Compactly supported chern character

Question

It is a standard result that for a CW complex $X$, the chern character

$$\text{ch}: K^*(X)\otimes_{\mathbb{Z}} \mathbb{Q}\to H^*(X,\mathbb{Q})$$

induces an isomorphism. Suppose now that $X$ is an open manifold and consider the chern character with compact support

$$\text{ch}_{\text{cs}}: K^*_{\text{cs}}(X)\otimes_{\mathbb{Z}} \mathbb{Q}\to H^{*}_{\text{cs}}(X,\mathbb{Q})$$

Is it still true that it is an isomorphism? It seems to be so for the case of $\text{Tot}(E\to S)$ where $S$ is a compact space and $E$ is a vector bundle. I suppose this is well known, but I couldn't find a reference.

Answer 1

Yes, this is true. For any generalized cohomology theory $E$, the compactly supported $E$-cohomology of a space $X$ is

$$E_{\mathit{cs}}^*(X) := \varinjlim\limits_{K\subseteq X:\text{ $K$ compact}} E^*(X, X\setminus K).$$

The Chern character is a natural isomorphism of cohomology theories, so is compatible with the limit above, hence induces an isomorphism on compactly supported rational $K$-theory.

One reference for this definition of compactly supported generalized cohomology is Ranicki-Roe, “Surgery for Amateurs”, Remark 2.1.