WolfgangLückBonnGermanyemailwolfgang.lueck@him.uni-bonn.dehttp://131.220.77.52/lueck/Berlin,August2013 K andWhiteheadtorsion(LectureII)

K

and Whitehead torsion (Lecture II)

Wolfgang Lück Bonn Germany

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

Berlin, August 2013

Outline

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

We define the Whitehead torsion of a homotopy equivalence of finite connectedCW-complexes

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

IntroducenegativeK-theoryand theBass-Heller-Swan decomposition.

The Whitehead group

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].

LetGLn(R)be the group of invertible(n,n)-matrices.

We get a sequence of inclusions

R^×=GL₁(R)⊆GL₂(R)⊆ · · · ⊆GL_n(R)⊆GL_n+1(R)⊆ · · · by sending an(n,n)-matrixAto the(n+1,n+1)-matrix

A 0 0 1

. LetGL(R) :=S

n≥1GL_n(R).

The obvious mapsGLn(R)→K₁(R)induce an epimorphism GL(R)→K₁(R)).

It induces 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)).

Exercise

Show for a commutative ring R that the determinant induces an epimorphism

det:K₁(R)→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}.

Lemma

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

In contrast toKe₀(ZG)the Whitehead group Wh(G)is computable for finite groupsG.

Exercise

Show that t−1−t⁻¹∈Z[Z/5]for t ∈Z/5the generator is a unit and hence defines an elementη inWh(Z/5). Prove that we obtain a well-defined map

Wh(Z/5)→R

by sending the class represented by theZ[Z/5]-automorphism

f:Z[Z/5]ⁿ→Z[Z/5]ⁿtoln(|det(f)|), where f:Cⁿ→Cⁿis theC-linear map f ⊗_Z[Z/5]id_C:Z[Z/5]ⁿ⊗_Z[Z/5]C→Z[Z/5]ⁿ⊗_Z[Z/5]Cwith respect to theZ/5-action onCgiven by multiplication withexp(2πi/5). Finally show thatηgenerates an infinite cyclic subgroup inWh(Z/5).

Whitehead torsion

LetC_∗ be a contractible finite based freeR-chain complex.

Choose a chain contractionγ∗.

Then(c∗+γ∗)_odd:C_odd→C_evis an isomorphism of finitely generated based freeR-modules and hence defines an element calledReidemeister torsion

ρ(C_∗):= [(c_∗+γ∗)_odd] ∈K₁(R).

Next we show that it is independent of the choice ofγ∗.

Letδ∗be another chain contraction.

DefineR-homomorphisms

(e∗+γ∗)_odd:E_odd → E_ev; (e∗+δ∗)_ev:E_ev → E_odd. Put

µn := (γ_n+1−δ_n+1)◦δn; νn := (δn+1−γn+1)◦γn.

One easily checks that

(id+µ∗)_odd, (id+ν∗)ev

and both compositions

(c∗+γ∗)_odd◦(id+µ∗)_odd◦(c∗+δ∗)ev

(c∗+δ∗)ev◦(id+ν∗)ev◦(c∗+γ∗)_odd

are given by upper triangular matrices whose diagonal entries are identity maps.

In particular they represent zero inK₁(R).

This implies[(c∗+γ∗)odd] =−[(c∗+δ∗)ev]inK₁(R)and hence that [(c∗+γ∗)_odd]∈K₁(R)does not depend on the choice ofγ.

Consider aR-chain homotopy equivalencef∗:C∗ →D∗of finite based freeR-chain complexes.

Then its mapping cone cone(f∗)is a contractible finite based free R-chain complexes.

Define theWhitehead torsionoff∗ to be

τ(f∗):=ρ(cone(f∗)) ∈K₁(R) Exercise

Let r ∈Qbe a rational number. Show that the following finite based freeQ-chain complex concentrated in dimensions2,1and0is contractible and compute its Reidemeister torsion

Q





0 r





−−−→Q⊕Q

1 0

−−−−−→Q

Definition (Whitehead torsion of maps)

Given a homotopy equivalence of connected finiteCW-complexes f:X →Y, define itsWhitehead torsion

τ(f)∈Wh(π₁(Y)) as follows.

Letef:Xe →Ye be a lift off to the universal coverings.

It is aπ₁(Y)-equivariant homotopy equivalence and hence induces aZ[π1(Y)]-chain homotopy equivalenceC∗(ef) :C∗(Xe)→C∗(Ye).

