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Invariant and Coinvariant Morse Homologies for Orbifolds

Erkao Bao, Lina Liu

TL;DR

The work addresses the challenge of defining Morse-type homology for orbifolds with integer coefficients by stabilizing Morse critical points to obtain a well-behaved Morse–Smale framework. It constructs two chain complexes, the coinvariant C_co and the invariant C_in, leveraging stabilized data to ensure ∂^2=0; the coinvariant homology recovers the integral homology of the underlying space H_*(𝔛), while the invariant homology encodes orbifold-specific information via stabilizers. The paper provides explicit orientation schemes and moduli-space results, and demonstrates with examples how stabilizers influence the invariant theory, including torsion phenomena arising from gcds of stabilizer orders. Overall, it advances integral Morse theory for orbifolds beyond global quotient cases and clarifies how stabilization affects comparison with classical topological invariants.

Abstract

In this note, we construct invariant and coinvariant Morse chain complexes with integer coefficients for any compact effective orbifold. We conjecture that the homology of the coinvariant chain complex computes the homology of the underlying topological space over $\mathbb{Z}$, improving a construction of \cite{cho2014orbifold}, which is isomorphic to the homology of the underlying topological space over $\mathbb{Q}$. In contrast, the homology of the invariant Morse chain complex is sensitive to the orbifold structure itself.

Invariant and Coinvariant Morse Homologies for Orbifolds

TL;DR

The work addresses the challenge of defining Morse-type homology for orbifolds with integer coefficients by stabilizing Morse critical points to obtain a well-behaved Morse–Smale framework. It constructs two chain complexes, the coinvariant C_co and the invariant C_in, leveraging stabilized data to ensure ∂^2=0; the coinvariant homology recovers the integral homology of the underlying space H_*(𝔛), while the invariant homology encodes orbifold-specific information via stabilizers. The paper provides explicit orientation schemes and moduli-space results, and demonstrates with examples how stabilizers influence the invariant theory, including torsion phenomena arising from gcds of stabilizer orders. Overall, it advances integral Morse theory for orbifolds beyond global quotient cases and clarifies how stabilization affects comparison with classical topological invariants.

Abstract

In this note, we construct invariant and coinvariant Morse chain complexes with integer coefficients for any compact effective orbifold. We conjecture that the homology of the coinvariant chain complex computes the homology of the underlying topological space over , improving a construction of \cite{cho2014orbifold}, which is isomorphic to the homology of the underlying topological space over . In contrast, the homology of the invariant Morse chain complex is sensitive to the orbifold structure itself.

Paper Structure

This paper contains 7 sections, 11 theorems, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Morse functions on orbifolds
  3. The moduli space of Morse flow lines
  4. Invariant and Coinvariant Chain Complexes
  5. Orientation
  6. Coinvariant Chain Complex
  7. Invariant Chain Complex

Key Result

Theorem 2.2

Suppose $\mathsf X$ is compact. For any integer $k \ge 2$, a generic $C^k$-function on $\mathbb X$ is Morse.

Figures (2)

  • Figure 1: Stabilization of the critical point $p$.
  • Figure 2: The bean shaped $S^2$ quotient by $\pi$ rotation.

Theorems & Definitions (28)

  • Example 1.1
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3: wasserman1969equivariant, Lemma 4.8; see also bao2024morse, Theorem 1.2
  • Proposition 2.4: bao2024morse, Theorem 3.2
  • proof : Outline of the proof of Theorem \ref{['thm: Morse functions are generic']}
  • Definition 2.5: Stable critical point
  • Remark 2.6
  • Definition 2.7: Stable Morse function
  • Theorem 2.8: mayer1989Ginvariant; bao2024morse, Theorem 1.7
  • ...and 18 more