Isomorphism Conjectures in K - and L-theory Joint UMI/DMV-meeting

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Isomorphism Conjectures in K - and L-theory Joint UMI/DMV-meeting

Wolfgang L¨uck M¨unster Germany

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

Perugia, June 2007

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Outline and goal

Explain the K-theoretic andL-theoreticFarrell-Jones Conjectureand Baum-Connes Conjecture at least for torsionfree groups

Discuss applications and thepotential of these conjectures.

State our main theorems.

Wolfgang L¨uck (M¨unster, Germany) Isomorphism Conjectures Perugia, June 2007 2 / 20

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Outline and goal

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Outline and goal

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Outline and goal

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Conjecture (Farrell-Jones)

The K -theoretic Farrell-Jones Conjecturewith coefficients in the regular ring R for the torsionfree group G predicts that the assembly map

Hn(BG;K_R)→Kn(RG)

is bijective for all n ∈Z.

K_n(RG) is the algebraicK-theory of the group ringRG;

K_R is the (non-connective) algebraicK-theory spectrum of the ringR.

Hn(pt;K_R)∼=πn(K_R)∼=Kn(R).

BGis the classifying space of the group G. ExampleG =Z: K_n(R[Z])∼=K_n(R)⊕Kn−1(R).

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Hn(BG;K_R)→Kn(RG) is bijective for all n ∈Z.

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K0(R) is the Grothendieck construction applied to the abelian semigroup of finitely generated projectiveR-modules.

Let G be a finite group andF be a field of characteristic zero. Then therepresentation ringR_F(G)is the same as K₀(FG).

The assignmentP 7→[P]∈K₀(R) is theuniversal additive invariant or dimension functionfor finitely generated projective R-modules.

K₁(R) is the abelianizationGL(R)/[GL(R),GL(R)] of GL(R) =S

n≥1GLn(R).

The assignmentA7→[A]∈K₁(R) for A∈GL_n(R) is the universal determinant forR.

Ke_n(R) is the cokernel ofK_n(Z)→K_n(R).

TheWhitehead group of a group G is defined to be Wh(G)=K₁(ZG)/{±g |g ∈G}.

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n≥1GLn(R).

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n≥1GLn(R).

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n≥1GLn(R).

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n≥1GLn(R).

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n≥1GLn(R).

TheWhitehead group of a groupG is defined to be Wh(G)=K₁(ZG)/{±g |g ∈G}.

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n≥1GLn(R).

TheWhitehead group of a groupG is defined to be Wh(G)=K₁(ZG)/{±g |g ∈G}.

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Definition (h-cobordism)

An h-cobordismover a closed manifold M0 is a compact manifold W whose boundary is the disjoint union M₀qM₁ such that both inclusions M0→W andM1 →W are homotopy equivalences.

Theorem (s-Cobordism Theorem (Barden, Mazur, Stallings, Kirby-Siebenmann)

Let M₀ be a closed manifold of dimension n≥5 with fundamental group G =π1(M0). Let(W;M0,M1) be an h-cobordism over M0. Then W is trivial over M0 if and only if its Whitehead torsionτ(W,M0)∈Wh(G) vanishes.

Thes-Cobordism Theorem implies the Poincar´e Conjecture in dimension≥5.

It is a key ingredient in the surgery programfor the classification of closed manifolds due to Browder, Novikov, Sullivanand Wall.

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In order to illustrate the depth of the Farrell-Jones Conjecture, we present some conclusions which are interesting in their own right.

Let FJ_K(R)be the class of groups which satisfy theK-theoretic Farrell-Jones Conjecture for the coefficient ringR.

Lemma

Let R be a regular ring, for instance Z, a field or a principal ideal domain.

Suppose that G is torsionfree and G ∈ FJ_K(R). Then K_n(RG) = 0 for n≤ −1;

The change of rings map K₀(R)→K₀(RG) is bijective. In particular Ke0(RG) is trivial if and only ifKe0(R) is trivial.

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Lemma

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Lemma

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Lemma

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Lemma

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Lemma

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Lemma

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We get for a torsionfree group G ∈ FJ_K(Z) Kn(ZG) = 0 for n≤ −1;

Ke0(ZG) = 0;

Wh(G) = 0;

Every compact h-cobordism W = (W;M₀,M₁) of dimension ≥6 with π1(W)∼=G is trivial.

IfG belongs to FJ_K(Z), then it is of type FF if and only if it is of type FP. (Serre’sproblem)

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Ke0(ZG) = 0;

Wh(G) = 0;

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Ke0(ZG) = 0;

Wh(G) = 0;

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Ke0(ZG) = 0;

Wh(G) = 0;

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Ke0(ZG) = 0;

Wh(G) = 0;

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Ke0(ZG) = 0;

Wh(G) = 0;

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Conjecture (Kaplansky)

The Kaplansky Conjecture says for a torsionfree group G and an integral domain R that 0and 1are the only idempotents in RG .

Theorem (Bartels-L.-Reich (2007))

Let F be a skew-field and let G be a group with G ∈ FJ_K(F). Suppose that one of the following conditions is satisfied:

F is commutative and has characteristic zero and G is torsionfree;

G is torsionfree and sofic, e.g., residually amenable;

The characteristic of F is p, all finite subgroups of G are p-groups and G is sofic.

