Cascades and perturbed Morse-Bott functions

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Let f:M → R be a Morse-Bott function on a finite dimensional closed smooth manifold M. Choosing an appropriate Riemannian metric on M and Morse-Smale functions fⱼ:Cⱼ → R on the critical submanifolds Cⱼ, one can construct a Morse chain complex whose boundary operator is defined by counting cascades FraTheA. Similar data, which also includes a parameter ε> 0 that scales the Morse-Smale functions fⱼ, can be used to define an explicit perturbation of the Morse-Bott function f to a Morse-Smale function hε:M → R AusMor BanDyn. In this paper we show that the Morse-Smale-Witten chain complex of hε is the same as the Morse chain complex defined using cascades for any ε>0 sufficiently small. That is, the two chain complexes have the same generators, and their boundary operators are the same (up to a choice of sign). Thus, the Morse Homology Theorem implies that the homology of the cascade chain complex of f:M → R is isomorphic to the singular homology H(M;Z).