Singular and CW-Homology with Local Coefficients

In this chapter we prove the Twisted Morse Homology Theorem, which says that a twisted Morse chain complex on a closed finite dimensional smooth manifold M with coefficients in a bundle of abelian groups G computes the singular homology of M with coefficients in G. This result is not surprising, but the proof is more technical than one might expect using CW homology. The main difficulty is that the CW-chain complex with local coefficients is not well-defined for general CW-complexes. To get a well-defined CW-chain complex with local coefficients one must restrict to a subclass of complexes, such as the regular CW-complexes. We prove a new theorem that shows that on a closed finite dimensional smooth manifold it is always possible to find a Riemannian metric and a Morse-Smale function whose unstable manifolds determine a regular CW-structure. In fact, it is always possible to find a Morse-Smale pair \((f,\mathsf {g})\) whose unstable manifolds coincide with a smooth triangulation (Theorem 4.12). This new result is essential for the proof of the Twisted Morse Homology Theorem using classical techniques contained in this chapter, and it may also be of independent interest in combinatorial Morse theory.