The Isomorphism Conjectures for arbitrary groups (Lecture V)

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The Isomorphism Conjectures for arbitrary groups (Lecture V)

Wolfgang Lück Münster Germany

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

Hangzhou, July 2007

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Flashback

We have introduced classifying spacesEF(G)for a familyF of subgroups.

We have introduced the notion of anequivariant homology theory.

We have formulated theFarrell-Jones Conjectureand the Baum-Connes Conjecture.

We have already discussed application for torsionfree groups such as to theKaplansky Conjectureand theBorel Conjecture.

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Question (Consequences)

What are the consequences of the Farrell-Jones Conjecture and the Baum-Connes Conjecture?

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Outline

We give a review of the Farrell-Jones and the Baum-Connes Conjecture.

We discuss the difference between the familiesF IN andVCYC.

We discuss consequences of the Farrell-Jones and the Baum-Connes Conjecture.

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Outline

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Outline

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Outline

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Review of the Isomorphism Conjectures

Gwill always be a discrete group.

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 ofEF(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).

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Review of the Isomorphism Conjectures

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Review of the Isomorphism Conjectures

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Review of the Isomorphism Conjectures

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:

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Review of the Isomorphism Conjectures

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Review of the Isomorphism Conjectures

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Review of the Isomorphism Conjectures

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Review of the Isomorphism Conjectures

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Review of the Isomorphism Conjectures

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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.

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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

Bijectivity

Functoriality inα;

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homomorphisms

Bijectivity

Functoriality inα;

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homomorphisms

Bijectivity

Functoriality inα;

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homomorphisms

Bijectivity

Functoriality inα;

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homomorphisms

Bijectivity

Functoriality inα;

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homomorphisms

Bijectivity

Functoriality inα;

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homomorphisms

Bijectivity

Functoriality inα;

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

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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) =Kn(C_r^∗(G)) is bijective for all n∈Z.

All assembly maps are the maps induced by the projection EF(G)→pt.

These Conjecture can be thought of a kind ofgeneralized induction theorem. They allow to compute the value of a functor such asK_n(RG)onGin terms of its values for all virtually cyclic subgroups ofG.

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These Conjecture can be thought of a kind ofgeneralized induction theorem.They allow to compute the value of a functor such asK_n(RG)onGin terms of its values for all virtually cyclic subgroups ofG.

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Changing the family

Theorem (Transitivity Principle)

LetF ⊆ Gbe two families of subgroups of G. LetH^?_∗be an equivariant homology theory. Assume that for every element H ∈ Gand n∈Zthe assembly map

H^H_n(E_{F |}_H(H))→ H_n^H(pt) is bijective, whereF |_H ={K∩H |K ∈ F }.

Then therelative assembly mapinduced by the up to G-homotopy unique G-map EF(G)→EG(G)

H_n^G(EF(G))→ H^G_n(EG(G)) is bijective for all n∈Z.

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Changing the family

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Changing the family

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Changing the family

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Changing the family

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Changing the family

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Example (Passage fromF IN toVCYC for the Baum-Connes Conjecture)

The Baum-Connes Conjecture is known to be true for virtually cyclic groups. The Transitivity Principle implies that the relative assembly

K_n^G(E G)−^∼⁼→K_n^G(E_VCYC(G))

is bijective for alln∈Z.

Hence it does not matter in the context of the Baum-Connes Conjecture whether we consider the familyF IN orVCYC.

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The Baum-Connes Conjecture is known to be true for virtually cyclic groups.The Transitivity Principle implies that the relative assembly

K_n^G(E G)−^∼⁼→K_n^G(E_VCYC(G))

is bijective for alln∈Z.

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K_n^G(E G)−^∼⁼→K_n^G(E_VCYC(G)) is bijective for alln∈Z.

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K_n^G(E G)−^∼⁼→K_n^G(E_VCYC(G)) is bijective for alln∈Z.

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In general the relative assembly maps

H_n^G(E G;K_R) → H_n^G(EVCYC(G);K_R);

H_n^G(E G;L^h−∞i_R ) → H_n^G(EVCYC(G);L^h−∞i_R ), are not bijective.

Hence in the Farrell-Jones setting one has to pass toVCYCand cannot use the easier to handle familyF IN.

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In general the relative assembly maps

H_n^G(E G;K_R) → H_n^G(EVCYC(G);K_R);

H_n^G(E G;L^h−∞i_R ) → H_n^G(EVCYC(G);L^h−∞i_R ), are not bijective.

