Applications of the Farrell-Jones Conjecture (Lecture V)

Applications of the Farrell-Jones Conjecture (Lecture V)

Wolfgang Lück Bonn Germany

email wolfgang.lueck@him.uni-bonn.de http://131.220.77.52/lueck/

Oberwolfach, October 2017

Outline

Review of theFarrell-Jones Conjectureand of some applications.

TheNovikov Conjecture TheBorel Conjecture TheMoody’s Conjecture TheThe Bass Conjecture

Hyperbolic groups with spheres as boundary Computational aspects.

Review

Conjecture (The Farrell-Jones-Conjecture)

TheFarrell-Jones Conjecturewith coefficients in R for the group G predicts that theassembly maps, which are induced by the projection EVCYC(G)→pt,

H_n^G(EVCYC(G),K_R) → H_n^G(pt,K_R) =Kn(RG);

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

A more general version,the Full Farrell-Jones Conjecture, holds for large class of groups and has good inheritance properties.

The Farrell-Jones Conjecture implies the following version for torsionfree groups

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

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

Conjecture (L-theoretic Farrell-Jones Conjecture for torsionfree groups)

TheL-theoretic Farrell-Jones Conjecturewith coefficients in the ring with involution R for the torsionfree group G predicts that theassembly map

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

Theorem (Some applications of theK-theoretic Farrell-Jones Conjecture)

The K -theoretic Farrell-Jones Conjecture implies:

K_n(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 of dimension≥6withπ1(W)∼=G is trivial;

Serre’sConjecture: The group G is of type FF if and only if it is of type FP;

Kaplansky’sIdempotent Conjecture: Let R be an integral domain and let G be a torsionfree group. Then all idempotents of RG are trivial, i.e., equal to0or1.

The Novikov Conjecture

Definition (L-class)

TheL-classof a closed manifoldM is a certain rational polynomial in the rational Pontryagin classes

L(M)=L(p₁(M),p₂(M), . . .)∈M

i≥0

H⁴ⁱ(M;Q).

Itsi-th component is denoted byL_i(M)∈H⁴ⁱ(M;Q).

L₁(M) = 1

3 ·p₁(M);

L₂(M) = 1 45 ·

7·p₂(M)−p₁(M)²

; L₃(M) = 1

945 ·

62·p₃(M)−13·p₁(M)∪p₂(M) +2·p₁(M)³ .

Theorem (Signature Theorem,Hirzebruch)

If M is a4k -dimensional closed oriented manifold M, then we get for its signature

sign(M) =hL_k(M),[M]i.

Exercise (Homotopy invariance ofLk(M)for 4k)

Show thatL_k(M)for n=4k is a homotopy invariant of closed orientable4k -dimensional manifolds.

One can show that a polynomial in the Pontryagin classes gives a homotopy invariant if and only if it is a multiple of thek-thL-class.

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:M_i →BG with f₁◦g 'f₂ we have

sign_x(M₀,f₀) =sign_x(M₁,f₁).

Exercise (Novikov Conjecture for closed aspherical manifolds) The Novikov Conjecture predicts for a homotopy equivalence f:M→N of closed aspherical manifolds

f^∗(L(N)) =L(M).

This is surprising since this is not true in general and in many cases one can detect that two specific closed homotopy

equivalent manifolds cannot be diffeomorphic by the failure of this equality to be true.

A deep theorem ofNovikovpredicts thatf^∗(L(N)) =L(M)holds for a homeomorphism of closed manifolds.

Hence an explanation, why the Novikov Conjecture may be true for closed aspherical manifolds, comes from the next conjecture.

The Borel Conjecture

Conjecture (Borel Conjecture)

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

In particular M and N are homeomorphic.

This is the topological version ofMostow rigidity. One version of Mostow rigidity says that any homotopy equivalence between hyperbolic closed Riemannian manifolds of dimension≥3 is homotopic to an isometric diffeomorphism. In particular they are isometrically diffeomorphic if and only if their fundamental groups are isomorphic.

The Borel Conjecture is in general false in the smooth category. A counterexample isTⁿforn≥5.

In some sense the Borel Conjecture is opposed to thePoincaré Conjecture. Namely, in the Borel Conjecture the fundamental group can be complicated but there are no higher homotopy groups, whereas in the Poincaré Conjecture there is no fundamental group but complicated higher homotopy groups.

