## Algebraic & Geometric Topology

### Cascades and perturbed Morse–Bott functions

#### 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 $H∗(M;ℤ)$.

#### Article information

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

Dates
Accepted: 30 August 2012
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715497

Digital Object Identifier
doi:10.2140/agt.2013.13.237

Mathematical Reviews number (MathSciNet)
MR3031642

Zentralblatt MATH identifier
1261.57029

#### Citation

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. https://projecteuclid.org/euclid.agt/1513715497

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