Morse-Bott-Smale chain complex

Banyaga and Hurtubise defined the Morse-Bott-Smale chain complex as a quotient of a large chain complex by introducing five degeneracy relations. However, their five degeneracy relations are in fact redundant. In the present paper, we unify these five conditions into a single degeneracy condition and resolve the issue of the well-definedness of the Morse-Bott-Smale chain complex. This provides an appropriate definition of the Morse-Bott homology and more computable examples. Moreover, we show that our chain complex for a Morse-Smale function is quasi-isomorphic to the usual Morse-Smale-Witten chain complex. As a consequence, we obtain an alternative proof of the Morse Homology Theorem.