The projective class group

The role of lower and middle K-theory in topology (Lecture I)

Wolfgang Lück Münster Germany

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

Hangzhou, July 2007

Outline

Introduce theprojective class groupK₀(R).

Discuss its algebraic and topological significance (e.g.,finiteness obstruction).

IntroduceK₁(R)and theWhitehead group Wh(G).

Discuss its algebraic and topological significance (e.g., s-cobordism theorem).

IntroducenegativeK-theoryand theBass-Heller-Swan decomposition.

The projective class group

Definition (ProjectiveR-module)

AnR-moduleP is calledprojectiveif it satisfies one of the following equivalent conditions:

P is a direct summand in a freeR-module;

The following lifting problem has always a solution M ^{p !!}N ^!!0

f P

""

!!

!! ^f

##

If 0→M₀→M₁→M₂→0 is an exact sequence ofR-modules, then 0→hom_R(P,M₀)→hom_R(P,M₁)→hom_R(P,M₂)→0 is

Over a field or, more generally, over a principal ideal domain every projective module is free.

IfRis a principal ideal domain, then a finitely generatedR-module is projective (and hence free) if and only if it is torsionfree.

For instanceZ/nis forn≥2 never projective asZ-module.

LetR andS be rings andR×Sbe their product. ThenR× {0}is a finitely generated projectiveR×S-module which is not free.

Example (Representations of finite groups)

LetF be a field of characteristicp forpa prime number or 0. LetGbe a finite group.

ThenF with the trivialG-action is a projectiveFG-module if and only if p=0 orp does not divide the order ofG. It is a freeFG-module only if Gis trivial.

Definition (Projective class groupK₀(R))

LetRbe an (associative) ring (with unit). Define itsprojective class group

K₀(R)

to be the abelian group whose generators are isomorphism classes[P]

of finitely generated projectiveR-modulesP and whose relations are [P₀] + [P₂] = [P₁]for every exact sequence 0→P₀→P₁→P₂→0 of finitely generated projectiveR-modules.

This is the same as theGrothendieck constructionapplied to the abelian monoid of isomorphism classes of finitely generated projectiveR-modules under direct sum.

Thereduced projective class groupK!₀(R)is the quotient ofK₀(R) by the subgroup generated by the classes of finitely generated freeR-modules, or, equivalently, the cokernel ofK₀(Z)→K₀(R).

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

K!₀(R)measures thedeviationof finitely generated projective R-modules from being stably finitely generated free.

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

Induction

Letf:R→Sbe a ring homomorphism. Given anR-moduleM, let f_∗Mbe theS-moduleS⊗RM. We obtain a homomorphism of abelian groups

f_∗:K₀(R)→K₀(S), [P]'→[f_∗P].

Compatibility with products

The two projections fromR×StoRandSinduce isomorphisms K₀(R×S)−→^∼= K₀(R)×K₀(S).

Morita equivalence

LetR be a ring andMn(R)be the ring of(n,n)-matrices overR.

We can considerRⁿas aM_n(R)-R-bimodule and as a R-M_n(R)-bimodule.

Tensoring with these yields mutually inverse isomorphisms K₀(R) −→^∼= K₀(Mn(R)), [P] '→ [_Mn(R)Rⁿ_R⊗RP];

K₀(Mn(R)) −→^∼= K₀(R), [Q] '→ [_RRⁿ_Mn(R)⊗Mn(R)Q].

Example (Principal ideal domains)

IfRis a principal ideal domain. LetF be its quotient field. Then we obtain mutually inverse isomorphisms

Z −→^∼= K₀(R), n '→ [Rⁿ];

K₀(R) −→^∼= Z, [P] '→ dim_F(F ⊗RP).

Example (Representation ring)

LetGbe a finite group and letF be a field of characteristic zero. Then therepresentation ringR_F(G)is the same asK₀(FG). Taking the character of a representation yields an isomorphism

R_C(G)⊗_ZC=K₀(CG)⊗_ZC−→^∼= class(G,C), whereclass(G;C)is the complex vector space ofclass functions G→C, i.e., functions, which are constant on conjugacy classes.

Example (Dedekind domains)

LetR be a Dedekind domain, for instance the ring of integers in an algebraic number field.

