The Homology Determined by the Isomorphism Class of G

In this chapter we prove an invariance theorem which shows that on a closed finite dimensional smooth Riemannian manifold the homology of the twisted Morse-Smale-Witten chain complex is independent of the Morse-Smale pair and depends only on the isomorphism class of the bundle of abelian groups G. The proof of the twisted invariance theorem follows standard arguments used to prove invariance for the Morse-Smale-Witten chain complex with integer coefficients. This chapter contains all the details needed to extend the proof from integer coefficients to local coefficients.