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

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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

Wolfgang Lück (Münster, Germany) Lower and middle K-theory in topology Hangzhou, July 2007 1 / 30

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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.

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Outline

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Outline

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Outline

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Outline

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Outline

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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.

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The projective class group

M ^p ^//N ^//0

P

f

``@@

@@ ^f

OO

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The projective class group

M ^p ^//N ^//0

P

f

``@@

@@ ^f

OO

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The projective class group

M ^p ^//N ^//0

P

f

``@@

@@ ^f

OO

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The projective class group

The following lifting problem has always a solution M ^p ^//N ^//0

P

f

``@@

@@ ^f

OO

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The projective class group

The following lifting problem has always a solution M ^p ^//N ^//0

P

f

``@@

@@ ^f

OO

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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.

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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.

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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 freeR-modules, or, equivalently, the cokernel ofK₀(Z)→K₀(R).

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

K₀(R)

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K₀(R)

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K₀(R)

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K₀(R)

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).

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K₀(R)

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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].

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Induction

f∗:K₀(R)→K₀(S), [P]7→[f∗P].

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Induction

f∗:K₀(R)→K₀(S), [P]7→[f∗P].

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Induction

f∗:K₀(R)→K₀(S), [P]7→[f∗P].

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Induction

f∗:K₀(R)→K₀(S), [P]7→[f∗P].

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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].

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Induction

f∗:K₀(R)→K₀(S), [P]7→[f∗P].

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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→ [_M_n_(R)Rⁿ_R⊗_RP];

K₀(Mn(R)) −^∼⁼→ K₀(R), [Q] 7→ [_RRⁿ_M_n_(R)⊗_M_n_(R)Q].

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Morita equivalence

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Morita equivalence

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Morita equivalence

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Morita equivalence

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Morita equivalence

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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.

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Z

∼=

−→ K₀(R), n 7→ [Rⁿ];

K₀(R) −→^∼⁼ Z, [P] 7→ dim_F(F ⊗_RP).

∼=

−→class(G,C),

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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).

∼=

−→class(G,C),

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Z

∼=

−→ K₀(R), n 7→ [Rⁿ];

K₀(R) −→^∼⁼ Z, [P] 7→ dim_F(F ⊗_RP).

∼=

−→class(G,C),

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Z

∼=

−→ K₀(R), n 7→ [Rⁿ];

K₀(R) −→^∼⁼ Z, [P] 7→ dim_F(F ⊗_RP).

∼=

−→class(G,C),

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Z

∼=

−→ K₀(R), n 7→ [Rⁿ];

K₀(R) −→^∼⁼ Z, [P] 7→ dim_F(F ⊗_RP).

∼=

−→class(G,C),

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Z

∼=

−→ K₀(R), n 7→ [Rⁿ];

K₀(R) −→^∼⁼ Z, [P] 7→ dim_F(F ⊗_RP).

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

∼=

−→class(G,C),

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Z

∼=

−→ K₀(R), n 7→ [Rⁿ];

K₀(R) −→^∼⁼ Z, [P] 7→ dim_F(F ⊗_RP).

∼=

−→class(G,C), whereclass(G;C)is the complex vector space ofclass functions G→C, i.e., functions, which are constant on conjugacy classes.

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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)−→^∼⁼ 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.

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C(R)−→^∼⁼ Ke₀(R), [I]7→[I].

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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

C(R)−→^∼⁼ Ke₀(R), [I]7→[I].

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C(R)−→^∼⁼ Ke₀(R), [I]7→[I].

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C(R)−→^∼⁼ Ke₀(R), [I]7→[I].

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C(R)−→^∼⁼ Ke₀(R), [I]7→[I].

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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) =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)).

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TopologicalK-theory

K⁰(X)−^∼⁼→K₀(C(X)).

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TopologicalK-theory

K⁰(X)−^∼⁼→K₀(C(X)).

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TopologicalK-theory

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

K⁰(X)−^∼⁼→K₀(C(X)).

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TopologicalK-theory

K⁰(X)−^∼⁼→K₀(C(X)).

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TopologicalK-theory

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

K⁰(X)−^∼⁼→K₀(C(X)).

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TopologicalK-theory

K⁰(X)−^∼⁼→K₀(C(X)).

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TopologicalK-theory

K⁰(X)−^∼⁼→K₀(C(X)).

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TopologicalK-theory

K⁰(X)−^∼⁼→K₀(C(X)).

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TopologicalK-theory

K⁰(X)−^∼⁼→K₀(C(X)).

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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?

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Wall’s finiteness obstruction

Problem

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Wall’s finiteness obstruction

Problem

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Wall’s finiteness obstruction

Problem

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Wall’s finiteness obstruction

Problem

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Wall’s finiteness obstruction

Problem

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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).