# Duality between K-theory and K-homology in the non-compact, spin$^c$ case

Let $M$ be a compact spin$^c$ manifold, so that it has a fundamental class $[M] \in K_n(M)$. It is well-known that the cap product with $[M]$ induces Poincare duality isomorphisms $K^\ast(M) \cong K_\ast(M)$ between $K$-theory and $K$-homology.

What about the non-compact case? Do we have $K^\ast_{\text{cpt}}(M) \cong K_\ast(M)$ via the cap product with $[M]$, where $K_\ast(M)$ now stands for the $K$-homology of $C_0(X)$? Any references / proofs?

I'm wondering, since this is never mentioned anywhere.

-
You could look into papers on noncommutative geometry on noncompact spaces, since this is a relevant problem in this field. I did a quick search, and could so far only find a comment in arxiv.org/abs/hep-th/0307241 , where they state that at the time of writing, the details of formulating Poincare duality for noncompact spaces in K-theory and K-homology were somewhat cloudy. –  Jan Jitse Venselaar May 21 '13 at 20:52