Introduction to the Farrell-Jones Conjecture

Introduction to the Farrell-Jones Conjecture

Wolfgang Lück Bonn Germany

email wolfgang.lueck@him.uni-bonn.de http://www.him.uni-bonn.de/lueck/

Bloomington, April 2019

K

(R) and the Idempotent Conjecture

Given a ringRand a groupG, denote byRGorR[G]thegroup ring.

AnRG-module is the same asG-representationwith coefficients inR, i.e., anR-module withG-action byR-linear maps.

IfX →X is aG-covering of aCW-complexX, then the cellular chain complex ofX is a freeZG-chain complex.

Ifghas finite order|g|andF is a field of characteristic zero, then we get an idempotent inFGby

x = 1

|g|·

|g|−1

X

i=0

gⁱ. Are there other idempotents?

Conjecture (Idempotent Conjecture)

TheKaplansky Conjecturesays that for a torsionfree group G and a field F of characteristic zero the elements0and1are the only idempotents in FG.

Definition (Projective class groupK0(R)) Define theprojective class groupof a ringR

K₀(R) to be the following abelian group:

Generators are isomorphism classes[P]of finitely generated projectiveR-modulesP;

The relations are[P₀] + [P₂] = [P₁]for every exact sequence 0→P₀→P₁→P₂→0 of finitely generated projective R-modules.

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

Definition (Reduced Projective class groupKe0(R)) Thereduced projective class group

Ke₀(R)=cok K₀(Z)→K₀(R)

is the quotient ofK₀(R)by the subgroup generated by the classes of finitely generated freeR-modules.

LetP be a finitely generated projectiveR-module. It isstably free, i.e.,P⊕R^m ∼=Rⁿfor somem,n∈Z, if and only if[P] =0 in Ke₀(R).

Conjecture (Vanishing of reduced projective class group for torsionfreeG)

If G is torsionfree, thenKe₀(ZG)andKe₀(FG)for a field F of characteristic zero vanish.

Wh(G) and the h-Cobordism Theorem

Definition (K1-groupK1(R)) Define theK₁-group of a ringR

K₁(R)

to be the abelian group whose generators are conjugacy classes[f]of automorphismsf:P→P of finitely generated projectiveR-modules with the following relations:

Given an exact sequence 0→(P₀,f₀)→(P₁,f₁)→(P₂,f₂)→0 of automorphisms of finitely generated projectiveR-modules, we get [f₀] + [f₂] = [f₁];

[g◦f] = [f] + [g].

PutGL(R) :=S

n≥1GLn(R). The obvious mapsGLn(R)→K₁(R) induce an isomorphism

GL(R)/[GL(R),GL(R)]−→^∼= K₁(R).

An invertible matrixA∈GL(R)can be reduced byelementary row and column operationsand(de-)stabilizationto the trivial empty matrix if and only if[A] =0 holds in thereducedK₁-group

Ke₁(R):=K₁(R)/{±1}=cok(K₁(Z)→K₁(R)). The assignmentA7→[A]∈K₁(R)can be thought of as the universal determinant forR.

Definition (Whitehead group)

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

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

Let M be a closed smooth or topological manifold of dimension≥5.

Then the so called Whitehead torsion yields a bijection τ:H(M)−→^∼= Wh(π₁(M))

whereH(M)is the set of h-cobordisms over M modulo diffeomorphisms or homeomorphisms relative M.

Conjecture (Vanishing of Wh(G)for torsionfreeG) If G is torsionfree, then

Wh(G) ={0}.

Lemma

Let G be finitely presented and d ≥5be any natural number. Then the following statements are equivalent:

The Whitehead groupWh(G)vanishes;

For one closed manifold M of dimension d with G∼=π₁(M)every h-cobordism over M is trivial;

For every closed manifold M of dimension d with G∼=π₁(M)every h-cobordism over M is trivial.

Conjecture (Poincaré Conjecture)

Let M be an n-dimensional topological manifold which is a homotopy sphere, i.e., homotopy equivalent to Sⁿ.

Then M is homeomorphic to Sⁿ.

