The Farrell-Jones Conjecture (Lecture II)

(1)

The Farrell-Jones Conjecture (Lecture II)

Wolfgang Lück Bonn Germany

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

Göttingen, June 22, 2011

Wolfgang Lück (Bonn, Germany) The Farrell-Jones Conjecture Göttingen, June 22, 2011 1 / 32

(2)

Outline

We briefly explainhomology theoriesand how they arise from spectra.

We state theFarrell-Jones-Conjectureand theBaum-Connes Conjecturefor torsionfree groups.

We discuss applications of these conjectures such as the Kaplansky Conjecture,Novikov Conjectureand theBorel Conjecture.

We explain that the formulations for torsionfree groups cannot extend to arbitrary groups and state the general versions.

We give a report about the status of the Farrell-Jones Conjecture.

(3)

Homology theory

Definition (Homology theory)

Ahomology theoryH_∗is a covariant functor from the category of CW-pairs to the category ofZ-graded abelian groups together with natural transformations

∂n(X,A) :H_n(X,A)→ H_n−1(A) forn∈Zsatisfying the following axioms:

Homotopy invariance

Long exact sequence of a pair Excision

If(X,A)is aCW-pair andf:A→Bis a cellular map , then H_n(X,A)−^∼⁼→ H_n(X ∪_f B,B).

(4)

Homology theory

Definition (Homology theory)

Ahomology theoryH_∗is a covariant functor from the category of CW-pairs to the category ofZ-graded abelian groups together with natural transformations

∂n(X,A) :H_n(X,A)→ H_n−1(A) forn∈Zsatisfying the following axioms:

Homotopy invariance

Long exact sequence of a pair Excision

If(X,A)is aCW-pair andf:A→Bis a cellular map , then H_n(X,A)−^∼⁼→ H_n(X ∪_f B,B).

(5)

Definition (continued) Disjoint union axiom

M

i∈I

H_n(X_i)−→ H^∼⁼ _n a

i∈I

X_i

! .

If theCW-complexX is the union of two subcomplexesX₁andX₂ and we putX₀=X₁∩X₂, then there is a long exactMayer-Vietoris sequence

· · · → H_n+1(X₀)→ H_n+1(X₁)⊕ H_n+1(X₂)→ H_n+1(X)

→ H_n(X₀)→ H_n(X₁)⊕ H_n(X₂)→ H_n(X)→ · · ·.

(6)

Definition (continued) Disjoint union axiom

M

i∈I

H_n(X_i)−→ H^∼⁼ _n a

i∈I

X_i

! .

If theCW-complexX is the union of two subcomplexesX₁andX₂ and we putX₀=X₁∩X₂, then there is a long exactMayer-Vietoris sequence

· · · → H_n+1(X₀)→ H_n+1(X₁)⊕ H_n+1(X₂)→ H_n+1(X)

→ H_n(X₀)→ H_n(X₁)⊕ H_n(X₂)→ H_n(X)→ · · ·.

(7)

Theorem (Homology theories and spectra)

LetEbe a spectrum. Then we obtain a homology theoryH∗(−;E)by Hn(X,A;E) :=πn((X ∪_Acone(A))∧E).

It satisfies

H_n(pt;E) =π_n(E).

Any homology theory arises in this way.

The following conjectures are motivated by computations which reveal a homological flavour ofK andL-theory of group rings.

(8)

It satisfies

H_n(pt;E) =π_n(E).

(9)

It satisfies

H_n(pt;E) =π_n(E).

(10)

The Isomorphism Conjectures for torsionfree groups

Conjecture (Baum-Connes Conjecture for torsionfree groups) TheBaum-Connes Conjecturefor the torsionfree group predicts that theassembly map

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

BGis theclassifying spaceof the groupG.

K_n(BG)is the topologicalK-homology ofBG.

K_n(C_r^∗(G))is the topologicalK-theory of the reduced complex groupC^∗-algebraC_r^∗(G)ofGwhich is the closure in the norm topology ofCGconsidered as subalgebra ofB(l²(G)).

There is also areal versionof the Baum-Connes Conjecture

KO_n(BG)→K_n(C_r^∗(G;R)).

(11)

The Isomorphism Conjectures for torsionfree groups

(12)

The Isomorphism Conjectures for torsionfree groups

(13)

The Isomorphism Conjectures for torsionfree groups

(14)

The Isomorphism Conjectures for torsionfree groups

There is also areal versionof the Baum-Connes Conjecture KO_n(BG)→K_n(C_r^∗(G;R)).

(15)

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

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 all n∈Z.

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

K_Ris the (non-connective) algebraicK-theory spectrum ofR;

H_n(pt;K_R)∼=πn(K_R)∼=K_n(R)forn∈Z.

(16)

(17)

(18)

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.