Then 0 and1 are the only idempotents in FG .

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The K -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.

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The K -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.

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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

OrFin(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

OrFin(G)K0(FH)[1/p]→K0(FG)[1/p]

is bijective.

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colim

is bijective.

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colim

is bijective.

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colim

is bijective.

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colim

is bijective.

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The surjectivity of the map colim

OrFin(G)K₀(CH)→K₀(CG)

plays a role (33 %) in a program to prove theAtiyah Conjecture. It predicts that for a closed Riemannian manifoldM with torsionfree fundamental group the p-thL²-Betti numberof its universal covering

b⁽²⁾p (Me)= lim

t→∞

Z

F

tr

e^−t^∆^f^p(ex,ex)

dvol_M_e is an integer.

There are further applications of the Farrell-Jones Conjecture to L²-torsion, theL²-version of analytic torsion due toRay-Singer, and to the Bass Conjecture, which predicts that character theory for finite groups can be extended to infinite groups.

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t→∞

Z

F

tr

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t→∞

Z

F

tr

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t→∞

Z

F

tr

There are further applications of the Farrell-Jones Conjecture to L²-torsion, theL²-version of analytic torsion due toRay-Singer,and to the Bass Conjecture, which predicts that character theory for finite groups can be extended to infinite groups.

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t→∞

Z

F

tr

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Let FJ_K(R)be the class of groups which satisfy the (Fibered) Farrell-Jones Conjecture for algebraicK-theory with (G-twisted) coefficients in R.

Every hyperbolic group and every virtually nilpotent group belongs to FJ_K(R);

If G1 and G2 belong to FJ_K(R), then G1×G2 belongs toFJ_K(R);

Let {G_i |i ∈I}be a directed system of groups (with not necessarily injective structure maps) such that G_i ∈ FJ_K(R) for i ∈I . Then colim_i∈IG_i belongs toFJ_K(R);

If H is a subgroup of G and G ∈ FJ_K(R), then H ∈ FJ_K(R).

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We emphasize that this result holds for all rings R.

The groups above are certainly wild in Bridson’suniverse.

Gromov’s groups with expanders, for which the Baum-Connes Conjecture with coefficients fails byHigson-Lafforgue-Skandalis, belong to FJ_K(R) for all R.

Bartels and L. have a program to proveG ∈ FJ_K(R) ifG acts properly and cocompact on a CAT(0)-space.

This would yield the same result for all subgroups of cocompact lattices in almost connected Lie groups.

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The L-theoretic Farrell-Jones Conjecturewith coefficients in the ring with involution R for the torsionfree group G predicts that the assembly map

Hn(BG;L^h−∞i_R )→L^h−∞in (RG) is bijective for all n ∈Z.

This is the Farrell-Jones Conjecture for K-theory if one replaces K-theory by L-theory with decorationh−∞i.

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The L-theoretic Farrell-Jones Conjecturewith coefficients in the ring with involution R for the torsionfree group G predicts that the assembly map

Hn(BG;L^h−∞i_R )→L^h−∞in (RG) is bijective for all n ∈Z.

This is the Farrell-Jones Conjecture for K-theory if one replaces K-theory by L-theory with decorationh−∞i.

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Conjecture (Novikov)

The Novikov Conjecturepredicts the homotopy invariance of higher signatures

Conjecture (Borel)

The Borel Conjecture for G predicts for two closed aspherical manifolds M and N with π1(M)∼=π1(N)∼=G that any homotopy equivalence M →N is homotopic to a homeomorphism and in particular that M and N are homeomorphic.

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Conjecture (Borel)

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Conjecture (Borel)

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Conjecture (Borel)

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TheL-theoretic Farrell-Jones Conjecture for a group G in the case R =Zimplies the Novikov Conjecture in dimension≥5.

If theK- andL-theoretic Farrell-Jones Conjecture hold forG in the caseR =Z, then the Borel Conjecture is true in dimension ≥5 and in dimension 4 ifG is good in the sense ofFreedman.

As in the case of algebraicK-theory there is also an analogous version of the L-theoretic Farrell-Jones Conjecture for arbitrary groupsG. Bartels and L.have a program to extend our result for theK-theoretic Farrell-Jones Conjecture also to theL-theoretic version. This would imply the Novikov and the Borel Conjecture for such groups.

Bartels and L. have a program to proveG ∈ FJ_L(R) ifG acts properly and cocompact on a CAT(0)-space. This would yield the same result for all subgroups of cocompact lattices in almost connected Lie groups.

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Conjecture (Baum-Connes)

The Baum-Connes Conjecture for the torsionfree group predicts that the assembly map

K_n(BG)→K_n(C_r^∗(G)) is bijective for all n ∈Z.

K_n(BG) is the topological K-homology ofBG.

Kn(C_r^∗(G))is the topological K-theory of the reduced complex group C^∗-algebraC_r^∗(G) ofG;

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There is also a version for arbitrary groups K_n^G(EFin(G))→K_n(C_r^∗(G)).

TheBost Conjecture is the analogue forl¹(G), i.e., it concerns the assembly map.

K_n^G(EFin(G))→Kn(l¹(G)).

Its composition with the canonical mapKn(l¹(G))→Kn(C_r^∗(G)) is the Baum-Connes assembly map.

Both Conjectures have versions, where coefficients in aG-C^∗-algebra are allowed.