Hence in the Farrell-Jones setting one has to pass toVCYCand cannot use the easier to handle familyF IN.

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Example (Passage fromF IN toVCYC for the Farrell-Jones Conjecture)

For instance theBass-Heller Swan decomposition

K_n−1(R)⊕K_n(R)⊕NK_n(R)⊕NK_n(R))−^∼⁼→K_n(R[t,t⁻¹])∼=K_n(R[Z])

and the isomorphism

H_n^Z(EZ;K_R) =H_n^Z(EZ;K_R) =H_n^{1}(S¹,K_R) =K_n−1(R)⊕K_n(R)

show that

H_n^Z(EZ;K_R)→H_n^Z(pt;K_R) =K_n(RZ) is bijective if and only if NK_n(R) =0.

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K_n−1(R)⊕K_n(R)⊕NK_n(R)⊕NK_n(R))−^∼⁼→K_n(R[t,t⁻¹])∼=K_n(R[Z]) and the isomorphism

H_n^Z(EZ;K_R) =H_n^Z(EZ;K_R) =H_n^{1}(S¹,K_R) =K_n−1(R)⊕K_n(R)

show that

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H_n^Z(EZ;K_R) =H_n^Z(EZ;K_R) =H_n^{1}(S¹,K_R) =K_n−1(R)⊕K_n(R) show that

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H_n^Z(EZ;K_R) =H_n^Z(EZ;K_R) =H_n^{1}(S¹,K_R) =K_n−1(R)⊕K_n(R) show that

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An infinite virtually cyclic groupGis called oftypeIif it admits an epimorphism ontoZand oftypeII otherwise. A virtually cyclic group is of typeII if and only if admits an epimorphism ontoD∞. LetVCYC_I orVCYC_II respectively be the family of subgroups which are either finite or which are virtually cyclic of typeIorII respectively.

Theorem (L. (2004), Quinn (2007), Reich (2007)) The following maps are bijective for all n ∈Z

H_n^G(EVCYC_I(G);K_R) → H_n^G(EVCYC(G);K_R);

H_n^G(E G;L^h−∞i_R ) → H_n^G(E_VCYC_I(G);L^h−∞i_R ).

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Theorem (Cappell (1973), Grunewald (2005), Waldhausen (1978))

The following maps are bijective for all n∈Z.

H_n^G(E G;K_Z)⊗_ZQ → H_n^G(E_VCYC(G);K_Z)⊗_ZQ;

H_n^G(E G;L^h−∞i_R ) 1

2

→ H_n^G(EVCYC(G);L^h−∞i_R ) 1

2

; If R is regular andQ⊆R, then for all n∈Zwe get a bijection

H_n^G(E G;K_R)→H_n^G(E_VCYC(G);K_R).

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2

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2

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Theorem (Bartels (2003)) For every n∈Zthe two maps

H_n^G(E G;K_R) → H_n^G(E_VCYC(G);K_R);

H_n^G(E G;L^h−∞i_R ) → H_n^G(EVCYC(G);L^h−∞i_R ), are split injective.

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Theorem (Bartels (2003)) For every n∈Zthe two maps

H_n^G(E G;K_R) → H_n^G(E_VCYC(G);K_R);

H_n^G(E G;L^h−∞i_R ) → H_n^G(EVCYC(G);L^h−∞i_R ), are split injective.

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Hence we get (natural) isomorphisms H_n^G(EVCYC(G);K_R)

∼=H_n^G(E G;K_R)⊕H_n^G(EVCYC(G),E G;K_R);

H_n^G(EVCYC(G);L^h−∞i_R )

∼=H_n^G(E G;L^h−∞i_R )⊕H_n^G(E_VCYC(G),E G;L^h−∞i_R ).

The analysis of the termsH_n^G(EVCYC(G),E G;K_R)and

H_n^G(EVCYC(G),E G;L^h−∞i_R )boils down to investigatingNil-terms andUNil-termsin the sense ofWaldhausenandCappell.

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Hence we get (natural) isomorphisms H_n^G(EVCYC(G);K_R)

∼=H_n^G(E G;K_R)⊕H_n^G(EVCYC(G),E G;K_R);

H_n^G(EVCYC(G);L^h−∞i_R )

∼=H_n^G(E G;L^h−∞i_R )⊕H_n^G(E_VCYC(G),E G;L^h−∞i_R ).

