The Isomorphism Conjectures in the torsion free case (Lecture III)

The Isomorphism Conjectures in the torsion free case (Lecture III)

Wolfgang Lück Bonn Germany

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

Bonn, August 2013

Flashback

We introducedK_n(R)forn∈Z.

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

We stated the conjectures thatKe₀(ZG)and Wh(G)vanish for torsion freeG.

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

Cliffhanger

Question (K-theory of group rings and group homology)

Is there a relationship 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 torsion free groups.

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

We explain that the formulations for torsion free 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).

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).

IfX is a pointed space andEis a spectrum, then we obtain a new spectrum byX∧E.

Exercise

Show Sⁿ⁺¹∼=Sⁿ∧S¹.

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 byZ

· · ·−−−−−→^σ(k−1)∗ π_k+n(E(k))−−−→^σ(k)∗ π_k+n+1(E(k +1))−−−−−→ · · ·^σ(k+1)∗ . Notice that a spectrum, in contrast to a space, can have non-trivial negative homotopy groups.

AlgebraicK-theory spectrum

For a ringR, there is thealgebraicK-theory spectrumK_Rwith the property

πn(K_R) =Kn(R) forn∈Z.

AlgebraicL-theory spectrum

For a ring with involutionR, there is thealgebraicL-theory spectrumL^h−∞i_R with the property

πn(L^h−∞i_R ) =L^h−∞i_n (R) forn∈Z.

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;

Disjoint union axiom.

Theorem (Homology theories and spectra) LetEbe a spectrum.

Then we obtain a homology theoryH∗(−;E)by

H_n(X,A;E) :=πn((X ∪_Acone(A))∧E). It satisfies

Hn(pt;E) =πn(E).

The Isomorphism Conjectures for torsion free groups

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

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

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

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

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

Hn(pt;K_R)∼=πn(K_R)∼=Kn(R)forn∈Z. BGis theclassifying spaceof the groupG.

Conjecture (L-theoretic Farrell-Jones Conjecture for torsion free groups)

TheL-theoretic Farrell-Jones Conjecturewith coefficients in the ring with involution R for the torsion free group G predicts that the assembly map

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

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

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

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

K_n(BG)is the topologicalK-homology ofBG.

Kn(C_r^∗(G))is the topologicalK-theory of the reduced complex groupC^∗-algebraC_r^∗(G)ofG.

Exercise

Let G be the fundamental group of a closed orientable2-manifold.

Compute Kn(BG).

Conclusions of the Isomorphism Conjectures for torsion free 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.

LetFJ_K(R), respectivelyFJ_L(R), be the class of groups that satisfy theK-theoretic, respectivelyL-theoretic, Farrell-Jones Conjecture for the coefficient ringR.

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

Lemma

Let R be a regular ring. Suppose that G is torsion free 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 torsion free 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)).

In particular, for a torsion free groupG∈ FJ_K(Z)we get:

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;

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

Mini-Break

Mathematicians!

Conjecture (Kaplansky Conjecture)

TheKaplansky Conjecturesays that for a torsion free group G and an integral domain R the elements0and1are the only idempotents in RG.

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

Let F be a skew-field and let G be a group with G ∈ FJ_K(F). Suppose that one of the following conditions is satisfied:

F is commutative and has characteristic zero, and G is torsion free;

G is torsion free and sofic;

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.

We only treat the case of fields of characteristic zero.

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

Denote by:FG→F the augmentation homomorphism sending P

g∈Gr_g·g toP

g∈Gr_g. It suffices to showp=0 under the assumption that(p) =0.

Let(p)⊆FGbe the ideal generated byp, which 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.

Proof (continued).

If we now applyi∗◦∗and use◦i =id,i∗◦∗(FG^l)∼=FG^l and (p) =0, then 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.

Exercise

Let p be a prime. Find all idempotents in R[Z/p]for R=Z, R =Cand R =Fp.

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 and in particular that M and N are homeomorphic.

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

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.

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

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.

Exercise

Prove the Borel Conjecture in dimensions1and2.

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◦g'f₀.

Theorem

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

Theorem (Ranicki)

There is an exact sequence of abelian groups, calledthe algebraic surgery exact sequence, for 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 sequence due toSullivan and Wallin high dimensions.

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

H_k(M;Lh1i)→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 are false for finite groups unless the group is trivial.

For instance the version of the Baum-Connes Conjecture above would predict that 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 torsion free, then the version of theK-theoretic Farrell-Jones Conjecture predicts

Hn(BZ;K_R) =Hn(S¹;K_R) =Hn(pt;K_R)⊕Hn−1(pt;K_R)

=Kn(R)⊕Kn−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.

Cliffhanger

Question (Arbitrary groups and rings)

Are there versions of the Farrell-Jones Conjecture for arbitrary groups and rings and of the Baum-Connes Conjecture for arbitrary groups?

To be continued Stay tuned

Next talk: Friday 9:15