Call two idealsIandJ inRequivalent if there exists non-zero elementsr andsinRwithrI =sJ. Theideal class group C(R)is the abelian group of equivalence classes of ideals under

multiplication of ideals.

Then we obtain an isomorphism

C(R)−→^∼= K!₀(R), [I]'→[I].

The structure of the finite abelian group

C(Z[exp(2πi/p)])∼=K!₀(Z[exp(2πi/p)])∼=K!₀(Z[Z/p]) is only known for small prime numbersp.

Theorem (Swan (1960))

If G is finite, thenK!₀(ZG)is finite.

TopologicalK-theory

LetX be a compact space. LetK⁰(X)be the Grothendieck group of isomorphism classes of finite-dimensional complex vector bundles overX.

This is the zero-th term of a generalized cohomology theory K^∗(X)calledtopologicalK-theory. It is 2-periodic, i.e.,

Kⁿ(X) =Kⁿ⁺²(X), and satisfiesK⁰(pt) =ZandK¹(pt) ={0}. LetC(X)be the ring of continuous functions fromX toC. Theorem (Swan (1962))

There is an isomorphism

K⁰(X)−→^∼= K₀(C(X)).

Wall’s finiteness obstruction

Definition (Finitely dominated)

ACW-complexX is calledfinitely dominatedif there exists a finite (=

compact)CW-complexY together with mapsi:X →Y andr:Y →X satisfyingr ◦i+id_X.

A finiteCW-complex is finitely dominated.

A closed manifold is a finiteCW-complex.

Problem

Is a given finitely dominated CW -complex homotopy equivalent to a finite CW -complex?

Definition (Wall’sfiniteness obstruction)

A finitely dominatedCW-complexX defines an element o(X)∈K₀(Z[π₁(X)])

called itsfiniteness obstructionas follows.

LetX! be the universal covering. The fundamental group π =π₁(X)acts freely onX!.

LetC_∗(X!)be the cellular chain complex. It is a freeZπ-chain complex.

SinceX is finitely dominated, there exists a finite projective Zπ-chain complexP_∗ withP_∗ +Zπ C_∗(X!).

Define

o(X) :="

n

(−1)ⁿ·[P_n]∈K₀(Zπ).

Theorem (Wall (1965))

A finitely dominated CW -complex X is homotopy equivalent to a finite CW -complex if and only if its reduced finiteness obstruction

o(X! )∈K!₀(Z[π₁(X)])vanishes.

A finitely dominated simply connectedCW-complex is always homotopy equivalent to a finiteCW-complex sinceK!₀(Z) ={0}. Given a finitely presented groupGandξ∈K₀(ZG), there exists a finitely dominatedCW-complexX withπ₁(X)∼=Gando(X) =ξ.

Theorem (Geometric characterization ofK!₀(ZG) = {0})

The following statements are equivalent for a finitely presented group G:

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

K!₀(ZG) ={0}.

Conjecture (Vanishing ofK!0(ZG)for torsionfreeG) If G is torsionfree, then

K!₀(ZG) ={0}.

The Whitehead group

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

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].

This is the same asGL(R)/[GL(R),GL(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 thereduced K₁-group

K!₁(R):=K₁(R)/{±1}=cok(K₁(Z)→K₁(R)). IfRis commutative, the determinant induces an epimorphism

det:K₁(R)→R^×, which in general is not bijective.

The assignmentA'→[A]∈K₁(R)can be thought of theuniversal determinant forR.

Definition (Whitehead group)

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

We haveWh({1}) ={0}. Proof.

The ringZpossesses anEuclidean algorithm.

Hence every invertible matrix overZcan be reduced via elementary row and column operations and destabilization to a (1,1)-matrix(±1).

This implies that any element inK₁(Z)is represented by±1.

LetGbe a finite group. Then:

LetF beQ,RorC.

Definer_F(G)to be the number of irreducibleF-representations of G.

This is the same as the number ofF-conjugacy classes of elements ofG.

Hereg₁∼Cg₂if and only ifg₁∼g₂, i.e.,gg₁g⁻¹=g₂for some g ∈G. We haveg₁∼Rg₂if and only ifg₁∼g₂org₁∼g₂⁻¹holds.

We haveg₁∼_Qg₂if and only if,g₁-and,g₁-are conjugated as subgroups ofG.