The Borel Conjecture in dimension 1 and 2 is obviously true.

Thurston’s Geometrization Conjectureimplies the Borel Conjecture in dimension 3.

Exercise (The Borel Conjecture forT²) Prove the Borel Conjecture for the2-torus T².

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

If the K -theoretic and the L-theoretic Farrell-Jones Conjecture hold for the group G, then the Borel Conjecture holds for any n-dimensional aspherical closed manifold withπ₁(M)∼=G, provided that n≥5.

Next we sketch the proof. Therefore we recall:

Conjecture (L-theoretic Farrell-Jones Conjecture for torsionfree groups)

TheL-theoretic Farrell-Jones Conjecturewith coefficients in the ring with involution R for the torsionfree group G predicts that theassembly map

H_n BG;L^h−∞i_R

→L^h−∞i_n (RG) is bijective for all n∈Z.

Definition (Structure set)

The(topological) structure setS^top(M)of a manifoldM consists of equivalence classes of homotopy equivalencesN→M with a manifold N as source.

Two such homotopy equivalencesf₀:N₀→Mandf₁:N₁→Mare equivalent if there exists a homeomorphismg:N₀→N₁with f₁◦g'f₀.

Theorem (Topological rigidity and the structure set)

The Borel Conjecture holds for a closed manifold M if and only if S^top(M)consists of one element.

Theorem (Algebraic surgery sequence,Ranicki)

There is an exact sequence of abelian groups calledalgebraic surgery exact sequencefor an n-dimensional closed manifold M

. . .−^σ−−ⁿ⁺¹→H_n+1(M;Lh1i)−−−→^An+1 L_n+1(Zπ₁(M))−^∂−−ⁿ⁺¹→

S^top(M)−→^σn H_n(M;Lh1i)−→^An L_n(Zπ₁(M))−→^∂n . . . It can be identified with the classical geometric surgery exact

sequence due toSullivan and Wallin high dimensions.

HereLh1iis the1-connective coverof theL-theory spectrumL_Z. It comes with a natural map of spectrai:Lh1i →L_Zwhich induces onπ_i^san isomorphism fori ≥1, and we haveπ^s_i(Lh1i) =0 for i ≤0.

There are natural identifications

N(M)∼= [M,G/O]∼=Hn(M;Lh1i).

We can writeAnas the composite An:N(M) =Hn(M;Lh1i)

Hn(idM;i)

−−−−−→H_n(M;L_Z) =H_n(BG;L_Z) =H_n^G(EG)→L_n(ZG), where the second map is assembly map for the familyT R.

The mapAncan be identified with the map given by the surgery obstruction in the geometric surgery exact sequence.

This gives an interestinginterpretationof the homotopy theoretic assembly map in geometric terms. Its proof is non-trivial.

The analog statement aboutA_m holds in all degreesm≥n.

S^top(M)consists of one element if and only ifA_n+1is surjective andA_nis injective.

An easy spectral sequence argument shows that H_m(id_M;i) :H_n(M;Lh1i)→H_m(M;L_Z) is bijective form≥n+1 and injective form=n.

This finishes the proof since the Farrell-Jones Conjecture implies form=n,n+1 the bijectivity of

Hn(M;L_Z) =Hn(BG;L_Z) =H_n^G(EG)→Lm(ZG).

TheK-theoretic Farrell-Jones Conjecture is needed in the proof above since it implies that Wh(G),Ke₀(ZG)andKe_n(ZG)forn≤ −1 vanish and hence the decorations in theL-groups do not matter.

Theorem (The Farrell-Jones Conjecture implies the Novikov Conjecture)

Consider the following statements for a group G.

1 The L-theoretic assembly map for the familyVCYC

H_n^G(EVCYC(G),L^h−∞i_R )→H_n^G(pt,L^h−∞i_R ) =L^h−∞i_n (RG) is rationally injective.

2 The L-theoretic assembly map for the familyT R

H_n^G(ET R(G),L^h−∞i_R ) =H_n(BG;L^h−∞i_R )→L^h−∞i_n (RG) is rationally injective;

3 For every n≥5the Novikov Conjecture holds for G.

Then: (1) =⇒ (2) =⇒ (3).

