Morse homology on noncompact manifolds

This is joint work with Kai Cieliebak. We study Morse functions on noncompact manifolds whose moduli spaces of gradient flow lines for each action window are compact up to breaking. For each action window we get a Morse homology group. There are three ways to define the full Morse homology. One is via Novikov sums and the other two via inverse and direct limits. We study the relation between these three definitions.