L^h−∞i_n (RG)is the algebraicL-theory ofRGwith decorationh−∞i;

L^h−∞i_R is the algebraicL-theory spectrum ofRwith decoration h−∞i;

Hn(pt;L^h−∞i_R )∼=πn(L^h−∞i_R )∼=L^h−∞i_n (R)forn∈Z.

(19)

(20)

(21)

Consequences of the Isomorphism Conjectures for torsionfree groups

LetFJ_K(R)andFJ_L(R)respectively be the class of groups which satisfy theK-theoretic andL-theoretic respectively Farrell-Jones Conjecture for the coefficient ringR.

LetBC be the class of groups which satisfy the Baum-Connes Conjecture.

(22)

Consequences of the Isomorphism Conjectures for torsionfree groups

LetFJ_K(R)andFJ_L(R)respectively be the class of groups which satisfy theK-theoretic andL-theoretic respectively Farrell-Jones Conjecture for the coefficient ringR.

LetBC be the class of groups which satisfy the Baum-Connes Conjecture.

(23)

Lemma

Supose that R is a regular ring, G is torsionfree and G∈ FJ_K(R).

Then

K_n(RG) =0for n≤ −1;

The change of rings map K₀(R)→K₀(RG)is bijective. In particularKe₀(RG)is trivial if and only ifKe₀(R)is trivial.

Lemma

Suppose that G is torsionfree and G ∈ FJ_K(Z). Then the Whitehead groupWh(G)is trivial.

Proof.

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)).

(24)

Lemma

Then

Lemma

Proof.

(25)

Lemma

Then

Lemma

Proof.

(26)

In particular we get for a torsionfree groupG∈ FJ_K(Z):

Kn(ZG) =0 forn≤ −1;

Ke₀(ZG) =0;

Wh(G) =0;

Every finitely dominatedCW-complexX withG=π₁(X)is homotopy equivalent to a finiteCW-complex;

Every compacth-cobordismW of dimension≥6 withπ₁(W)∼=G is trivial.

(27)

Ke₀(ZG) =0;

Wh(G) =0;

(28)

Ke₀(ZG) =0;

Wh(G) =0;

(29)

Ke₀(ZG) =0;

Wh(G) =0;

(30)

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,Bartels-L.-Reich(2007))

Let F be a field and let G be a torsionfree group with G∈ FJ_K(F).

Then0and1are the only idempotents in FG.

(31)

(32)

(33)

(34)

Proof.

Letpbe an idempotent inFG. We want to showp∈ {0,1}.

Denote by:FG→F the augmentation homomorphism sending P

g∈Grg·g toP

g∈Grg. Obviously(p)∈F is 0 or 1. Hence it suffices to showp=0 under the assumption that(p) =0.

Let(p)⊆FGbe the ideal generated bypwhich is a finitely generated projectiveFG-module.

SinceG∈ FJ_K(F), we can conclude that

i∗:K₀(F)⊗_ZQ→K₀(FG)⊗_ZQ is surjective.

Hence we can find a finitely generated projectiveF-modulePand integersk,m,n≥0 satisfying

(p)^k⊕FG^m∼=_FGi_∗(P)⊕FGⁿ.

(35)

Proof (continued).

If we now applyi_∗◦∗and use◦i =id,i_∗◦∗(FG^l)∼=FG^l and (p) =0 we obtain

FG^m ∼=i∗(P)⊕FGⁿ. Inserting this in the first equation yields

(p)^k ⊕i∗(P)⊕FGⁿ∼=i∗(P)⊕FGⁿ.

Our assumptions onF andGimply thatFGisstably finite, i.e., ifA andBare square matrices overFGwithAB=I, thenBA=I.

This implies(p)^k =0 and hencep=0.

(36)

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).

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

The Novikov Conjecture is true if the assembly map appearing in the Baum-Connes Conjecture or in the L-theoretic Farrell-Jones

Conjecture are rationally injective.

(37)

(38)

(39)

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

f∗(L(M)) =L(N).

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

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

A deep theorem ofNovikov (1965)predicts thatf∗(L(M)) =L(N) holds for a homeomorphism of closed manifolds.

Hence an explanation why the Novikov Conjecture may be true for closed aspherical manifolds is due to the next conjecture.

(40)

f∗(L(M)) =L(N).

(41)

f∗(L(M)) =L(N).

(42)

f∗(L(M)) =L(N).

(43)

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.

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(1989).

There are also non-aspherical manifolds which are topologically rigid in the sense of the Borel Conjecture (seeKreck-L. (2005)).

(44)

(45)

(46)

(47)

(48)

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

If the K - and L-theoretic Farrell-Jones Conjecture hold for G in the case R =Z, then the Borel Conjecture is true in dimension≥5and in dimension4if G is good in the sense of Freedman.

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

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

(49)

(50)

(51)

(52)

What happens for groups with torsion?