The class of groups for which the Novikov Conjecture holds is larger than the class for which theL-theoretic Farrell-Jones Conjecture is known. For instance it contains all linear groups.

Exercise (Homotopy groups of G/TOP) Show thatπ_n(G/TOP)vanishes for odd n≥5.

Moody’s Induction Conjecture

Conjecture (Moody’s Induction Conjecture)

Let R be a regular ring withQ⊆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. Then the map

colim

OrF IN(G)K₀(FH)[1/p]→K₀(FG)[1/p]

is bijective.

Theorem (Bredon homology) Consider any covariant functor

M:OrG→Λ-Modules.

Then there is up to natural equivalence of G-homology theories precisely one G-homology theoryH_∗^G(−,M), calledBredon homology, with the property that the covariant functor

H_n^G:OrG→Λ-Modules, G/H 7→H_n^G(G/H) is trivial for n6=0and naturally equivalent to M for n=0.

Bredon homology plays the role of cellular of singular homology in the equivariant setting.

LetH^G_∗ be aG-homology theory. Then there is an equivariant version of theAtiyah-Hirzebruch spectral sequenceconverging to H^G_p+q(X). ItsE²-term

E_p,q² =H_p^G(X;H_q^G(?))

is given by the Bredon homology associated to the covariant functor

OrG→Λ-Modules, G/H 7→ H^G_q(G/H).

LetM be the constant functor with value theΛ-moduleA. Then we get for everyG-CW-complexX

H_n^G(X;M)∼=ΛH_n(X/G;A).

We haveH₀(EF(G);M)∼=colim_OrF(G)M.

Theorem (Farrell-Jones implies Moody)

The K -theoretic Farrell-Jones Conjecture implies Moody’s Induction Conjecture.

Proof.

The Transitivity Principle implies that the canonical maps H_n^G(EF IN(G);K_R) → H_n^G(EVCYC(G);K_R);

H_n^G(EF IN(G);K_F)[1/p] → H_n^G(EVCYC(G);K_F)[1/p], are bijective.

Hence the Farrell-Jones Conjecture implies H₀^G(EF IN(G);K_R) → K₀(RG);

H₀^G(E_{F IN}(G);K_F)[1/p] → K₀(FG)[1/p].

Proof continued.

SinceKn(RH)andKn(FH)]1/p]vanish forn≤ −1, the equivariant Atiyah-Hirzebruch spectral sequence implies

H₀^G(EF IN(G);K_R) → H₀^G(EF IN(G);K₀(R?));

H₀^G(E_{F IN}(G);K_F)[1/p] → H₀^G(E_{F IN}(G);K₀(F?))[1/p], for the covariant functors fromOrGtoZ-Modulesand

Z[1/p]-Modulesrespectively sendingG/H toK₀(RH)and K₀(FH)[1/p]respectively.

Now the claim follows from the identifications H₀^G(EF IN(G);K(R?) ∼= colim

OrF IN(G)K₀(R?);

H₀^G(E_{F IN}(G);K(F?))[1/p] ∼= colim

OrF IN(G)K₀(F?)[1/p].

The Bass Conjecture

Letcon(G)be the set of conjugacy classes(g)of elementsg ∈G.

LetR be a commutative ring. Letclass(G,R)be the free R-module with the set con(G)as basis. This is the same as the R-module ofR-valued class functionsf:G→Rfor which {(g)∈con(G)|f(g)6=0}is finite.

Define theuniversalR-trace

tr^u_RG:RG→class(G,R), X

g∈G

rg·g 7→X

g∈G

rg·(g).

It extends to a function tr^u_RG:M_n(RG)→class(G,R)on (n,n)-matrices overRGby taking the sum of the traces of the diagonal entries.

LetP be a finitely generated projectiveRG-module. Choose a matrixA∈M_n(RG)such thatA²=Aand the image of the RG-mapr_A:RGⁿ→RGⁿgiven by right multiplication withAis RG-isomorphic toP. Define theHattori-Stallings rankofPto be

HS_RG(P) =tr^u_RG(A)∈class(G,R).

The Hattori-Stallings rank depends only on the isomorphism class of theRG-moduleP.