Theorem (Freedman, Perelman, Smale) The Poincaré Conjecture is true.

Proof.

We sketch the proof forn≥6.

LetM be an-dimensional homotopy sphere.

LetW be obtained fromMby deleting the interior of two disjoint embedded disksDⁿ₁andD₂ⁿ. ThenW is a simply connected h-cobordism.

Since Wh({1})is trivial, we can find a homeomorphism f:W −→^∼= ∂Dⁿ₁×[0,1]which is the identity on∂D₁ⁿ=D₁ⁿ× {0}.

By theAlexander trickwe can extend the homeomorphism f|_Dn

1×{1}:∂Dⁿ₂−^∼=→∂D₁ⁿ× {1}to a homeomorphismg:D₁ⁿ→D₂ⁿ. The three homeomorphismsid_Dⁿ

1,f andgfit together to a homeomorphismh:M →D₁ⁿ∪_∂Dⁿ

1×{0}∂Dⁿ₁×[0,1]∪_∂Dⁿ

1×{1}D₁ⁿ. The target is obviously homeomorphic toSⁿ.

Figure (Proof of the Poincaré Conjecture)

homotopy n-sphere Sⁿ⁻¹×[0,1]

embedded disks

f

id_Sⁿ⁻¹

∼=

The argument above does not imply that for a smooth manifoldM we obtain a diffeomorphismg:M →Sⁿ since the Alexander trick does not work smoothly.

Indeed, there exist so calledexotic spheres, i.e., closed smooth manifolds which are homeomorphic but not diffeomorphic toSⁿ. Thes-cobordism theorem is a key ingredient in theSurgery Programfor the classification of closed manifolds due toBrowder, Novikov, SullivanandWall.

Motivation and Statement of the Farrell-Jones Conjecture for torsionfree groups

There areK-groupsK_n(R)for everyn∈Z.

Can one identifyK_n(RG)with more accessible terms?

IfG₀andG₁are torsionfree andRis regular, one gets isomorphisms

K_n(R[Z]) ∼= K_n(R)⊕K_n−1(R);

Ken(R[G₀∗G₁]) ∼= Ken(RG₀)⊕Ken(RG₁).

IfHis any (generalized) homology theory, then H_n(BZ) ∼= H_n(pt)⊕ H_n−1(pt);

He_n(B(G₀∗G₁)) ∼= He_n(BG₀)⊕He_n(BG₁).

Question: Can we findH_∗ withH_n(BG)∼=K_n(RG), provided that Gis torsionfree andRis regular.

Of course suchH∗has to satisfyH_n(pt) =K_n(R).

So the only reasonable candidate isH_n(−;K_R).

Conjecture (K-theoretic Farrell-Jones Conjecture for torsionfree groups and regular rings)

TheK -theoretic Farrell-Jones Conjecturewith coefficients in the regular ring R for the torsionfree group G predicts that theassembly map

H_n(BG;K_R)→K_n(RG) is bijective for every n∈Z.

There is also anL-theoryversion.

Applications of the Farrell-Jones Conjecture

The conjectures above about the vanishing ofKe₀(ZG)and Wh(G) for torsionfreeGdo follow from the Farrell-Jones Conjecture above.

The idea of the proof is to study theAtiyah-Hirzebruch spectral sequenceconverging toHn(BG;K_R)whoseE²-term is given by

E_p,q² =H_p(BG,K_q(R)), using

Kn(Z) =







{0} n≤ −1;

Z n=0;

{±1} n=1.

Definition (Topologically rigid)

A closed topological manifoldN is calledtopologically rigidif any homotopy equivalencef:M →Nwith a closed manifoldM as source is homotopic to a homeomorphism.

Conjecture (Borel Conjecture)

TheBorel Conjecture for Gpredicts that an aspherical closed manifold with fundamental group G is topologically rigid.

In particular the Borel Conjecture predicts that two aspherical closed manifolds are homeomorphic if and only if their

fundamental groups are isomorphic.

The Poincaré Conjecture is equivalent to the statement thatSⁿis

The Borel Conjecture can be viewed as the topological version of Mostow rigidity.