The analysis of the termsH_n^G(EVCYC(G),E G;K_R)and

H_n^G(EVCYC(G),E G;L^h−∞i_R )boils down to investigatingNil-terms andUNil-termsin the sense ofWaldhausenandCappell.

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The analysis of the termsH_n^G(E G;K_R)andH_n^G(E G;L^h−∞i_R )is using the methods of the previous lecture (e.g., Chern characters).

The results above imply that the versions of the Farrell-Jones Conjecture for torsionfree groups which we have presented in the second lecture follow from the general versions.

The latter is obvious for the Baum-Connes Conjecture since for torsionfreeGwe haveEG=E G.

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Consequence of the Isomorphism Conjectures

Conjecture (Novikov Conjecture)

TheNovikov Conjecture for Gpredicts for a closed oriented manifold M together with a map f:M→BG that for any x ∈H^∗(BG)thehigher signature

sign_x(M,f):=hL(M)∪f^∗x,[M]i

is an oriented homotopy invariant of(M,f), i.e., for every orientation preserving homotopy equivalence of closed oriented manifolds g:M₀→M₁and homotopy equivalence f_i:M₀→M₁with f₁◦g 'f₂ we have

sign_x(M₀,f₀) =sign_x(M₁,f₁).

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Consequence of the Isomorphism Conjectures

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Consequence of the Isomorphism Conjectures

is an oriented homotopy invariant of(M,f),i.e., for every orientation preserving homotopy equivalence of closed oriented manifolds g:M₀→M₁and homotopy equivalence f_i:M₀→M₁with f₁◦g 'f₂ we have

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Consequence of the Isomorphism Conjectures

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Theorem (The Farrell-Jones, the Baum-Connes and the Novikov Conjecture)

Suppose that one of the following assembly maps

H_n^G(EVCYC(G),L^h−∞i_R ) → H_n^G(pt,L^h−∞i_R ) =L^h−∞i_n (RG);

K_n^G(E G) =H_n^G(EF IN(G),K^top) → H_n^G(pt,K^top) =Kn(C_r^∗(G)), is rationally injective.

Then the Novikov Conjecture holds for the group G.

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Theorem (K0(RG)and induction from finite subgroups, Bartels-L.-Reich (2007))

Let R be a regular ring withQ⊆R. Suppose G∈ FJ_K(R). Then the map given by induction from finite subgroups of G

colim

OrF IN(G)K₀(RH)→K₀(RG) is bijective;

Let F be a field of characteristic p for a prime number p. Suppose that G∈ FJ_K(F). Then the map

colim

OrF IN(G)K₀(FH)[1/p]→K₀(FG)[1/p]

is bijective.

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colim

is bijective.

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Let R be a regular ring withQ⊆R. Suppose G∈ FJ_K(R).Then the map given by induction from finite subgroups of G

colim

is bijective.

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colim

is bijective.

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colim

Let F be a field of characteristic p for a prime number p. Suppose that G∈ FJ_K(F).Then the map

colim

is bijective.

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colim

is bijective.

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Theorem (Permutation Modules,Bartels-L.-Reich (2007)) Let F be a field of characteristic zero. Suppose that G∈ FJ_K(F).

Then for every finitely generated projective FG-module P there exists a positive integer k and finitely many finite subgroups H₁, H₂,. . ., Hr

such that

P^k ∼=FGF[G/H₁]⊕F[G/H₂]⊕. . .⊕F[G/Hr].

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such that

P^k ∼=FGF[G/H₁]⊕F[G/H₂]⊕. . .⊕F[G/Hr].

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such that

P^k ∼=FGF[G/H₁]⊕F[G/H₂]⊕. . .⊕F[G/Hr].

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such that

P^k ∼=FGF[G/H₁]⊕F[G/H₂]⊕. . .⊕F[G/Hr].

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LetR be commutative ring and letGbe a group.

Letclass(G,R)be theR-module ofclass functionsG→R, i.e., functionsG→Rwhich are constant on conjugacy classes.

Let tr_RG:RG→class(G,R)be the obviousR-homomorphism. It extends to a map

tr_RG:M_n(RG)→class(G,R)

by taking the sums of the values of the diagonal entries.

LetP be a finitely generatedRG-module. Choose a finitely generated projectiveRG-moduleQand an isomorphism φ:RGⁿ−→^∼⁼ P⊕Q. LetA∈Mn(RG)be a matrix such that φ⁻¹◦(f⊕idq)◦φ:RGⁿ→RGⁿis given byA.

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