The Morse Complex with Local Coefficients

In this chapter the Morse-Smale-Witten chain complex with coefficients in a bundle of abelian groups over a finite dimensional smooth Riemannian manifold is constructed. The twisted Morse chain complex twisted Morse homology is described in detail for \(S^1\) and real projective space. Relationships with Morse complexes on covering spaces covering space are discussed, and a Morse theoretic version of Eilenberg’s Theorem Eilenberg is proved.