A special case of Mostow rigidity says that any homotopy equivalence between closed hyperbolic manifolds of dimension

≥3 is homotopic to an isometric diffeomorphism.

The Borel Conjecture is not true in the smooth category by results ofFarrell-Jones.

The Borel Conjecture follows in dimension≥5 from the Farrell-Jones Conjecture as briefly explained next.

Definition (The structure set)

LetNbe a closed topological manifold of dimensionn. We call two homotopy equivalencesf_i:M_i →N from closed topological manifolds M_i of dimensionntoN fori =0,1 equivalent if there exists a

homeomorphismg:M₀→M₁such thatf₁◦g is homotopic tof₀. Thestructure setS_n(N)ofN is the set of equivalence classes of homotopy equivalencesM→X from closed topological manifolds of dimensionntoN. This set has a preferred base point, namely the class of the identity id:N→N.

A closed topological manifoldM is topologically rigid if and only if the structure setS_n(M)consists of exactly one point.

Theorem (The topological Surgery Exact Sequence,Browder, Novikov, Sullivan, Wall)

For a closed n-dimensional topological manifold N with n≥5, there is an exact sequence of abelian groups, calledsurgery exact sequence,

· · ·−→ N^η _n+1^top(N×[0,1],N× {0,1})−^σ−−ⁿ⁺¹→L^s_n+1(Zπ)−→ S^∂ _n^top(N)

−→ Nη _n^top(N)−→^σn L^s_n(Zπ)

It is the main ingredient in showing that the Farrell-Jones Conjecture implies the Borel Conjecture in dimension≥5.

The key idea is to use the Farrell-Jones Conjecture to show that the mapσn+1is surjective and the mapσnis injective.

Theorem (Bartels-Lück-Weinberger)

Let G be a torsionfree hyperbolic group and let n be an integer≥6.

Then the following statements are equivalent:

The boundary∂G is homeomorphic to Sⁿ⁻¹;

There is a closed aspherical topological manifold M such that G∼=π₁(M), its universal coveringM is homeomorphic toe Rⁿand the compactification ofM bye ∂G is homeomorphic to Dⁿ.

The manifold above is unique up to homeomorphism.

Theorem (Homotopy groups of automorphism groups of aspherical manifolds)

Let M be an orientable closed aspherical (smooth) manifold of dimension>10with fundamental group G. Suppose that G satisfies the K -and the L-theoretic Farrell Jones Conjecture.

Then for1≤i ≤(dimM−7)/3one has π_i(Top(M))⊗_ZQ=

center(G)⊗_ZQ if i =1;

0 if i >1,

and

π_i(Diff(M))⊗_ZQ=







center(G)⊗_ZQ if i =1;

L∞

j=1H_(i+1)−4j(M;Q) if i >1, dimM odd;

0 if i >1, dimM even.

There are many other applications of the Farrell-Jones Conjecture, for instance:

Novikov Conjecture.

Bass Conjecture.

Moody’s Induction Conjecture.

Serre’s Conjecture.

Classification of certain classes ofmanifoldswith infinite fundamental group.

Classification ofPoincaré duality groups.

κ-classesfor aspherical manifolds.

StableCannon Conjecture.

The general version the Farrell-Jones Conjecture

One can formulate a version of the Farrell-Jones Conjecture which makes sense for all groupsGand all ringsR.

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 every n∈Z.

There is also anL-theoryversion.

One can also allowtwisted group ringsandorientation characters.

In the sequel theFull Farrell-Jones Conjecturerefers to the most general version for bothK-theory andL-theory, namely, with coefficients in additiveG-categories (with involution) and finite wreath products.

All conjectures or results mentioned in this talk follow from the Full Farrell-Jones Conjecture.

