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.



Compactly supported chern character
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”](http://sites.psu.edu/surgeryforamateurs/files/2017/12/surgerybook2004-1i0fcys.pdf), Remark 2.1.