The Whitehead group Wh(G)is a finitely generated abelian group.

Its rank isr_R(G)−r_Q(G).

The torsion subgroup of Wh(G)is the kernel of the map K₁(ZG)→K₁(QG).

In contrast toK!₀(ZG)the Whitehead group Wh(G)is computable.

Whitehead torsion

Definition (h-cobordism)

Anh-cobordismover a closed manifoldM₀is a compact manifoldW whose boundary is the disjoint unionM₀.M₁such that both inclusions M₀→W andM₁→W are homotopy equivalences.

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

Let M₀be a closed (smooth) manifold of dimension≥5. Let (W;M₀,M₁)be an h-cobordism over M₀.

Then W is homeomorphic (diffeomorpic) to M₀×[0,1]relative M₀if and only if itsWhitehead torsion

τ(W,M₀)∈Wh(π₁(M₀))

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

For n≥5the 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|D₁ⁿ×{1}:∂Dⁿ₂−→^∼= ∂D₁ⁿ× {1}to a homeomorphismg:D₁ⁿ→D₂ⁿ. The three homeomorphismsid_D1ⁿ,f andgfit together to a homeomorphismh:M →D₁ⁿ∪∂D₁ⁿ×{0}∂Dⁿ₁×[0,1]∪∂D₁ⁿ×{1}D₁ⁿ. The target is obviously homeomorphic toSⁿ.

The argument above does not imply that for a smooth manifoldM we obtain a diffeomorphismg:M →Sⁿ.

The Alexander trick does not work smoothly.

Indeed, there exists 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.

Given a finitely presented groupG, an elementξ∈Wh(G)and a closed manifoldMof dimensionn≥5 withG∼=π₁(M), there exists anh-cobordismW overMwithτ(W,M) =ξ.

Theorem (Geometric characterization of Wh(G) ={0})

The following statements are equivalent for a finitely presented group G and a fixed integer n ≥6

Every compact n-dimensional h-cobordism W with G∼=π₁(W)is trivial;

Wh(G) ={0}.

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

Wh(G) ={0}.

Negative K -theory

Definition (Bass-Nil-groups) Define forn=0,1

NKn(R):=coker(Kn(R)→Kn(R[t])).

Theorem (Bass-Heller-Swan decomposition forK1 (1964)) There is an isomorphism, natural in R,

K₀(R)⊕K₁(R)⊕NK₁(R)⊕NK₁(R)−→^∼= K₁(R[t,t⁻¹]) =K₁(R[Z]).

Definition (NegativeK-theory) Define inductively forn=−1,−2, . . .

K_n(R):=coker#

K_n+1(R[t])⊕K_n+1(R[t⁻¹])→K_n+1(R[t,t⁻¹])$ .

Define forn=−1,−2, . . .

NK_n(R):=coker(K_n(R)→K_n(R[t])).

Theorem (Bass-Heller-Swan decomposition for negative K-theory)

For n≤1there is an isomorphism, natural in R,

K_n−1(R)⊕K_n(R)⊕NK_n(R)⊕NK_n(R)−→^∼= K_n(R[t,t⁻¹]) =K_n(R[Z]).

Definition (Regular ring)

A ringRis calledregularif it is Noetherian and every finitely generated R-module possesses a finite projective resolution.

Principal ideal domains are regular. In particularZand any field are regular.

IfRis regular, thenR[t]andR[t,t⁻¹] =R[Z]are regular.

IfRis regular, thenRGin general is not Noetherian or regular.

Theorem (Bass-Heller-Swan decomposition for regular rings) Suppose that R is regular. Then

Kn(R) = 0 for n≤ −1;

NK_n(R) = 0 for n≤1,

and the Bass-Heller-Swan decomposition reduces for n ≤1to the natural isomorphism

K_n−1(R)⊕K_n(R)−→^∼= K_n(R[t,t⁻¹]) =K_n(R[Z]).

There are alsohigher algebraicK-groupsK_n(R)forn≥2 due to Quillen (1973).

They are defined as homotopy groups of certain spaces or spectra. We refer to the lectures ofGrayson.

Most of the well known features ofK₀(R)andK₁(R)extend to both negative and higher algebraicK-theory.

For instance the Bass-Heller-Swan decomposition holds also for higher algebraicK-theory.

