Invariant and Coinvariant Morse Homologies for Orbifolds

TL;DR

This paper develops invariant and coinvariant Morse homology theories for compact effective orbifolds, establishing their invariance and exploring their relation to classical topological invariants.

Contribution

It introduces Morse chain complexes for orbifolds with integer coefficients and demonstrates their invariance, extending previous rational coefficient results.

Findings

Homologies of the chain complexes are orbifold invariants.

Coinvariant homology potentially computes the singular homology of the underlying space.

Invariant homology captures orbifold-specific information.

Abstract

In this note, we construct invariant and coinvariant Morse chain complexes with integer coefficients for any compact effective orbifold. We show that the homologies of these two chain complexes are invariants of the orbifold. We conjecture that the homology of the coinvariant chain complex computes the singular homology of the underlying topological space with $\mathbb{Z}$-coefficients, thereby refining the construction by Cho-Hong, which recovers the homology over $\mathbb{Q}$. In contrast, the homology of the invariant Morse chain complex is sensitive to the orbifold structure.