The versions of the Farrell-Jones Conjecture and the Baum- Connes Conjecture above become false for finite groups unless the group is trivial.

For instance the version of the Baum-Connes Conjecture above would predict for a finite groupG

K₀(BG)∼=K₀(C_r^∗(G))∼=R_C(G).

However,K₀(BG)⊗_ZQ∼=_Q K₀(pt)⊗_ZQ∼=_Q Qand R_C(G)⊗_ZQ∼=_QQholds if and only ifGis trivial.

Next we formulate a general version.

(53)

What happens for groups with torsion?

K₀(BG)∼=K₀(C_r^∗(G))∼=R_C(G).

(54)

What happens for groups with torsion?

K₀(BG)∼=K₀(C_r^∗(G))∼=R_C(G).

(55)

What happens for groups with torsion?

K₀(BG)∼=K₀(C_r^∗(G))∼=R_C(G).

(56)

Classifying spaces for families

Definition (Family of subgroups)

AfamilyF of subgroupsofGis a set of (closed) subgroups ofGwhich is closed under conjugation and finite intersections.

Examples forF are:

T R = {trivial subgroup};

F IN = {finite subgroups};

VCYC = {virtually cyclic subgroups};

ALL = {all subgroups}.

(57)

Classifying spaces for families

Definition (Family of subgroups)

AfamilyF of subgroupsofGis a set of (closed) subgroups ofGwhich is closed under conjugation and finite intersections.

Examples forF are:

T R = {trivial subgroup};

F IN = {finite subgroups};

VCYC = {virtually cyclic subgroups};

ALL = {all subgroups}.

(58)

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-complexE_F(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 →EF(G).

EF IN(G)is also called theclassifying space for properG-actions.

(59)

(60)

(61)

Theorem (Homotopy characterization ofEF(G)) LetF be a family of subgroups.

There exists a model for EF(G)for any familyF;

Two model for EF(G)are G-homotopy equivalent;

A G-CW -complex X is a model for E_F(G)if and only if all its isotropy groups belong toF and for each H∈ F the H-fixed point set X^H is weakly contractible.

(62)

Theorem (Homotopy characterization ofEF(G)) LetF be a family of subgroups.

There exists a model for EF(G)for any familyF;

Two model for EF(G)are G-homotopy equivalent;

A G-CW -complex X is a model for E_F(G)if and only if all its isotropy groups belong toF and for each H∈ F the H-fixed point set X^H is weakly contractible.

(63)

A model forEALL(G)isG/G;

EG→BG:=G\EGis theuniversalG-principal bundlefor G-CW-complexes.

Example (Infinite dihedral group)

LetD∞=Z o Z/2=Z/2∗Z/2 be the infinite dihedral group.

A model forED∞is the universal covering ofRP^∞∨RP^∞. A model forE D∞isRwith the obviousD∞-action.

(64)

(65)

(66)

(67)

(68)

(69)

The general formulation 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) =K_n(RG) is bijective for all n∈Z.

H_n^G(−,K_R)is aG-homology theory defined forG-CW-complexes which satisfiesH_n^G(G/H,K_R)∼=K_n(RH)for all subgroupsH ⊆G;

The assembly map is the map induced by the projection E_VCYC(G)→pt.

(70)

The general formulation of the Isomorphism Conjectures

(71)

The general formulation of the Isomorphism Conjectures

(72)

The general formulation of the Isomorphism Conjectures

(73)

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

H_n^G(−,L^h−∞i_R )is aG-homology theory defined for

G-CW-complexes which satisfiesH_n^G(G/H,L^h−∞i_R )∼=L^h−∞i_n (RH) for all subgroupsH ⊆G;

The assembly map is the map induced by the projection EVCYC(G)→pt.

(74)

(75)

(76)

Conjecture (Baum-Connes Conjecture)

TheBaum-Connes Conjecturepredicts that the assembly map K_n^G(E G)→K_n^G(pt) =Kn(C_r^∗(G))

is bijective for all n∈Z.

K_n^G(−)is aG-homology theory defined forG-CW-complexes which satisfiesK_n^G(G/H)∼=Kn(C_r^∗(H))for all subgroupsH ⊆G;

(77)

(78)

(79)

Status of the Farrell-Jones Conjectures

There are more general versions of the Farrell-Jones Conjecture, where one allowstwisted coefficientswhich can actually be additiveG- categories. In the sequel we refer to this general version.

Theorem (Main Theorem

(Bartels-Echterhoff-Farrell-Lück-Reich-Rüping-Wegner (2008-2012))

LetFJ be the class of groups for which both the K -theoretic and the L-theoretic Farrell-Jones Conjectures holds. It has the following properties:

Hyperbolic groups belong toFJ;

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

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

(80)

Status of the Farrell-Jones Conjectures

(81)