The Hattori-Stallings rank extends the notion of a character of a representation of a finite group to infinite groups.

Theorem (Hattori-Stallings rank and characters)

Let G be a finite group and F be a field of characteristic zero. Let V be a finitely generated FG-module which is the same as a

finite-dimensional G-representation over F . Letχ_V be its character. If C_G(g)denotes the centralizer of the element g∈G, then show

χ_V(g⁻¹) =|C_G(g)| ·HS_FG(P)(g).

The Hattori-Stallings rank induces anR-homomorphism, the Hattori-Stallings homomorphism,

HS_RG:K₀(RG)⊗_ZR→class(G,R), [P]⊗r 7→r ·HS_RG(P).

Exercise (Bijectivity of the Hattori-Stallings rank)

Show that for a group satisfying Moody’ Induction Conjecture, the map induced by the Hattori-Stallings rank

HS_CG:K₀(CG)⊗_ZC→class(G,C)

is injective and has as image the class functions which vanish on elements of infinite order.

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 its Hattori-Stallings rankHS_RG(P) at(g)is trivial.

Exercise (The character of a rationalized finitely generated ZG-module)

Let G be a finite group and M be a finitely generated projective ZG-module. Suppose that G satisfies the Bass Conjecture.

Show thatQ⊗_ZM is a finitely generated freeQG-module.

Actually, the conclusion appearing in the exercise above is a theorem ofSwanwhich is older than and was one motivation for the Bass Conjecture.

Theorem (Farrell-Jones implies Bass)

The K -theoretic Farrell-Jones Conjecture implies the Bass Conjecture

Proof.

We give only the proof in the caseR=Zusing the fact that the K-theoretic Farrell-Jones Conjecture implies Moody’s Induction Conjecture.

The claim follows from Moody’s Induction Conjecture applied toQ ifghas infinite order, since the Hattori-Stallings rank is compatible with induction.

In the case, whereg is finite and not invertible inR, Moody’s Induction Conjecture applied toQreduces the claim to the special case, whereGis finite, which follows fromSwan’sTheorem.

Word hyperbolic groups with spheres as boundary

Conjecture (Gromov)

Let G be a torsionfree hyperbolic group whose boundary is a sphere Sⁿ⁻¹. Then there is a closed aspherical manifold M withπ₁(M)∼=G.

Theorem (Bartels-Lück-Weinberger)

For n≥6the Conjecture is true, and the manifold M is unique up to homeomorphism.

Conjecture (Cannon’sConjecture)

A hyperbolic group G has S²as boundary if and only if it acts properly, cocompactly and isometrically onH³.

Theorem (Ferry-Lück-Weinberger (in preparation))

Let G be a torsionfree hyperbolic group with S²as boundary and l≥3 be natural number.

Then there is a closed aspherical(k +l)-dimensional manifold M with an isomorphism u_M:π₁(M)−→^∼= G×Z^l.

If M⁰is another closed aspherical manifold M⁰ with an isomorphism u_M⁰:π₁(M⁰)−→^∼= G×Z^l, then there is a homeomorphism f:M→M⁰ withπ1(f) =u⁻¹_M0 ◦u_M.

The Cannon Conjecture is now equivalent to the statement that there exists a closed 3-manifoldNsuch thatM is homeomorphic toN×T^l.

Equivariant Chern characters

Theorem (Dold (1962))

LetH_∗ be a generalized homology theory with values inΛ-modules for Q⊆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).

This means that theAtiyah-Hirzebruch spectral sequence collapses in the strongest sense.

The assumptionQ⊆Λis necessary.

Dolds’ Chern character for aCW-complexX is given by the following composite

chn: M

p+q=n

Hp(X;H_q(∗))←−^α M

p+q=n

Hp(X;Z)⊗_ZH_q(∗)

L

p+q=nhur⊗id ∼=

←−−−−−−−−−−−− M

p+q=n

π^s_p(X₊,∗)⊗_ZH_q(∗)

L

p+q=nDp,q

−−−−−−−−→ H_n(X).

We want to extend this to the equivariant setting.

This requires an extra structure on the coefficients of an equivariant homology theoryH^?_∗.

We define a covariant functor calledinduction ind:FGINJ→Λ- Mod

from the categoryFGINJof finite groups with injective group homomorphisms as morphisms to the category ofΛ-modules as follows.

