose isotropy groups belong toF, there is up toG-homotopy precisely oneG-mapY →X.

We abbreviateE G:=EF IN(G)and call it theuniversal G-CW -complex for proper G-actions.

We also writeEG=ET R(G).

A model forEF(G)exists and is unique up toG-homotopy.

Definition (ClassifyingG-CW-complex for a family of subgroups) LetF be a family of subgroups ofG. A model for theclassifying G-CW -complex for the familyF is aG-CW-complexEF(G)which has the following properties:

All isotropy groups ofE_F(G)belong toF;

For anyG-CW-complexY, whose isotropy groups belong toF, there is up toG-homotopy precisely oneG-mapY →X.

We abbreviateE G:=EF IN(G)and call it theuniversal G-CW -complex for proper G-actions.

We also writeEG=ET R(G).

A model forEF(G)exists and is unique up toG-homotopy.

Definition (ClassifyingG-CW-complex for a family of subgroups) LetF be a family of subgroups ofG. A model for theclassifying G-CW -complex for the familyF is aG-CW-complexEF(G)which has the following properties:

All isotropy groups ofE_F(G)belong toF;

For anyG-CW-complexY, whose isotropy groups belong toF, there is up toG-homotopy precisely oneG-mapY →X.

We abbreviateE G:=EF IN(G)and call it theuniversal G-CW -complex for proper G-actions.

We also writeEG=ET R(G).

A model forEF(G)exists and is unique up toG-homotopy.

Cliffhanger

Question (Homological computations based on nice models for E G)

Can nice geometric models for E G be used to compute the group homology and more general homology and cohomology theories of a group G?

Question (K-theory of group rings and group homology) Is there a relation between Kn(RG)and the group homology of G?

Question (Isomorphism Conjectures and classifying spaces of families)

Can classifying spaces of families be used to formulate a version of the Farrell-Jones Conjecture and the Baum-Connes Conjecture which may hold for all groups and all rings?

Cliffhanger

Question (Homological computations based on nice models for E G)

Can nice geometric models for E G be used to compute the group homology and more general homology and cohomology theories of a group G?

Question (K-theory of group rings and group homology) Is there a relation between Kn(RG)and the group homology of G?

Question (Isomorphism Conjectures and classifying spaces of families)

Can classifying spaces of families be used to formulate a version of the Farrell-Jones Conjecture and the Baum-Connes Conjecture which may hold for all groups and all rings?

Cliffhanger

Question (Homological computations based on nice models for E G)

Can nice geometric models for E G be used to compute the group homology and more general homology and cohomology theories of a group G?

Question (K-theory of group rings and group homology) Is there a relation between Kn(RG)and the group homology of G?

Question (Isomorphism Conjectures and classifying spaces of families)

Can classifying spaces of families be used to formulate a version of the Farrell-Jones Conjecture and the Baum-Connes Conjecture which may hold for all groups and all rings?

Cliffhanger

Question (Homological computations based on nice models for E G)

Can nice geometric models for E G be used to compute the group homology and more general homology and cohomology theories of a group G?

Question (K-theory of group rings and group homology) Is there a relation between Kn(RG)and the group homology of G?

Question (Isomorphism Conjectures and classifying spaces of families)

Can classifying spaces of families be used to formulate a version of the Farrell-Jones Conjecture and the Baum-Connes Conjecture which may hold for all groups and all rings?

Cliffhanger

Question (Homological computations based on nice models for E G)

Can nice geometric models for E G be used to compute the group homology and more general homology and cohomology theories of a group G?

Question (K-theory of group rings and group homology) Is there a relation between Kn(RG)and the group homology of G?

Question (Isomorphism Conjectures and classifying spaces of families)

Can classifying spaces of families be used to formulate a version of the Farrell-Jones Conjecture and the Baum-Connes Conjecture which may hold for all groups and all rings?

Cliffhanger

Question (Homological computations based on nice models for E G)

Can nice geometric models for E G be used to compute the group homology and more general homology and cohomology theories of a group G?

