K
0and Wall’s finiteness obstruction (Lecture I)
Wolfgang Lück Bonn Germany
email wolfgang.lueck@him.uni-bonn.de http://131.220.77.52/lueck/
Bonn, August 2013
Outline
Introduce theprojective class groupK0(R).
Discuss examples.
StateSwan’s Theorem.
Discuss its algebraic and topological significance (e.g.,finiteness obstruction).
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→homR(P,M0)→homR(P,M1)→homR(P,M2)→0 is
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 ifGis trivial.
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⊕Rm ∼=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 andMn(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→ dimF(F ⊗RP).
Example (Representation ring)
LetGbe a finite group and letF be a field of characteristic zero.
Then therepresentation ringRF(G)is the same asK0(FG).
K0(FG)∼=RF(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.
Mini-Break
Solutions to the exercises
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.
Exercise
Show that the C(S2)-module of sections of the tangent bundle TS2is finitely generated projective and and even stably finitely generated free, but not finitely generated free.
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'idX.
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).
Letf∗:C∗→D∗ be aR-chain homotopy equivalence of finite projectiveR-chain complexes. We want to show that
X
n
(−1)n·[Cn] =X
n
(−1)n·[Dn].
Define themapping conecone(f∗)off∗to be the chain complex whosen-th differential is
cone(f∗)n:=Cn−1⊕Dn
−cn−1 0 fn−1 dn
−−−−−−−−−−−→cone(f∗)n−1:=Cn−2⊕Dn−1 It is contractible if and only iff∗ is aR-chain homotopy
equivalence.
LetE∗ be any contractibleR-chain complex.
Letγ andδ be two chain contractions.
DefineR-homomorphisms
(e∗+γ∗)odd:Eodd → Eev; (e∗+δ∗)ev:Eev → Eodd. Put
µn := (γn+1−δn+1)◦δn; νn := (δn+1−γn+1)◦γn.
One easily checks that
(id+µ∗)odd, (id+ν∗)ev
and both compositions
(e∗+γ∗)odd◦(id+µ∗)odd◦(e∗+δ∗)ev (e∗+δ∗)ev◦(id+ν∗)ev◦(e∗+γ∗)odd
are given by upper triangular matrices whose diagonal entries are identity maps.
In particular these four maps are isomorphisms.
This implies that(e∗+γ∗)odd:Eodd→Eevis an isomorphism.
HenceP
n(−1)n·[En] =0 inK0(R).
If we apply this toE∗=cone(f∗), we get inK0(R) X
n
(−1)n·[Cn−1⊕Dn] =X
n
(−1)n· [Cn−1] + [Dn]
=0.
This implies inK0(R) X
n
(−1)n·[Cn] =X
n
(−1)n·[Dn].
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}.
Cliffhanger
Question What is K1(R)?
