Cascades and perturbed Morse-Bott functions

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 f_j: C_j → ℝ on the critical submanifolds C_j, one can construct a Morse chain complex whose boundary operator is defined by counting cascades [Int. Math. Res. Not. 42 (2004) 2179-2269]. Similar data, which also includes a parameter ε > 0 that scales the Morse-Smale functions f_j, can be used to define an explicit perturbation of the Morse-Bott function f to a Morse-Smale function hε: M → ℝ [Progr. Math. 133 (1995) 123-183; Ergodic Theory Dynam. Systems 29 (2009) 1693-1703]. 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).