Notice the following formulas for a regular ringRand a generalized homology theoryH∗, which look similar:

K_n(R[Z]) ∼= K_n(R)⊕K_n−1(R);

Hn(BZ) ∼= Hn(pt)⊕ Hn−1(pt).

IfGandK are groups, then we have the following formulas, which look similar:

K!_n(Z[G∗K]) ∼= K!_n(ZG)⊕K!_n(ZK);

H!n(B(G∗K)) ∼= H!n(BG)⊕H!n(BK).

Question (K-theory of group rings and group homology) Is there a relation between K_n(RG)and group homology of G?

To be continued

Stay tuned

The Isomorphism Conjectures in the torsionfree case (Lecture II)

Wolfgang Lück Münster Germany

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

Hangzhou, July 2007

Flashback

We have introducedK_n(R)forn∈Z,n≤1.

We have discussed the topological relevance ofK₀(RG)and the Whitehead group Wh(G), e.g.,the finiteness obstructionand the s-cobordism theorem.

We have stated the conjectures thatK!₀(ZG)and Wh(G)vanish for torsionfreeG.

We have presented theBass-Heller-Swan decompositionand indicated some similarities betweenK_n(RG)andgroup homology.

Cliffhanger

Question (K-theory of group rings and group homology) Is there a relation between K_n(RG)and the group homology of G?

Outline

We introducespectraand how they yieldhomology theories.

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

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

We explain that the formulations for torsionfree groups cannot extend to arbitrary groups.

Homology theories and spectra

Definition (Spectrum) Aspectrum

E={(E(n), σ(n))|n∈Z}

is a sequence of pointed spaces{E(n)|n∈Z}together with pointed maps calledstructure maps

σ(n) :E(n)∧S¹−→E(n+1).

Amap of spectra

f:E→E^!

is a sequence of mapsf(n) :E(n)→E^!(n)which are compatible with the structure mapsσ(n), i.e.,f(n+1)◦σ(n) = σ^!(n)◦(f(n)∧id_S1) holds for alln∈Z.

Given two pointed spacesX = (X,x₀)andY = (Y,y₀), their one-point-unionand theirsmash productare defined to be the pointed spaces

X∨Y := {(x,y₀)|x ∈X} ∪ {(x₀,y)|y ∈Y} ⊆X×Y; X∧Y := (X×Y)/(X∨Y).

We haveSⁿ⁺¹∼=Sⁿ∧S¹.

Thesphere spectrumShas asn-th spaceSⁿand asn-th structure map the homeomorphismSⁿ∧S¹−→^∼= Sⁿ⁺¹.

LetX be a pointed space. Itssuspension spectrumΣ^∞X is given by the sequence of spaces{X ∧Sⁿ|n≥0}with the

homeomorphism(X ∧Sⁿ)∧S¹∼=X ∧Sⁿ⁺¹as structure maps.

We haveS= Σ^∞S⁰.

Definition (Ω-spectrum)

Given a spectrumE, we can consider instead of the structure map σ(n) :E(n)∧S¹→E(n+1)its adjoint

σ^!(n) :E(n)→ΩE(n+1) =map(S¹,E(n+1)).

We callEanΩ-spectrumif each mapσ^!(n)is a weak homotopy equivalence.

Definition (Homotopy groups of a spectrum)

Given a spectrumE, define forn∈Zitsn-th homotopy group π_n(E):=colim

k→∞ π_k+n(E(k))

to be the abelian group which is given by the colimit over the directed system indexed byZwithk-th structure map

π_k+n(E(k))−−−→^σ!(k) π_k+n(ΩE(k+1)) =π_k+n+1(E(k+1)).

Notice that a spectrum can have in contrast to a space non-trivial negative homotopy groups.

IfEis anΩ-spectrum, thenπ_n(E) =π_n(E(0))for alln≥0.

Eilenberg-MacLane spectrum

LetAbe an abelian group. Then-thEilenberg-MacLane space EM(A,n)associated toAforn≥0 is aCW-complex with π_m(EM(A,n)) =Aform=nandπ_m(EM(A,n)) ={0}form-=n.

The associatedEilenberg-MacLane spectrumH(A)has asn-th spaceEM(A,n)and asn-th structure map a homotopy

equivalenceEM(A,n)→ΩEM(A,n+1).

