#### Volume 13, issue 1 (2013)

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### Augustin Banyaga and David E Hurtubise

Algebraic & Geometric Topology 13 (2013) 237–275
##### 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 ${H}_{\ast }\left(M;ℤ\right)$.

##### Keywords
Morse homology, Morse–Bott, critical submanifold, cascade, exchange lemma
##### Mathematical Subject Classification 2010
Primary: 57R70
Secondary: 37D05, 37D15, 58E05