# K-Theory

# Definitions

If XX is any space, the set Vect(X)\Vect(X) has the structure of an abelian semigroup, where the additive structure is defined by the direct sum. If AA is any abelian semigroup, we can associate to AA an abelian group K(A)K(A) with the following property:
there is a semigroup homomorphism α:AK(A)\alpha : A \ra K(A) such that if GG is any group, γ:AG\gamma : A \ra G any semigroup homomorphism, there is a unique homomorphism χ:K(A)G\chi : K(A) \ra G such that γ=χα\gamma = \chi \circ \alpha. If such a K(A)K(A) exists, it must be unique.

Definition

The group K(A)K(A) is defined via the following construction: let F(A)F(A) be the free abelian group generated by the elements of AA, let E(A)E(A) be the subgroup of F(A)F(A) generated by those elements of the form

a+a(aa), a + a^{\prime} - (a \oplus a^{\prime}),

where \oplus is the addition in AA, a,aAa, a^{\prime} \in A. Then

K(A):=F(A)/E(A) K(A) := F(A) / E(A)

has the universal property described above, with α:AK(A)\alpha: A \ra K(A) being the obvious map.

Alternative Definition

A different construction of K(A)K(A) which is often useful is the following: let Δ:AA×A\Delta: A \ra A \times A be the diagonal homomorphism of semigroups, and let K(A)K(A) denote the set of cosets of Δ(A)\Delta(A) in A×AA \times A. It is a quotient semigroup, but interchanging the factors in A×AA \times A induces an inverse in K(A)K(A) so that K(A)K(A) is a group.

We then define αA:AK(A)\alpha_{A} : A \ra K(A) to be the composition of a(a,0)a \mapsto (a,0) with the natural projection A×AK(A)A \times A \ra K(A) (assuming that AA has a zero for simplicity). The pair (K(A),αA)(K(A), \alpha_{A}) is a functor of AA so that if γ:AB\gamma: A \ra B is a semigroup homomorphism, we have a commutative diagram (insert diagram).

If BB is a group then αB\alpha_{B} is an isomorphism, showing that K(A)K(A) has the required universal property.

Remark

If AA is also a semi-ring, then K(A)K(A) is a ring too.