Introduction to Algebraic K-theory (Lecture I)

Introduction to Algebraic K-theory (Lecture I)

Wolfgang Lück Bonn Germany

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

Göttingen, June 21, 2011

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

P

f

``@@

@@ ^f

OO

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 exact.

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.

Definition (Group ring)

LetGbe a group andRbe a ring. Thegroup ring RGis theR-algebra whose underlyingR-module is the freeR-module withGas basis and whose multiplicative structure is induced by the group structure

The cellular chain complexC∗(Xe)of the universal coveringXe of a CW-complexX with fundamental groupπis a freeZπ-chain complex.

LetF be a field of characteristicpforpa prime number or 0. LetG be a finite group. Afinite-dimensionalG-representationofGwith coefficients inF is the same as a finitely generatedFG-module.

F with the trivialG-action is a projectiveFG-module if and only if p=0 orpdoes not divide the order ofG, and is a freeFG-module if and only ifGis 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 groupKe₀(R)is the quotient ofK₀(R) by the subgroup generated by the classes of finitely generated

→

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.

The assignmentP 7→[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]7→[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] 7→ [_Mn(R)Rⁿ_R⊗_RP];

K₀(Mn(R)) −^∼=→ K₀(R), [Q] 7→ [_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 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).

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.

Theideal class group C(R)is the abelian group of equivalence classes ideals under multiplication 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.

Theorem (Swan (1960))

If G is finite, thenKe₀(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) =Z andK¹(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.

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

LetC_∗(Xe)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∗(Xe).

Define

o(X) :=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(Xe )∈Ke₀(Z[π1(X)])vanishes.

A finitely dominated simply connectedCW-complex is always homotopy equivalent to a finiteCW-complex sinceKe₀(Z) ={0}.

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 torsionfreeG) If G is torsionfree, then

Ke₀(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

Ke₁(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 assignmentA7→[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}.

In contrast toKe₀(ZG)the Whitehead group Wh(G)can be computed for finite groups.

For instance we ge for an odd prime numberp Wh(Z/p)∼=Z^(p−1)/2.

Ifp=5, a generator of Wh(Z/5)∼=Zis given by the unit

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 (diffeomorpic) 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 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|_Dn

1×{1}:∂Dⁿ₂−^∼=→∂D₁ⁿ× {1}to a homeomorphismg:D₁ⁿ→D₂ⁿ. The three homeomorphismsid_Dⁿ

1,f andgfit together to a homeomorphismh:M →D₁ⁿ∪_∂Dⁿ

1×{0}∂Dⁿ₁×[0,1]∪_∂Dⁿ

1×{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, since 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 constructed byQuillen (1973)as homotopy groups of certain spaces or spectra.

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

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

Kn(R[Z]) ∼= Kn(R)⊕Kn−1(R);

H_n(BZ) ∼= H_n(pt)⊕ H_n−1(pt).

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

Ke_n(Z[G∗K]) ∼= Ke_n(ZG)⊕Ke_n(ZK);

He_n(B(G∗K)) ∼= He_n(BG)⊕He_n(BK).

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