AlgebraicK-theory spectrum

For a ringR there is thealgebraicK-theory spectrumKRwith the property

π_n(KR) =K_n(R) forn∈Z.

AlgebraicL-theory spectrum

For a ring with involutionRthere is thealgebraicL-theory spectrumL^%−∞'_R with the property

π_n(L^%−∞'_R ) =L^%−∞'_n (R) forn∈Z.

TopologicalK-theory spectrum

ByBott periodicitythere is a homotopy equivalence β:BU×Z−→⁽ Ω²(BU×Z).

ThetopologicalK-theory spectrumK^tophas in even degrees BU×Zand in odd degreesΩ(BU×Z).

The structure maps are given in even degrees by the mapβ and in odd degrees by the identity id: Ω(BU×Z)→Ω(BU×Z).

Definition (Homology theory)

LetΛbe a commutative ring, for instanceZorQ.

Ahomology theoryH∗with values inΛ-modules is a covariant functor from the category ofCW-pairs to the category ofZ-gradedΛ-modules together with natural transformations

∂_n(X,A) :Hn(X,A)→ Hn−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 Hn(X,A)−→ H^∼= n(X ∪f B,B).

Definition (continued) Disjoint union axiom

"

i∈I

Hn(X_i)−→ H^∼= n

#$

i∈I

X_i

% .

Definition (Smash product)

LetEbe a spectrum andX be a pointed space. Define thesmash productX ∧Eto be the spectrum whosen-th space isX∧E(n)and whosen-th structure map is

X ∧E(n)∧S^{1 id}−−−−−−→^X∧σ(n) X ∧E(n+1).

Theorem (Homology theories and spectra)

LetEbe a spectrum. Then we obtain a homology theoryH_∗(−;E)by Hn(X,A;E) :=π_n((X ∪Acone(A))∧E).

It satisfies

Hn(pt;E) =π_n(E).

Example (Stable homotopy theory)

The homology theory associated to the sphere spectrumSisstable homotopyπ^s_∗(X). The groupsπ^s_n(pt)are finite abelian groups forn-=0 by a result ofSerre (1953). Their structure is only known for smalln.

Example (Singular homology theory with coefficients)

The homology theory associated to the Eilenberg-MacLane spectrum H(A)issingular homology with coefficients inA.

Example (TopologicalK-homology)

The homology theory associated to the topologicalK-theory spectrum K^topisK-homologyK_∗(X). We have

K_n(pt)∼=

&

Z neven;

{0} nodd.

The Isomorphism Conjectures for torsionfree groups

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

H_n(BG;KR)→K_n(RG) is bijective for all n∈Z.

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

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

Hn(pt;KR)∼=π_n(KR)∼=Kn(R)forn∈Z.

BGis theclassifying spaceof the groupG, i.e., the base space of the universalG-principalG-bundleG→EG→BG. Equivalently, BG=EM(G,1). The spaceBGis unique up to homotopy.

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^%−∞'_R )→L^%−∞'_n (RG) is bijective for all n∈Z.

L^%−∞'_n (RG)is the algebraicL-theory ofRGwith decoration.−∞0; L^%−∞'_R is the algebraicL-theory spectrum ofRwith decoration .−∞0;

H_n(pt;L^%−∞'_R )∼=π_n(L^%−∞'_R )∼=L^%−∞'_n (R)forn∈Z.

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.

K_n(BG)is the topologicalK-homology ofBG, where

K_∗(−) =H_∗(−;K^top)forK^topthe topologicalK-theory spectrum.

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)).

Consequences of the Isomorphism Conjectures for torsionfree groups

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

LetFJK(R)andFJL(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.

Lemma

Let R be a regular ring. Suppose that G is torsionfree and G∈ FJK(R). Then

K_n(RG) =0for n≤ −1;

The change of rings map K₀(R)→K₀(RG)is bijective. In particularK!₀(RG)is trivial if and only ifK!₀(R)is trivial.

Lemma

Suppose that G is torsionfree and G ∈ FJK(Z).

Then the Whitehead groupWh(G)is trivial.

Proof.

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

E_p,q² =H_p(BG,K_q(R)).

Proof (continued).

