Cascades and perturbed Morse–Bott functions

Banyaga, Augustin; Hurtubise, David E

doi:10.2140/agt.2013.13.237

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Cascades and perturbed Morse–Bott functions

Augustin Banyaga and David E Hurtubise

Algebraic & Geometric Topology 13 (2013) 237–275

DOI: 10.2140/agt.2013.13.237

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 $ε > 0$ 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 $h_{ε} : M \to ℝ$ [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 $h_{ε}$ 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 $f : M \to ℝ$ is isomorphic to the singular homology $H_{*} (M; ℤ)$ .

Keywords

Morse homology, Morse–Bott, critical submanifold, cascade, exchange lemma

Mathematical Subject Classification 2010

Primary: 57R70

Secondary: 37D05, 37D15, 58E05

References

Publication

Received: 22 March 2012

Accepted: 30 August 2012

Published: 16 February 2013

Authors

Augustin Banyaga
Department of Mathematics Penn State University University Park, PA 16802 USA
http://www.math.psu.edu/banyaga/
David E Hurtubise
Department of Mathematics and Statistics Penn State Altoona 3000 Ivyside Park Altoona, PA 16601-3760 USA
http://www.personal.psu.edu/dxh40/

Volume 13, issue 1 (2013)