It sendsGtoH^G_n(pt)and an injection of finite groupsα:H →Gto the morphism given by the induction structure

H^H_n(pt)−−→ H^{indα G}_n(ind_αpt) ^H

G n(pr)

−−−−→ H^G_n(pt).

Definition (Mackey extension)

We say thatH^?_∗ has aMackey extensionif for everyn∈Zthere is a contravariant functor calledrestriction

res:FGINJ→Λ- Mod

such that these two functors ind and res agree on objects and satisfy thedouble coset formula,i.e., we have for two subgroupsH,K ⊂Gof the finite groupG

res^K_G◦ind^G_H = X

KgH∈K\G/H

ind_c(g):H∩g−1Kg→K◦res^H∩g_H ^−1Kg,

wherec(g)is conjugation withg, i.e.,c(g)(h) =ghg⁻¹.

In every case we will consider such a Mackey extension does exist and is given by an actual restriction.

For instance forK₀^G(?) =R_Cinduction functor is just the classical induction of representations. The restriction functor is given by the classical restriction of representations.

Analogous statements hold forKn(R?)andL^h−∞i_n (R?).

Theorem (Lück)

LetH_∗^?be a proper equivariant homology theory with values in

Λ-modules forQ⊆Λ. Suppose thatH^?_∗ has a Mackey extension. Let I be the set of conjugacy classes(H)of finite subgroups H of G.

Then there is for every group G, every proper G-CW -complex X and every n∈Za natural isomorphism calledequivariant Chern character

ch^G_n: M

p+q=n

M

(H)∈I

Hp(C_GH\X^H; Λ)⊗_Λ[W

GH]S_H

H^H_q(∗)−∼=→ H^G_n(X)

C_GH is the centralizer andN_GH the normalizer ofH ⊆G;

W_GH:=N_GH/H·C_GH(This is always a finite group);

S_H H_q^H(∗)

:=cok

L

K⊂H K6=H

ind^H_K :L

K⊂H K6=H

H^K_q(∗)→ H^H_q(∗)

. ch^?_∗ is an equivalence of equivariant homology theories.

Theorem (Lück)

Let G be a group. Let T be the set of conjugacy classes(g)of elements g ∈G of finite order. There is a commutative diagram

L

p+q=n

L

(g)∈T H_p(BC_Ghgi;C)⊗_ZK_q(C) ^//

K_n(CG)⊗_ZC

L

p+q=n

L

(g)∈T H_p(BC_Ghgi;C)⊗_ZK_q^top(C) ^//K_n^top(C_r^∗(G))⊗_ZC The vertical arrows come from the obvious change of rings and of K-theory maps.

The horizontal arrows can be identified with the assembly maps occurring in the Farrell-Jones Conjecture and the Baum-Connes Conjecture by the equivariant Chern character.

Exercise (Infinite dihedral groupD∞) Compute Kn(CD∞)⊗_ZCfor n≤1.

Hyperbolic groups

Theorem (Hyperbolic groups)

Let G be a hyperbolic group. LetMbe a complete system of representatives of the conjugacy classes of maximal infinite virtual cyclic subgroups of G.

For n∈Zthere is an isomorphism H_n^G E G;K_R

⊕ M

V∈M

H^V_n E V →pt;K_R ^∼₌

−→ Kn(RG);

For n∈Zthere is an isomorphism H_n E G;L^h−∞i_R

⊕ M

V∈M

H^V_n E V →pt;L^h−∞i_R −→^∼= L^h−∞i_n (RG).

Theorem (Torsionfree hyperbolic groups)

If G is a torsionfree hyperbolic group, then we get isomorphisms

Hn(BG;K_R)⊕









M

(C),C⊆G,C6=1 C maximal cyclic

NKn(R)⊕NKn(R)









∼=

−→ Kn(RG),

and

Hn(BG;L^h−∞i_R ) −^∼=→ L^h−∞i_n (RG).

Exercise (Fundamental groups of hyperbolic closed 3-manifolds) Let G be the fundamental group of a hyperbolic closed3-manifold.

Compute L_n(ZG)for all decorationsin terms of H₁(M;Z).

The end of these lecture series

Thank you for your attention!