Cascades and perturbed Morse–Bott functions

Abstract

Let $f:M\to ℝ$ be a Morse–Bott function on a finite-dimensional closed smooth manifold $M$. Choosing an appropriate Riemannian metric on $M$ and Morse-Smale functions $f_j:C_j\to ℝ$ 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 $\epsilon >0$ 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 $h_{\epsilon }:M\to ℝ$ [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 $h_{\epsilon }$ is the same as the Morse chain complex defined using cascades for any $\epsilon >0$ 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 $f:M\to ℝ$ is isomorphic to the singular homology $H_{\ast }\left(M;\mathbb{Z}\right)$.