Abstract: The quantization commutes with reduction problem for Hamiltonian actions of compact Lie groups was solved by Meinrenken in the mid-1990s, and solved again afterwards by many other people using different methods. In this talk, I will consider a generalization of [Q, R]=0 theorem when the manifold is non-compact. In this case, the main issue is that how to quantize a non-compact manifold. I will adopt some ideas from geometric K-homology introduced by Baum and Douglas in 1980s and examine this problem from a topological perspective. One of the applications is that it provides a geometric model for the Kasparov KK group KK(C*(G, X), C).

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