Complex K-theory KU

Remarks

Example: for $X$ a point, a complex vector bundle is a finite-dimensional complex vector space, determined up to isomorphism by its dimension. The monoid is $\N$, so $K^0(\text{point}) = \Z$. Also $K^{-1}(\text{point}) = \tilde{K}^0(S^1) = 0$, since every complex vector bundle over $S^1$ is trivial.

Non-example: the monoid of bundles itself is not $K^0(X)$, because it is not a group. Over a point, no vector space $V$ satisfies $V \oplus \C \cong 0$, so the trivial line bundle has no inverse before passing to the Grothendieck group.

For compact $X$, $K^0(X)$ is the set of homotopy classes of maps $X \to \Z \times BU$, where $BU$ is the classifying space of the infinite unitary group. So $KU$ is represented by a spectrum with $\Z \times BU$ in even degrees and $U$ in odd degrees.