Invariant and Coinvariant Morse Homologies for Orbifolds

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.