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Cascades and perturbed Morse–Bott functions

Augustin Banyaga and David E Hurtubise

Algebraic & Geometric Topology 13 (2013) 237–275

Let f : M 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 on the critical submanifolds Cj, 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 fj, can be used to define an explicit perturbation of the Morse-Bott function f to a Morse-Smale function hε: M [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 is isomorphic to the singular homology H(M; ).

Morse homology, Morse–Bott, critical submanifold, cascade, exchange lemma
Mathematical Subject Classification 2010
Primary: 57R70
Secondary: 37D05, 37D15, 58E05
Received: 22 March 2012
Accepted: 30 August 2012
Published: 16 February 2013
Augustin Banyaga
Department of Mathematics
Penn State University
University Park, PA 16802
David E Hurtubise
Department of Mathematics and Statistics
Penn State Altoona
3000 Ivyside Park
Altoona, PA 16601-3760