Cascades and perturbed Morse-Bott functions

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Abstract

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 [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 → 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).

Original languageEnglish (US)
Pages (from-to)237-275
Number of pages39
JournalAlgebraic and Geometric Topology
Volume13
Issue number1
DOIs
StatePublished - Feb 16 2013

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

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