Question (K-theory of group rings and group homology) Is there a relation between Kn(RG)and the group homology of G?

Question (Isomorphism Conjectures and classifying spaces of families)

Can classifying spaces of families be used to formulate a version of the Farrell-Jones Conjecture and the Baum-Connes Conjecture which may hold for all groups and all rings?

Cliffhanger

Question (Homological computations based on nice models for E G)

Can nice geometric models for E G be used to compute the group homology and more general homology and cohomology theories of a group G?

Question (K-theory of group rings and group homology) Is there a relation between Kn(RG)and the group homology of G?

Question (Isomorphism Conjectures and classifying spaces of families)

Can classifying spaces of families be used to formulate a version of the Farrell-Jones Conjecture and the Baum-Connes Conjecture which may hold for all groups and all rings?

Outline

We intoduce the notion of anequivariant homology theory.

We present the general formulation of theFarrell-Jones Conjectureand theBaum-Connes Conjecture.

We discussequivariant Chern characters.

We present some explicitcomputationsof equivariant topological K-groups and of homology groups associated to classifying spaces of groups.

Outline

We intoduce the notion of anequivariant homology theory.

We present the general formulation of theFarrell-Jones Conjectureand theBaum-Connes Conjecture.

We discussequivariant Chern characters.

We present some explicitcomputationsof equivariant topological K-groups and of homology groups associated to classifying spaces of groups.

Outline

We intoduce the notion of anequivariant homology theory.

We present the general formulation of theFarrell-Jones Conjectureand theBaum-Connes Conjecture.

We discussequivariant Chern characters.

We present some explicitcomputationsof equivariant topological K-groups and of homology groups associated to classifying spaces of groups.

Outline

We intoduce the notion of anequivariant homology theory.

We present the general formulation of theFarrell-Jones Conjectureand theBaum-Connes Conjecture.

We discussequivariant Chern characters.

We present some explicitcomputationsof equivariant topological K-groups and of homology groups associated to classifying spaces of groups.

Outline

We intoduce the notion of anequivariant homology theory.

We present the general formulation of theFarrell-Jones Conjectureand theBaum-Connes Conjecture.

We discussequivariant Chern characters.

We present some explicitcomputationsof equivariant topological K-groups and of homology groups associated to classifying spaces of groups.

Equivariant homology theories

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.

Equivariant homology theories

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.

Equivariant homology theories

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.

Equivariant homology theories

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.

Equivariant homology theories

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.

Equivariant homology theories

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.

Equivariant homology theories

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.

Equivariant homology theories

Definition (G-homology theory)

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

∂_n^G(X,A) :H^G_n(X,A)→ H^G_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.

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.

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.

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.

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.

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.

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.

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.

Example (Equivariant homology theories)

Given a non-equivariant homology theoryK_∗, put

H^G_∗(X) := K_∗(X/G);

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

Equivariant bordismΩ^?_∗(X);

Equivariant topologicalK-theoryK_∗^?(X).

Theorem (L.-Reich (2005))

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

H^H_n(pt)∼=H^G_n(G/H)∼=πn(E(H)).

Example (Equivariant homology theories)

Given a non-equivariant homology theoryK_∗, put H^G_∗(X) := K_∗(X/G);

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

Equivariant bordismΩ^?_∗(X);

Equivariant topologicalK-theoryK_∗^?(X).

Theorem (L.-Reich (2005))

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

H^H_n(pt)∼=H^G_n(G/H)∼=πn(E(H)).

Example (Equivariant homology theories)

Given a non-equivariant homology theoryK_∗, put H^G_∗(X) := K_∗(X/G);

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

Equivariant bordismΩ^?_∗(X);

Equivariant topologicalK-theoryK_∗^?(X).

Theorem (L.-Reich (2005))

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

H^H_n(pt)∼=H^G_n(G/H)∼=πn(E(H)).

Example (Equivariant homology theories)

Given a non-equivariant homology theoryK_∗, put H^G_∗(X) := K_∗(X/G);

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

Equivariant bordismΩ^?_∗(X);

Equivariant topologicalK-theoryK_∗^?(X).

