Middle algebraic K-theory (Lecture I)

Middle algebraic K-theory (Lecture I)

Wolfgang Lück Bonn Germany

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

Berlin, June 18, 2012

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 (Projective class groupK0(R))

Define theprojective class groupof an (associative) ringR(with unit) K₀(R)

to be the following abelian group:

Generators are isomorphism classes[P]of finitely generated projectiveR-modulesP;

The relations are[P₀] + [P₂] = [P₁]for every exact sequence 0→P₀→P₁→P₂→0 of finitely generated projective R-modules.

Exercise

Show that K₀(R)is the same as theGrothendieck constructionapplied to the abelian monoid of isomorphism classes of finitely generated projective R-modules under direct sum.

A ring homomorphismf:R→Sinduces a homomorphism of abelian groups

f_∗:K₀(R)→K₀(S), [P]7→[f_∗P].

The assignmentP 7→[P]∈K₀(R)is theuniversal additive invariantordimension functionfor finitely generated projective R-modules.

Thereduced projective class groupKe₀(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 Ke₀(R).

Ke₀(R)measures thedeviationof finitely generated projective R-modules from being stably finitely generated free.

Compatibility with products

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

Morita equivalence

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

Then there is a natural isomorphism

K₀(R)−^∼=→K₀(Mn(R)).

Example (Principal ideal domains)

IfRis a principal ideal domain andF is its quotient field, then we obtain mutually inverse isomorphisms

Z

∼=

−→ K₀(R), n 7→ [Rⁿ];

K₀(R) −→^∼= Z, [P] 7→ 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).

K₀(FG)∼=R_F(G)is the finitely generated free abelian group with the irreducibleG-representations as basis.

For instanceKo(C[Z/n])∼=Zⁿ. Exercise

Compute K₀(C[S₃]).

Example (Dedekind domains)

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

Theideal class groupC(R)is the abelian group of equivalence classes of ideals.

Then we obtain an isomorphism

C(R)−→^∼= Ke₀(R), [I]7→[I].

The structure of the finite abelian group

C(Z[exp(2πi/p)])∼=Ke₀(Z[exp(2πi/p)])∼=Ke₀(Z[Z/p]) is only known for small prime numbersp.

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, which 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.

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:

LetXe be the universal covering. The fundamental group π =π₁(X)acts freely onXe.

LetC∗(Xe)be the cellular chain complex, which is a freeZπ-chain complex.

SinceX is finitely dominated, there exists a finite projective Zπ-chain complexP∗ withP∗ '_Zπ C∗(Xe).

Define

o(X) :=X

n

(−1)ⁿ·[Pn]∈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(Xe )∈Ke₀(Z[π₁(X)])vanishes.

Exercise

Show that a finitely dominated simply connected CW -complex is always homotopy equivalent to a finite CW -complex.

Given a finitely presented groupGandξ∈K₀(ZG), there exists a finitely dominatedCW-complexX withπ₁(X)∼=Gando(X) =ξ.

Theorem (Geometric characterization ofKe0(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;

Ke₀(ZG) ={0}.

Conjecture (Vanishing ofKe0(ZG)for torsion freeG) If G is torsion free, then

Ke₀(ZG) ={0}.

Mini-Break

Solutions to the exercises

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

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]∈K (R)can be thought of as the

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.

Whitehead torsion

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))

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. 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 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

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 E has such a functor S and hence K_n(E) =0for n∈Z.

Notice the similiarity 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: Tomorrow at 10:20