SinceRis regular by assumption, we getK_q(R) =0 forq ≤ −1.

Hence the edge homomorphism yields an isomorphism K₀(R) =H₀(pt,K₀(R))−→^∼= H₀(BG;KR)∼=K₀(RG).

We haveK₀(Z) =ZandK₁(Z) ={±1}. We get an exact sequence

0→H₀(BG;K₁(Z)) ={±1} →H₁(BG;K_Z)∼=K₁(ZG)

→H₁(BG;K₀(Z)) =G/[G,G]→0.

This implies Wh(G) :=K₁(ZG)/{±g|g ∈G} ∼=0.

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

K_n(ZG) =0 forn≤ −1;

K!₀(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;

IfGbelongs toFJK(Z), then it is of type FF if and only if it is of type FP (Serre’sproblem).

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 skew-field and let G be a group with G ∈ FJK(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.

Then0and1are the only idempotents in FG.

Proof.

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

Denote by%:FG→F the augmentation homomorphism sending '

g∈Grg·g to'

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∈ FJK(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.

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.

Theorem (The Baum-Connes Conjecture and the Kaplansky Conjecture)

Let G be a torsionfree group with G∈ BC. Then0and1are the only idempotents inCG.

Proof.

There is a trace map

tr:C_r^∗(G)→C

which sendsf ∈C_r^∗(G)⊆ B(l²(G))to.f(e),e0l²(G).

TheL²-index theoremdue toAtiyah (1976)shows that the composite

K₀(BG)→K₀(C_r^∗(G))−→^tr C coincides with

K₀(BG)−−−−→^K0(pr) K₀(pt) =Z−→ⁱ C.

Proof (continued).

HenceG∈ BCimplies tr(p)∈Z.

Since tr(1) =1, tr(0) =0, 0≤p≤1 andp²=p, we get tr(p)∈R and 0≤tr(p)≤1.

We conclude tr(0) =tr(p)or tr(1) =tr(p).

This implies alreadyp=0 orp=1.

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)).

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.

Definition (Structure set)

Thestructure setS^top(M)of a manifoldM consists of equivalence classes of orientation preserving homotopy equivalencesN→M with a manifoldN as source.

Two such homotopy equivalencesf₀:N₀→Mandf₁:N₁→Mare equivalent if there exists a homeomorphismg:N₀→N₁with f₁◦g3f₀.

Theorem

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

Theorem (Ranicki (1992))

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

. . .−−−→^σn+1 H_n+1(M;L.10)−−−→^An+1 L_n+1(Zπ₁(M))−−−→^∂n+1

S^top(M)−→^σn H_n(M;L.10)−→^An L_n(Zπ₁(M))−→^∂n . . . It can be identified with the classical geometric surgery sequence due toSullivan and Wallin high dimensions.

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

H_k(M;L.10)→H_k(M;L)is bijective fork ≥n+1 and injective for k =n.

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.

IfGis torsionfree, then the version of theK-theoretic Farrell-Jones Conjecture predicts

Hn(BZ;KR) =Hn(S¹;KR) =Hn(pt;KR)⊕H_n−1(pt;KR)

=Kn(R)⊕K_n−1(R)∼=Kn(RZ).

In view of the Bass-Heller-Swan decomposition this is only possible if NKn(R)vanishes which is true for regular ringsRbut not for general ringsR.

We want to figure out what is needed for a general version which may be true for all groups.

Assembly

For a fieldF of characteristic zero and some groupsGone knows that there is an isomorphism

colim

|H|<∞H⊆G

K₀(FH)−→^∼= K₀(FG).

This indicates that one has at least to take into account the values for all finite subgroups to assembleKn(FG).

Degree Mixing

The Bass-Heller-Swan decomposition shows that theK-theory of finite subgroups in degreem≤ncan affect theK-theory in degreenand that at least in the Farrell-Jones setting finite subgroups are not enough.

In the Baum-Connes setting Nil-phenomena do not appear.

Namely, a special case of a result due toPimsner-Voiculescu (1982)says

Kn(C_r^∗(G×Z))∼=Kn(C_r^∗(G))⊕K_n−1(C_r^∗(G)).

Homological behaviour

There is still a lot of homological behaviour known forK_∗(C_r^∗(G)).