Theorem (L.-Reich (2005))

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

H^H_n(pt)∼=H^G_n(G/H)∼=πn(E(H)).

Example (Equivariant homology theories)

Given a non-equivariant homology theoryK_∗, put H^G_∗(X) := K_∗(X/G);

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

Equivariant bordismΩ^?_∗(X);

Equivariant topologicalK-theoryK_∗^?(X).

Theorem (L.-Reich (2005))

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

H^H_n(pt)∼=H^G_n(G/H)∼=πn(E(H)).

Example (Equivariant homology theories)

Given a non-equivariant homology theoryK_∗, put H^G_∗(X) := K_∗(X/G);

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

Equivariant bordismΩ^?_∗(X);

Equivariant topologicalK-theoryK_∗^?(X).

Theorem (L.-Reich (2005))

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

H^H_n(pt)∼=H^G_n(G/H)∼=πn(E(H)).

Example (Equivariant homology theories)

Given a non-equivariant homology theoryK_∗, put H^G_∗(X) := K_∗(X/G);

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

Equivariant bordismΩ^?_∗(X);

Equivariant topologicalK-theoryK_∗^?(X).

Theorem (L.-Reich (2005))

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

H^H_n(pt)∼=H^G_n(G/H)∼=πn(E(H)).

Example (Equivariant homology theories)

Given a non-equivariant homology theoryK_∗, put H^G_∗(X) := K_∗(X/G);

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

Equivariant bordismΩ^?_∗(X);

Equivariant topologicalK-theoryK_∗^?(X).

Theorem (L.-Reich (2005))

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

H^H_n(pt)∼=H^G_n(G/H)∼=πn(E(H)).

Example (Equivariant homology theories)

Given a non-equivariant homology theoryK_∗, put H^G_∗(X) := K_∗(X/G);

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

Equivariant bordismΩ^?_∗(X);

Equivariant topologicalK-theoryK_∗^?(X).

Theorem (L.-Reich (2005))

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

H^H_n(pt)∼=H^G_n(G/H)∼=πn(E(H)).

Theorem (Equivariant homology theories associated toK and L-theory,Davis-L. (1998))

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

L^h∞i_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 πn(K_R(G)) ∼= Kn(RG);

πn(L^h−∞i_R (G)) ∼= L^h−∞i_n (RG);

πn(K^top(G)) ∼= Kn(C_r^∗(G)).

Theorem (Equivariant homology theories associated toK and L-theory,Davis-L. (1998))

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

L^h∞i_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 πn(K_R(G)) ∼= Kn(RG);

πn(L^h−∞i_R (G)) ∼= L^h−∞i_n (RG);

πn(K^top(G)) ∼= Kn(C_r^∗(G)).

Theorem (Equivariant homology theories associated toK and L-theory,Davis-L. (1998))

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

L^h∞i_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 πn(K_R(G)) ∼= Kn(RG);

πn(L^h−∞i_R (G)) ∼= L^h−∞i_n (RG);

πn(K^top(G)) ∼= Kn(C_r^∗(G)).

Theorem (Equivariant homology theories associated toK and L-theory,Davis-L. (1998))

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

L^h∞i_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 πn(K_R(G)) ∼= Kn(RG);

πn(L^h−∞i_R (G)) ∼= L^h−∞i_n (RG);

πn(K^top(G)) ∼= Kn(C_r^∗(G)).

Theorem (Equivariant homology theories associated toK and L-theory,Davis-L. (1998))

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

L^h∞i_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 πn(K_R(G)) ∼= Kn(RG);

πn(L^h−∞i_R (G)) ∼= L^h−∞i_n (RG);

πn(K^top(G)) ∼= Kn(C_r^∗(G)).

Theorem (Equivariant homology theories associated toK and L-theory,Davis-L. (1998))

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

L^h∞i_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 πn(K_R(G)) ∼= Kn(RG);

πn(L^h−∞i_R (G)) ∼= L^h−∞i_n (RG);

πn(K^top(G)) ∼= Kn(C_r^∗(G)).

