TITLE: How does Morse theory on non-compact manifolds differ from compact manifolds?
QUESTION [6 upvotes]: What is the Morse homology of a non-compact manifold? When is it, as in the compact case, isomorphic to singular homology of the underlying manifold? What other constructions can be identified with the Morse homology of a non-compact manifold? Links to places where these questions are discussed would be appreciated.

REPLY [3 votes]: Schwarz' book discusses Morse homology for coercive Morse functions on non-compact manifolds. I believe one recovers the singular homology.
I think compact manifolds with boundary are a good playground to get intuition for what is happening in the non-compact case. Suppose $(M,f,g)$ is a manifolds with boundary, Morse function, and metric such that the gradient flow of $f$ is always transverse to the boundary. Then the Morse homology is well defined and it computes the singular homology as follows
$$
HM_*(M,f,g)\cong H_*(M,\partial M_-)
$$
Here $\partial M_-$ is the part of the boundary where the gradient points outwards. If one can connect $(f_0,g_0)$ and $(f_1,g_1)$ using a homotopy where the gradient flow stays transverse to the boundary the Morse homology does not change.
I discuss this more in the introduction of my thesis (http://dare.ubvu.vu.nl/bitstream/handle/1871/52012/complete_dissertation.pdf?sequence=1). See the introduction and the section of local Morse homology.