TheCW-structures induceZ[π₁(Y)]-basis onC∗(Xe)andC∗(Ye) which are unique up to multiplying a basis element with some element±w forw ∈π1(Y)and up to permutation of the basis elements.

Defineτ(f)to be the Whitehead torsion ofC∗(ef)considered in Wh(π₁(Y)).

Definition (simple homotopy equivalence)

A homotopy equivalence of connected finiteCW-complex is called simpleif its Whitehead torsion is trivial.

A homotopy equivalence is a simple homotopy equivalence if and only if it is homotopic to a composition of so calledexpansions andcollapses.

Any element in Wh(π₁(Y))can be realized asτ(f)for some homotopy equivalencef:X →Y.

Since there exist connected finiteCW-complexesY with

Wh(π₁(Y))6=0, there exists homotopy equivalences of connected finiteCW-complexes which are not simple.

Mini-Break

Carrying out mathematics

Definition (h-cobordism)

Anh-cobordismover a closed manifoldM₀is a compact manifoldW whose boundary is the disjoint unionM₀qM₁such 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 (diffeomorphic) to M₀×[0,1]relative M₀if and only if itsWhitehead torsion

τ(W,M₀)∈Wh(π₁(M₀)) vanishes.

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 disksDⁿ₀andD₁ⁿ. ThenW is a simply connected h-cobordism.

Since Wh({1})is trivial, we can find a homeomorphism f:W −→^∼= ∂Dⁿ₀×[0,1]that is the identity on∂D₀ⁿ=∂D₀ⁿ× {0}.

By theAlexander trickwe can extend the homeomorphism f|_∂Dⁿ

1:∂D₁ⁿ−→^∼= ∂D₀ⁿto a homeomorphismg:Dⁿ₁→D₀ⁿ. The three homeomorphismsid_Dⁿ

0,f andgfit together to a homeomorphismh:M →D₀ⁿ∪_∂Dⁿ

0×{0}∂Dⁿ₀×[0,1]∪_∂Dⁿ

0×{1}D₀ⁿ. The target is obviously homeomorphic toSⁿ.

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. 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.

Thes-Cobordism Theorem is one step in thesurgery program due toBrowder, Novikov, SullivanandWallto decide whether two closed manifoldsM andN are diffeomorphic what is in general a very hard question. It consists of the following steps.

1 Construct a simple homotopy equivalencef:M →N;

2 Construct a cobordism(W;M,N)and a map

(F,f,id) : (W;M,N)→(N×[0,1],N× {0},N× {1});

3 ModifyW andF relative boundary bysurgerysuch thatF becomes a simple homotopy equivalence and thusW becomes anh-cobordism whose Whitehead torsion is trivial.

4 Apply thes-Cobordism Theorem.

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 torsion freeG) If G is torsion free, then

Wh(G) ={0}.

Negative and higher K -theory

There existK-groupsKn(R)for everyn∈Z. The negative

K-groups were introduced byBass, the higher algebraicK-groups byQuillen.

Theorem (Bass-Heller-Swan decomposition) For n∈Zthere 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.

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

K_n(R) = 0 for n≤ −1;

NK_n(R) = 0 for n∈Z;

The Bass-Heller-Swan decomposition reduces for n∈Zto the natural isomorphism

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

Example (Eilenberg swindle)

Consider a ringR. LetP(R)be the additive category of finitely generated projectiveR-modules.

Suppose that there exists a functorS:P(R)→ P(R)of additive categories together with a natural equivalenceS⊕id_P(R)−→^∼= S.

ThenK_n(R) =0 forn∈Zsince

Kn(S) +id_Kn(R)=Kn(S⊕id_P(R)) =Kn(S)holds.

Exercise

Let R be a ring. Consider the ring E of R-endomorphisms ofL

i∈NR.

Show that K_n(E) =0for n∈Z.

Notice the similarity between following formulas for a regular ring Rand a generalized homology theoryH_∗:

K_n(R[Z]) ∼= K_n(R)⊕K_n−1(R);

H_n(BZ) ∼= H_n(pt)⊕ H_n−1(pt).

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

Ken(Z[G∗K]) ∼= Ken(ZG)⊕Ken(ZK);

He_n(B(G∗K)) ∼= He_n(BG)⊕He_n(BK).

Cliffhanger

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

Is there a relationship between Kn(RG)and the group homology of G?

To be continued Stay tuned

Next talk: Thursday 14:30