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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.

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    $\begingroup$ Compactly supported cohomology is the same as reduced cohomology of the one point compactification (for reasonable spaces at least). So the result for compactly supported is implied by the result for usual. $\endgroup$ Jul 27, 2020 at 16:38

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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.

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