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2013 Cascades and perturbed Morse–Bott functions
Augustin Banyaga, David E Hurtubise
Algebr. Geom. Topol. 13(1): 237-275 (2013). DOI: 10.2140/agt.2013.13.237


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;).


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Augustin Banyaga. David E Hurtubise. "Cascades and perturbed Morse–Bott functions." Algebr. Geom. Topol. 13 (1) 237 - 275, 2013.


Received: 22 March 2012; Accepted: 30 August 2012; Published: 2013
First available in Project Euclid: 19 December 2017

zbMATH: 1261.57029
MathSciNet: MR3031642
Digital Object Identifier: 10.2140/agt.2013.13.237

Primary: 57R70
Secondary: 37D05, 37D15, 58E05

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.13 • No. 1 • 2013
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