Theorem (Equivariant homology theories associated toK and L-theory,Davis-L. (1998))

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

L^h∞i_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 πn(K_R(G)) ∼= Kn(RG);

πn(L^h−∞i_R (G)) ∼= L^h−∞i_n (RG);

πn(K^top(G)) ∼= Kn(C_r^∗(G)).

Theorem (Equivariant homology theories associated toK and L-theory,Davis-L. (1998))

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

L^h∞i_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 πn(K_R(G)) ∼= Kn(RG);

πn(L^h−∞i_R (G)) ∼= L^h−∞i_n (RG);

πn(K^top(G)) ∼= Kn(C_r^∗(G)).

Theorem (Equivariant homology theories associated toK and L-theory,Davis-L. (1998))

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

L^h∞i_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 πn(K_R(G)) ∼= Kn(RG);

πn(L^h−∞i_R (G)) ∼= L^h−∞i_n (RG);

πn(K^top(G)) ∼= Kn(C_r^∗(G)).

Theorem (Equivariant homology theories associated toK and L-theory,Davis-L. (1998))

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

L^h∞i_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 πn(K_R(G)) ∼= Kn(RG);

πn(L^h−∞i_R (G)) ∼= L^h−∞i_n (RG);

πn(K^top(G)) ∼= Kn(C_r^∗(G)).

Example (Equivariant homology theories associated toK and L-theory)

We get equivariant homology theories

H_∗^?(−;K_R);

H_∗^?(−;L^h−∞i_R );

H_∗^?(−;K^top),

satisfying forH ⊆G

H_n^G(G/H;K_R) ∼= H_n^H(pt;K_R) ∼= K_n(RH);

H_n^G(G/H;L^h−∞i_R ) ∼= H_n^H(pt;L^h−∞i_R ) ∼= L^h−∞i_n (RH);

H_n^G(G/H;K^top) ∼= H_n^H(pt;K^top) ∼= Kn(C_r^∗(H)).

Example (Equivariant homology theories associated toK and L-theory)

We get equivariant homology theories H_∗^?(−;K_R);

H_∗^?(−;L^h−∞i_R );

H_∗^?(−;K^top),

satisfying forH ⊆G

H_n^G(G/H;K_R) ∼= H_n^H(pt;K_R) ∼= K_n(RH);

H_n^G(G/H;L^h−∞i_R ) ∼= H_n^H(pt;L^h−∞i_R ) ∼= L^h−∞i_n (RH);

H_n^G(G/H;K^top) ∼= H_n^H(pt;K^top) ∼= Kn(C_r^∗(H)).

Example (Equivariant homology theories associated toK and L-theory)

We get equivariant homology theories H_∗^?(−;K_R);

H_∗^?(−;L^h−∞i_R );

H_∗^?(−;K^top), satisfying forH ⊆G

H_n^G(G/H;K_R) ∼= H_n^H(pt;K_R) ∼= K_n(RH);

H_n^G(G/H;L^h−∞i_R ) ∼= H_n^H(pt;L^h−∞i_R ) ∼= L^h−∞i_n (RH);

H_n^G(G/H;K^top) ∼= H_n^H(pt;K^top) ∼= Kn(C_r^∗(H)).

The general formulation of the Isomorphism Conjectures

Conjecture (K-theoretic Farrell-Jones-Conjecture)

TheK -theoretic Farrell-Jones Conjecturewith 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 assembly map is the map induced by the projection EVCYC(G)→pt.

The general formulation of the Isomorphism Conjectures

Conjecture (K-theoretic Farrell-Jones-Conjecture)

TheK -theoretic Farrell-Jones Conjecturewith 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 assembly map is the map induced by the projection EVCYC(G)→pt.

The general formulation of the Isomorphism Conjectures

Conjecture (K-theoretic Farrell-Jones-Conjecture)

TheK -theoretic Farrell-Jones Conjecturewith 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 assembly map is the map induced by the projection EVCYC(G)→pt.