Status of the Full Farrell-Jones Conjecture

Theorem (Bartels, Bestvina, Farrell, Kammeyer, Lück, Reich, Rüping, Wegner)

LetFJ be the class of groups for which the Full Farrell-Jones Conjecture holds. ThenFJ contains the following groups:

Hyperbolic groups;

CAT(0)-groups;

Solvable groups;

(Not necessarily uniform) lattices in almost connected Lie groups;

Fundamental groups of (not necessarily compact) d -dimensional manifolds (possibly with boundary) for d ≤3;

Subgroups of GLn(Q)and of GLn(F[t])for a finite field F ; All S-arithmetic groups;

mapping class groups.

Theorem (continued)

Moreover,FJ has the following inheritance properties:

If G₁and G₂belong toFJ, then G₁×G₂and G₁∗G₂belong to FJ;

If H is a subgroup of G and G∈ FJ, then H ∈ FJ;

If H ⊆G is a subgroup of G with[G:H]<∞and H ∈ FJ, then G∈ FJ;

Let{G_i |i ∈I}be a directed system of groups (with not

necessarily injective structure maps) such that G_i ∈ FJ for i ∈I.

Thencolim_i∈IG_i belongs toFJ;

Many more mathematicians have made important contributions to the Farrell-Jones Conjecture, e.g.,Bökstedt, Carlsson, Jim Davis, Ferry, Hambleton, Gandini, Hsiang, Jones, Kasprowski, Linnell, Madsen, Nicas, Pedersen, Quinn, Ranicki, Rognes, Roushon,

The Farrell-Jones Conjecture is open for:

Out(Fn);

amenable groups;

Thompson’s groups;

G=F_no Z.

There are manyconstructions of groups with exotic properties which arise as colimits of hyperbolic groups.

One example is the construction ofgroups with expandersdue to Gromov, seeArzhantseva-Delzant. These yieldcounterexamples to theBaum-Connes Conjecture with coefficientsdue to

Higson-Lafforgue-Skandalis.

However, our results show that these groups do satisfy the Full Farrell-Jones Conjecture and hence also the other conjectures mentioned above.

Many groups of the region ‘Hic abundant leones’in the universe of groups in the sense ofBridsondo satisfy the Full Farrell-Jones Conjecture.

We have no good candidate for a group (or for a property of

Davis-Januszkiewiczhave constructed exotic aspherical closed manifolds usinghyperbolization techniques. For instance there are examples which donot admit a triangulationor whose universal covering is not homeomorphic to Euclidean space.

However, in all cases the universal coverings are CAT(0)-spaces and the fundamental groups are CAT(0)-groups. Hence they satisfy the Full Farrell-Jones Conjecture and in particular the Borel Conjecture in dimension≥5.

Ideas of proofs

The assembly map can be thought of anapproximationof the algebraicK- orL-theoryby a homology theory.

The basic feature between the left and right side of the assembly map is that on the left side one hasexcisionwhich is not present on the right side.

In general excision is available if one can makerepresenting cycles small.

A best illustration for this is the proof of excision for simplicial or singular homology based onsubdivisionwhose effect is to make the support of cycles arbitrary small.

Then the basic goal of the proof is obvious: Find a procedure to make the support of a representing cocycle as small as possible without changing its class.

Suppose thatG=π₁(M)for a closed Riemannian manifold with negative sectional curvature.

The idea is to use thegeodesic flowon the universal covering to gain the necessary control.

We will briefly explain this in the case, where the universal covering is the two-dimensional hyperbolic spaceH².

Consider two points with coordinates(x₁,y₁)and(x₂,y₂)in the upper half plane model of two-dimensional hyperbolic space. We want to use the geodesic flow to make their distance smaller in a functorial fashion. This is achieved by letting these points flow towards the boundary at infinity along the geodesic given by the vertical line through these points, i.e., towards infinity in the y-direction.

There is a fundamental problem: ifx₁=x₂, then the distance between these points is unchanged. Therefore we make the following prearrangement. Suppose thaty₁<y₂. Then we first let the point(x₁,y₁)flow so that it reaches a position wherey₁=y₂. Inspecting the hyperbolic metric, one sees that the distance between the two points(x₁, τ)and(x₂, τ)goes to zero ifτ goes to infinity. This is the basic idea n the negatively curved case to make the cycles small, or in other words, to gain control.