Summary, status and outlook (Lecture VI)

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Summary, status and outlook (Lecture VI)

Wolfgang Lück Münster Germany

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

Hangzhou, July 2007

Wolfgang Lück (Münster, Germany) Summary, status and outlook Hangzhou, July 2007 1 / 41

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

We have already discussed applications.

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Question (Status)

For which groups are the Farrell-Jones Conjecture and the Baum-Connes Conjecture known to be true? What are open interesting cases?

Question (Relations)

What are the relations between the Farrell-Jones Conjecture and the Baum-Connes Conjecture?

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Outline

We briefly review the Farrell-Jones and the Baum-Connes Conjecture.

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

We mention other versions of the Isomorphism Conjectures.

We explain relations between the Farrell-Jones and the Baum-Connes Conjecture.

We give a status report about the Farrell-Jones and the Baum-Connes Conjecture.

Miscellaneous.

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Review 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) =Kn(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^h−∞i_R ) =L^h−∞i_n (RG) is bijective for all n∈Z.

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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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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.

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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.

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Review of Applications

The following results or conjectures are consequences of the Farrell-Jones or Baum-Connes Conjecture.

FJ_K(R),FJ_L(R)orBCrespectively are the classes of groups which satisfy the Farrell-Jones Conjecture forK- orL-theory with coefficients inRor the Baum-Connes Conjecture respectively.

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Review of Applications

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Review of Applications

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Theorem (Kn(ZG)forn≤1 and torsionfreeG) We get for a torsionfree group G∈ FJ(Z):

Kn(ZG) =0for n≤ −1;

Ke₀(ZG) =0;

Wh(G) =0;

Every finitely dominated CW -complex X with G=π₁(X)is homotopy equivalent to a finite CW -complex;

Every compact h-cobordism W = (W;M₀,M₁)of dimension≥6 withπ1(W)∼=G is trivial.

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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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Ke₀(ZG) =0;

Wh(G) =0;

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Conjecture (Kaplansky Conjecture)

TheKaplansky Conjecturesays for a torsionfree group G and an integral domain R that0and1are the only idempotents in RG.

Theorem (The Farrell-Jones Conjecture and the Kaplansky Conjecture)

If F is a field of characteristic zero and the torsionfree group G belongs toFJ_K(F), then G and F satisfy the Kaplansky Conjecture.

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Conjecture (Borel Conjecture)

TheBorel Conjecture for Gpredicts for two closed aspherical manifolds M and N withπ₁(M)∼=π₁(N)∼=G that any homotopy equivalence M →N is homotopic to a homeomorphism and in particular that M and N are homeomorphic.

Theorem (The Farrell-Jones Conjecture and the Borel Conjecture)

If G belongs to bothFJ_K(Z)andFJ_L(Z), then the Borel Conjecture is true in dimension≥5and in dimension4if G is good in the sense of Freedman.

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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)

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

If G belongs toFJ_L(Z)or toBC, then the Novikov Conjecture holds for the group G.

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Theorem (K0(RG)and induction from finite subgroups) Let R be a regular ring withQ⊆R. Suppose G∈ FJ(R).

Then the map given by induction from finite subgroups of G colim

OrF IN(G)K₀(RH)→K₀(RG) is bijective.

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Conjecture (Bass Conjecture)

Let R be a commutative integral domain and let G be a group. Let g 6=1be an element in G. Suppose that either the order|g|is infinite or that the order|g|is finite and not invertible in R.

Then theBass Conjecturepredicts that for every finitely generated projective RG-module P the value of itsHattori-Stallings rankHS_RG(P) at(g)is trivial.

Theorem (The Farrell-Jones Conjecture and the Bass Conjecture)

Let G be a group. Suppose that G ∈ FJ(F)for every field F of prime characteristic.

Then the Bass Conjecture is satisfied for every integral domain R.

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Let G be a group. Suppose that G∈ FJ(F)for every field F of prime characteristic.

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Conjecture (Homotopy invariance ofL2-torsion)

If X and Y are det-L²-acyclic finite G-CW -complexes, which are G-homotopy equivalent, then their L²-torsion agree:

ρ⁽²⁾(X;N(G)) =ρ⁽²⁾(Y;N(G)).

