# # K-Theory

## # Definitions

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

Definition

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

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

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

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

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

Alternative Definition

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

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

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

Remark

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

Last Updated: 12/18/2019, 10:19:21 PM