Last time, I described the construction which assigns to every compact ${G}$-space ${X}$ (for ${G}$ a compact Lie group) the equivariant K-group ${K_G(X)}$. We saw that this was a functor from the (equivariant) homotopy category to commutative rings, using more or less the same arguments as in ordinary homotopy theory, only with small alterations.

The purpose of this post is to describe more of Segal’s paper. Actually, I won’t be covering any legitimate K-theory in this post; that’ll have to wait for a third. I’ll mostly be describing various classical constructions for vector bundles in the equivariant setting.

In the classical theory of (ordinary) vector bundles on compact spaces, a basic result is the Serre-Swan theorem, which identifies the category ${\mathrm{Vect}(X)}$ of (complex) vector bundles with the category of projective modules over the ring ${C(X; \mathbb{C})}$ of complex-valued continuous functions on ${X}$. This is essentially a reflection of the fact that any vector bundle on ${X}$, say ${E \rightarrow X}$, can be obtained as a retract of some trivial bundle ${\mathbb{C}^n \times X \rightarrow X}$. Taking retracts corresponds to choosing idempotents in the ring of ${n}$-by-${n}$ matrices in ${C(X; \mathbb{C})}$, and this description via idempotents applies as well to projective modules over ${C(X; \mathbb{C})}$ (or, in fact, any commutative ring).

The crucial statement here, that any vector bundle is a retract of a trivial one, fails in the equivariant case, simply because a vector bundle on which ${G}$ acts nontrivially can’t be a retract of a vector bundle with trivial action. But we have something reasonably close to it.

Definition 1 Given a ${G}$-representation ${V}$ and a ${G}$-space ${X}$, we can form a vector bundle ${V \times X \rightarrow X}$, which his naturally ${G}$-equivariant.

This bundle is, equivalently, formed by taking the equivariant map ${X \rightarrow \ast}$. ${G}$-vector bundles on ${\ast}$ are identified with ${G}$-representations, so we just have to pull back.

Anyway, the claim is:

Theorem 2 (Equivariant Serre-Swan (Segal)) Any ${G}$-vector bundle ${E \rightarrow X}$ is a direct summand of a bundle ${V \times X \rightarrow X}$ for some ${G}$-representation ${V}$. (more…)