Theorem

Suppose that G∈ FJ(Z). Then G satisfies the Conjecture above.

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ρ⁽²⁾(X;N(G)) =ρ⁽²⁾(Y;N(G)).

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ρ⁽²⁾(X;N(G)) =ρ⁽²⁾(Y;N(G)).

Theorem

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Conjecture ((Stable) Gromov-Lawson-Rosenberg Conjecture) Let M be a closed connectedSpin-manifold of dimension n≥5.

Then M ×B^k carries for some integer k ≥0a Riemannian metric with positive scalar curvature if and only if

ind_C_r^∗_(π₁_(M);R)(M) = 0 ∈KO_n(C_r^∗(π₁(M);R)).

Theorem (The Baum-Connes Conjecture and the stable Gromov-Lawson-Rosenberg Conjecture)

If G∈ BC, then the Stable Gromov-Lawson-Rosenberg Conjecture true for all closedSpin-manifolds of dimension≥5withπ₁(M)∼=G.

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Other versions of Isomorphism Conjectures

Conjecture (Isomorphism Conjecture)

LetH_∗^?be an equivariant homology theory. It satisfies theIsomorphism Conjecturefor the group G and the familyF if the projection

EF(G)→pt induces for all n∈Za bijection H^G_n(EF(G))→ H_n^G(pt).

Example

The Farrell-Jones Conjecture forK-theory orL-theory respectively with coefficients inR is the Isomorphism Conjecture for

H^?_∗ =H∗(−;K_R)orH^?_∗=H∗(−;L^h−∞i_R )respectively and F =VCYC.

The Baum-Connes Conjecture is the Isomorphism Conjecture for H^?_∗ =K_∗^?=H_∗^?(−;K^top)andF =F IN.

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Other versions of Isomorphism Conjectures

Example

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Other versions of Isomorphism Conjectures

Example

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Other versions of Isomorphism Conjectures

Example

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Other versions of Isomorphism Conjectures

Example

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Other versions of Isomorphism Conjectures

Example

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There are functorsP andAwhich assign to a spaceX thespace of pseudo-isotopiesand itsA-theory.

Composing it with the functor sending a groupoid to its classifying space yields functorsPandAfromGroupoidstoSpectra.

Thus we obtain equivariant homology theoriesH_∗^?(−;P)and H_∗^?(−;A). They satisfyH_n^G(G/H;P) =π_n(P(BH))and H_n^G(G/H;A) =πn(A(BH)).

Conjecture (The Farrell-Jones Conjecture for pseudo-isotopies andA-theory)

The Farrell-Jones Conjecture for pseudo-isotopies and A-theory respectively is the Isomorphism Conjecture for H_∗^?(−;P)and H_∗^?(−;A) respectively for the familyVCYC.

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Thus we obtain equivariant homology theoriesH_∗^?(−;P)and H_∗^?(−;A). They satisfyH_n^G(G/H;P) =π_n(P(BH))and H_n^G(G/H;A) =πn(A(BH)).

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Theorem (Relating pseudo-isotopy andK-theory)

The rational version of the K -theoretic Farrell-Jones Conjecture for coefficients inZis equivalent Farrell-Jones Conjecture for

Pseudoisotopies.

In degree n≤1this is even true integrally.

Pseudo-isotopy andA-theory are important theories. In particular they are closely related to thespace of selfhomeomorphismsand thespace of selfdiffeomorphismsof closed manifolds.

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Pseudoisotopies.

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Pseudoisotopies.

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Pseudoisotopies.

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Pseudoisotopies.

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There are functorsTHHandTCwhich assign to a ring (or more generally to anS-algebra) a spectrum describing itstopological Hochschild homology and itstopological cyclic homology.

These functors play an important role inK-theoretic computations.

Composing it with the functor sending a groupoid to a kind of group ring yields functorsTHH_R andTC_R fromGroupoidsto Spectra.

Thus we obtain equivariant homology theoriesH_∗^?(−;THH_R)and H_∗^?(−;TC_R). They satisfyH_n^G(G/H;THH_R) =THH_n(RH)and H_n^G(G/H;TC_R) =TC_n(RH).

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