Cascades and perturbed Morse–Bott functions

Banyaga, Augustin¹ ; Hurtubise, David E²

¹Department of Mathematics, Penn State University, University Park, PA 16802, USA
²Department of Mathematics and Statistics, Penn State Altoona, 3000 Ivyside Park, Altoona, PA 16601-3760, USA

Algebraic and Geometric Topology, Tome 13 (2013) no. 1, pp. 237-275

Résumé

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).

Détail

DOI : 10.2140/agt.2013.13.237

Classification : 57R70, 37D05, 37D15, 58E05
Keywords: Morse homology, Morse–Bott, critical submanifold, cascade, exchange lemma

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