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Morse homology and equivariance

Erkao Bao, Tyler Lawson

TL;DR

This work addresses the challenge of computing equivariant homology for finite-group actions by introducing a stability framework for equivariant Morse functions, ensuring descending manifolds are fixed by stabilizers and enabling a $G$-CW decomposition. It proves that stable equivariant Morse functions and generic equivariant metrics yield Morse–Smale data, and shows how to extract equivariant homology via a Morse spectral sequence that collapses to a Thom–Smale–Witten complex in the stable case. The paper provides a thorough exposition of equivariant (co)homology theories (Borel, Bredon, etc.) and their interaction with Morse theory, including how representation cells govern spectral sequences and how orbifold-like structures arise in computations. Overall, it unifies and extends Morse-theoretic techniques to equivariant settings, offering practical computational tools and connections to Smith inequalities and orbifold cohomology.

Abstract

In this paper, we develop methods for calculating equivariant homology from equivariant Morse functions on a closed manifold with the action of a finite group. We show how to alter $G$-equivariant Morse functions to a stable one, where the descending manifold from a critical point $p$ has the same stabilizer group as $p$, giving a better-behaved cell structure on $M$. For an equivariant, stable Morse function, we show that a generic equivariant metric satisfies the Morse--Smale condition. In the process, we give a proof that a generic equivariant function is Morse, and that equivariant, stable Morse functions form a dense subset in the $C^0$-topology within the space of all equivariant functions. Finally, we give an expository account of equivariant homology and cohomology theories, as well as their interaction with Morse theory. We show that any equivariant Morse function gives a filtration of $M$ that induces a Morse spectral sequence, computing the equivariant homology of $M$ from information about how the stabilizer group of a critical point acts on its tangent space. In the case of a stable Morse function, we show that this can be further reduced to a Thom-Smale-Witten complex.

Morse homology and equivariance

TL;DR

This work addresses the challenge of computing equivariant homology for finite-group actions by introducing a stability framework for equivariant Morse functions, ensuring descending manifolds are fixed by stabilizers and enabling a -CW decomposition. It proves that stable equivariant Morse functions and generic equivariant metrics yield Morse–Smale data, and shows how to extract equivariant homology via a Morse spectral sequence that collapses to a Thom–Smale–Witten complex in the stable case. The paper provides a thorough exposition of equivariant (co)homology theories (Borel, Bredon, etc.) and their interaction with Morse theory, including how representation cells govern spectral sequences and how orbifold-like structures arise in computations. Overall, it unifies and extends Morse-theoretic techniques to equivariant settings, offering practical computational tools and connections to Smith inequalities and orbifold cohomology.

Abstract

In this paper, we develop methods for calculating equivariant homology from equivariant Morse functions on a closed manifold with the action of a finite group. We show how to alter -equivariant Morse functions to a stable one, where the descending manifold from a critical point has the same stabilizer group as , giving a better-behaved cell structure on . For an equivariant, stable Morse function, we show that a generic equivariant metric satisfies the Morse--Smale condition. In the process, we give a proof that a generic equivariant function is Morse, and that equivariant, stable Morse functions form a dense subset in the -topology within the space of all equivariant functions. Finally, we give an expository account of equivariant homology and cohomology theories, as well as their interaction with Morse theory. We show that any equivariant Morse function gives a filtration of that induces a Morse spectral sequence, computing the equivariant homology of from information about how the stabilizer group of a critical point acts on its tangent space. In the case of a stable Morse function, we show that this can be further reduced to a Thom-Smale-Witten complex.
Paper Structure (13 sections, 29 theorems, 47 equations, 2 figures)

This paper contains 13 sections, 29 theorems, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statements and outline
  3. Morse polynomials
  4. Local Morse functions
  5. Equivariant tubular neighborhoods
  6. Global Morse functions
  7. Stable Morse functions
  8. Morse--Smale Metrics for Stable Morse Functions
  9. Equivariant homology theories
  10. Equivariant (co)homology and Bredon (co)homology
  11. Filtrations and spectral sequences
  12. Equivariant Morse homology
  13. Equivariant Thom-Smale-Witten complex

Key Result

Theorem 1.2

Equivariant Morse functions are generic in the $C^k$-topology for any $k\geq 2$.

Figures (2)

  • Figure 1: $G=C_3$ acts on $X=\mathbb{R}^2$ by rotation. $X^G = \{o\}$
  • Figure 2: $G=C_2$ acts on $X=\mathbb{R}^2$ by reflection along the $y$-axis. $X^G = \{x = 0\}$.

Theorems & Definitions (68)

  • Definition 1.1: Morse function
  • Theorem 1.2
  • Definition 1.3: Morse--Smale condition
  • Definition 1.4
  • Definition 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8: Equivariant Smale Theorem
  • Corollary 1.9: Smith inequality
  • Proposition 2.1
  • ...and 58 more