The general formulation of the Isomorphism Conjectures

Conjecture (K-theoretic Farrell-Jones-Conjecture)

TheK -theoretic Farrell-Jones Conjecturewith 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 assembly map is the map induced by the projection EVCYC(G)→pt.

Conjecture (L-theoretic Farrell-Jones-Conjecture)

TheL-theoretic Farrell-Jones Conjecturewith coefficients in R for the group G predicts that the assembly map

H_n^G(E_VCYC(G),L^h−∞i_R )→H_n^G(pt,L^h−∞i_R ) =L^h−∞i_n (RG) is bijective for all n∈Z.

Conjecture (L-theoretic Farrell-Jones-Conjecture)

TheL-theoretic Farrell-Jones Conjecturewith coefficients in R for the group G predicts that the assembly map

H_n^G(E_VCYC(G),L^h−∞i_R )→H_n^G(pt,L^h−∞i_R ) =L^h−∞i_n (RG) is bijective for all n∈Z.

Conjecture (Baum-Connes Conjecture)

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

We will discuss these conjectures and their applications in the next lecture.

We will now continue with equivariant homology theories.

Conjecture (Baum-Connes Conjecture)

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

We will discuss these conjectures and their applications in the next lecture.

We will now continue with equivariant homology theories.

Conjecture (Baum-Connes Conjecture)

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

We will discuss these conjectures and their applications in the next lecture.

We will now continue with equivariant homology theories.

Conjecture (Baum-Connes Conjecture)

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

We will discuss these conjectures and their applications in the next lecture.

We will now continue with equivariant homology theories.

Equivariant Chern characters

LetH_∗ be a (non-equivariant) homology theory. There is the Atiyah-Hirzebruch spectral sequencewhich converges toH_p+q(X) and has asE²-term

E_p,q² =H_p(X;H_q(pt)).

Rationally it collapses completely. Namely, one has the following result

Theorem (Non-equivariant Chern character,Dold (1962)) LetH_∗ be a homology theory with values inΛ-modules forQ⊆Λ.

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

M

p+q=n

Hp(X; Λ)⊗_ΛH_q(pt)−→ H^∼= _n(X).

Equivariant Chern characters

LetH_∗ be a (non-equivariant) homology theory.There is the Atiyah-Hirzebruch spectral sequencewhich converges toH_p+q(X) and has asE²-term

E_p,q² =H_p(X;H_q(pt)).

Rationally it collapses completely. Namely, one has the following result

Theorem (Non-equivariant Chern character,Dold (1962)) LetH_∗ be a homology theory with values inΛ-modules forQ⊆Λ.

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

M

p+q=n

Hp(X; Λ)⊗_ΛH_q(pt)−→ H^∼= _n(X).

Equivariant Chern characters

LetH_∗ be a (non-equivariant) homology theory. There is the Atiyah-Hirzebruch spectral sequencewhich converges toH_p+q(X) and has asE²-term

E_p,q² =H_p(X;H_q(pt)).

Rationally it collapses completely. Namely, one has the following result

Theorem (Non-equivariant Chern character,Dold (1962)) LetH_∗ be a homology theory with values inΛ-modules forQ⊆Λ.

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

M

p+q=n

Hp(X; Λ)⊗_ΛH_q(pt)−→ H^∼= _n(X).

Equivariant Chern characters

LetH_∗ be a (non-equivariant) homology theory. There is the Atiyah-Hirzebruch spectral sequencewhich converges toH_p+q(X) and has asE²-term

E_p,q² =H_p(X;H_q(pt)).

Rationally it collapses completely. Namely, one has the following result

Theorem (Non-equivariant Chern character,Dold (1962)) LetH_∗ be a homology theory with values inΛ-modules forQ⊆Λ.

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

M

p+q=n

Hp(X; Λ)⊗_ΛH_q(pt)−→ H^∼= _n(X).

Equivariant Chern characters

LetH_∗ be a (non-equivariant) homology theory. There is the Atiyah-Hirzebruch spectral sequencewhich converges toH_p+q(X) and has asE²-term

E_p,q² =H_p(X;H_q(pt)).

