Algebraic & Geometric Topology

Cascades and perturbed Morse–Bott functions

Augustin Banyaga and David E Hurtubise

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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;).

Article information

Algebr. Geom. Topol., Volume 13, Number 1 (2013), 237-275.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R70: Critical points and critical submanifolds
Secondary: 37D05: Hyperbolic orbits and sets 37D15: Morse-Smale systems 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)

Morse homology Morse–Bott critical submanifold cascade exchange lemma


Banyaga, Augustin; Hurtubise, David E. Cascades and perturbed Morse–Bott functions. Algebr. Geom. Topol. 13 (2013), no. 1, 237--275. doi:10.2140/agt.2013.13.237.

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