For instance there exists a long exact Mayer-Vietoris sequence associated to amalgamated productsG₁∗G0G₂and a

Wang-sequence associated to semi-direct productsG! Zby Pimsner-Voiculescu (1982).

Similar versions under certain restrictions exist inK-andL-theory due toCappell (1974)andWaldhausen (1978)if one makes certain assumptions onRor ignores certain Nil-phenomena.

Question (Classifying spaces for families)

Is there a versionE_F(G)of the classifying space EG which takes the structure of the family of finite subgroups or other familiesF of subgroups into account and can be used for a general formulation of the Farrell-Jones Conjecture?

Question (Equivariant homology theories)

Can one define appropriate G-homology theoriesH^G_∗ that are in some sense computable and yield when applied to E_F(G)a term which potentially is isomorphic to the groups Kn(RG), L^−%∞'(RG)or K_n(C_r^∗(G))?

In the torsionfree case they should reduce to H_n(BG;KR), Hn(BG;L^−%∞')and Kn(BG).

To be continued

Stay tuned

Classifying spaces for families (Lecture III)

Wolfgang Lück Münster Germany

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

Hangzhou, July 2007

Flashback

We have introduced theFarrell-Jones Conjectureand the Baum-Connes Conjecturefor torsionfree groups:

H_n(BG;KR) −→^∼= K_n(RG);

Hn(BG;L^"−∞%_R ) −→^∼= L^"−∞%_n (RG);

Kn(BG) −→^∼= Kn(C_r^∗(G)).

We have discussed applications of these conjectures such as to theKaplansky Conjectureand theBorel Conjecture.

Cliffhanger

Question (Classifying spaces for families)

Is there a versionE_F(G)of the classifying space EG which takes the structure of the family of finite subgroups or other familiesF of subgroups into account and can be used for a general formulation of the Farrell-Jones Conjecture?

Outline

We introduce the notion of theclassifying space of a familyF of subgroupsE_F(G)andJ_F(G).

In the case, whereF is the familyCOMof compact subgroups, we present some nice geometric models forE_F(G)and explain E_F(G)#J_F(G).

We discussfiniteness propertiesof these classifying spaces.

Classifying spaces for families of subgroups

Definition (G-CW-complex)

AG-CW -complexX is aG-space together with aG-invariant filtration

∅=X₋₁⊆X₀⊆. . .⊆X_n⊆. . .⊆ !

n≥0

X_n=X

such thatX carries thecolimit topologywith respect to this filtration, andX_nis obtained fromX_n−1for eachn≥0 byattaching equivariant n-dimensional cells, i.e., there exists aG-pushout

"

i∈InG/H_i×Sⁿ⁻¹

‘i∈Inq_iⁿ

!!

""

X_n−1

""

"

i∈InG/H_i×Dⁿ

‘i∈InQⁿ_i

!!Xn

Group meanslocally compact Hausdorff topological group with a countable basis for its topology, unless explicitly stated differently.

Example (Simplicial actions)

LetX be a simplicial complex. Suppose thatGacts simplicially onX. ThenGacts simplicially also on thebarycentric subdivisionX⁾, and all isotropy groups are open and closed. TheG-spaceX⁾ inherits the structure of aG-CW-complex.

Example (Smooth actions)

LetGbe a Lie group acting properly and smoothly on a smooth manifoldM.

ThenM inherits the structure ofG-CW-complex.

Definition (ProperG-action)

AG-spaceX is calledproperif for each pair of pointsx andy inX there are open neighborhoodsVx ofx andWy ofy inX such that the closure of the subset{g ∈G|gV_x∩W_y *=∅}ofGis compact.

Lemma

A proper G-space has always compact isotropy groups.

A G-CW -complex X is proper if and only if all its isotropy groups are compact.

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};

FIN = {finite subgroups};

VCYC = {virtually cyclic subgroups};

COM = {compact subgroups};

COMOP = {compact open subgroups};

ALL = {all subgroups}.

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 ofE_F(G)belong toF;

For anyG-CW-complexY, whose isotropy groups belong toF, there is up toG-homotopy precisely oneG-mapY →E_F(G).

We abbreviateEG:=E_COM(G)and call it theuniversal G-CW -complex for proper G-actions.

We also writeEG=E_{T R}(G).