Rationally it collapses completely. Namely, one has the following result

Theorem (Non-equivariant Chern character,Dold (1962)) LetH_∗ be a homology theory with values inΛ-modules forQ⊆Λ.

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

M

p+q=n

Hp(X; Λ)⊗_ΛH_q(pt)−→ H^∼= _n(X).

Equivariant Chern characters

LetH_∗ be a (non-equivariant) homology theory. There is the Atiyah-Hirzebruch spectral sequencewhich converges toH_p+q(X) and has asE²-term

E_p,q² =H_p(X;H_q(pt)).

Rationally it collapses completely. Namely, one has the following result

Theorem (Non-equivariant Chern character,Dold (1962)) LetH_∗ be a homology theory with values inΛ-modules forQ⊆Λ.

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

M

p+q=n

Hp(X; Λ)⊗_ΛH_q(pt)−→ H^∼= _n(X).

Equivariant Chern characters

LetH_∗ be a (non-equivariant) homology theory. There is the Atiyah-Hirzebruch spectral sequencewhich converges toH_p+q(X) and has asE²-term

E_p,q² =H_p(X;H_q(pt)).

Rationally it collapses completely. Namely, one has the following result

Theorem (Non-equivariant Chern character,Dold (1962)) LetH_∗ be a homology theory with values inΛ-modules forQ⊆Λ.

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

M

p+q=n

Hp(X; Λ)⊗_ΛH_q(pt)−→ H^∼= _n(X).

Dold’sChern characterfor aCW-complexX is given by the following composite:

ch_n: M

p+q=n

H_p(X;H_q(∗))−−→^α−1 M

p+q=n

H_p(X;Z)⊗_ZH_q(∗)

L

p+q=n(hur⊗id)⁻¹

−−−−−−−−−−−→ M

p+q=n

π^s_p(X₊,∗)⊗_ZH_q(∗)

L

p+q=nDp,q

−−−−−−−−→ H_n(X),

whereD_p,qsends[f: (S^p+k,pt)→(S^k ∧X₊,pt)]⊗ηto the image ofη under the composite

H_q(∗)∼=H_p+k+q(S^p+k,pt)−−−−−−→ H^Hp+k+q(f) _p+k+q(S^k∧X₊,pt)∼=H_p+q(X).

Dold’sChern characterfor aCW-complexX is given by the following composite:

ch_n: M

p+q=n

H_p(X;H_q(∗))−−→^α−1 M

p+q=n

H_p(X;Z)⊗_ZH_q(∗)

L

p+q=n(hur⊗id)⁻¹

−−−−−−−−−−−→ M

p+q=n

π^s_p(X₊,∗)⊗_ZH_q(∗)

L

p+q=nDp,q

−−−−−−−−→ H_n(X),

whereD_p,qsends[f: (S^p+k,pt)→(S^k ∧X₊,pt)]⊗ηto the image ofη under the composite

H_q(∗)∼=H_p+k+q(S^p+k,pt)−−−−−−→ H^Hp+k+q(f) _p+k+q(S^k∧X₊,pt)∼=H_p+q(X).

Dold’sChern characterfor aCW-complexX is given by the following composite:

ch_n: M

p+q=n

H_p(X;H_q(∗))−−→^α−1 M

p+q=n

H_p(X;Z)⊗_ZH_q(∗)

L

p+q=n(hur⊗id)⁻¹

−−−−−−−−−−−→ M

p+q=n

π^s_p(X₊,∗)⊗_ZH_q(∗)

L

p+q=nDp,q

−−−−−−−−→ H_n(X),

whereD_p,qsends[f: (S^p+k,pt)→(S^k ∧X₊,pt)]⊗ηto the image ofη under the composite

H_q(∗)∼=H_p+k+q(S^p+k,pt)−−−−−−→ H^Hp+k+q(f) _p+k+q(S^k∧X₊,pt)∼=H_p+q(X).