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

There exists a model for E_F(G)for any familyF; Two model for E_F(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.

A model forE_ALL(G)isG/G;

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

Example (Infinite dihedral group)

LetD_∞=Z ! Z/2=Z/2∗Z/2 be the infinite dihedral group.

A model forED_∞is the universal covering ofRP^∞∨RP^∞. A model forED_∞isRwith the obviousD_∞-action.

Lemma

If G is totally disconnected, then E_COMOP(G) =EG.

Definition (F-numerableG-space)

AF-numerable G-spaceis aG-space, for which there exists an open covering{U_i |i ∈I}byG-subspaces satisfying:

For eachi∈Ithere exists aG-mapU_i →G/G_i for someG_i ∈ F; There is a locally finite partition of unity{e_i |i∈I}subordinate to {U_i |i ∈I}byG-invariant functionse_i:X →[0,1].

Notice that we do not demand that the isotropy groups of a F-numerableG-space belong toF.

Iff:X →Y is aG-map andY isF-numerable, thenX is also F-numerable.

AG-CW-complex isF-numerable if and only if each isotropy group appears as a subgroup of an element inF.

There is also a versionJ_F(G)of a classifying space for F-numerableG-spaces.

It is characterized by the property thatJ_F(G)isF-numerable and for everyF-numerableG-spaceY there is up toG-homotopy precisely oneG-mapY →J_F(G).

We abbreviateJG=J_COM(G)andJG=J_{T R}(G).

JG→G\JGis theuniversalG-principal bundlefor numerable free properG-spaces.

Theorem (Comparison ofE_F(G)andJ_F(G),L. (2005)) There is up to G-homotopy precisely one G-map

φ:E_F(G)→J_F(G);

It is a G-homotopy equivalence if one of the following conditions is satisfied:

Each element inFis open and closed;

G is discrete;

FisCOM;

Let G be totally disconnected. Then EG→JG is a G-homotopy equivalence if and only if G is discrete.

Special models for EG

We want to illustrate that the spaceEG=JGoften hasvery nice geometric modelsandappear naturally in many interesting situations.

LetC₀(G)be the Banach space of complex valued functions ofG vanishing at infinity with the supremum-norm. The groupGacts isometrically onC₀(G)by(g·f)(x) :=f(g⁻¹x)forf ∈C₀(G)and g,x ∈G.

LetPC₀(G)be the subspace ofC₀(G)consisting of functionsf such thatf is not identically zero and has non-negative real numbers as values.

Theorem (Operator theoretic model,Abels (1978)) The G-space PC₀(G)is a model for JG.

Theorem

Let G be discrete. A model for JG is the space X_G =

#

f:G→[0,1]

$$

$$f has finite support, %

g∈G

f(g) =1

&

with the topology coming from the supremum norm.

Theorem (Simplicial Model)

Let G be discrete. Let P_∞(G)be the geometric realization of the simplicial set whose k-simplices consist of(k+1)-tupels

(g₀,g₁, . . . ,g_k)of elements g_iin G. This is a model for EG.

The spacesX_G andP_∞(G)have the same underlying sets but in general they have different topologies.

The identity map induces aG-mapP_∞(G)→X_Gwhich is a

G-homotopy equivalence, but in general not aG-homeomorphism.

Theorem (Almost connected groups,Abels (1978).)

Suppose that G isalmost connected, i.e., the group G/G⁰is compact for G⁰the component of the identity element.

Then G contains a maximal compact subgroup K which is unique up to conjugation, and the G-space G/K is a model for JG.

As a special case we get:

Theorem (Discrete subgroups of almost connected Lie groups) Let L be a Lie group with finitely many path components.

Then L contains a maximal compact subgroup K which is unique up to conjugation, and the L-space L/K is a model for EL.

If G ⊆L is a discrete subgroup of L, then L/K with the obvious left G-action is a finite dimensional G-CW -model for EG.

Theorem (Actions on CAT(0)-spaces)

Let G be a (locally compact Hausdorff) topological group. Let X be a proper G-CW -complex. Suppose that X has the structure of a complete simply connectedCAT(0)-space for which G acts by isometries.

Then X is a model for EG.

The result above contains as special caseisometricGactions on simply-connected complete Riemannian manifolds with

non-positive sectional curvatureandG-actions on trees.