Morse Inequalities for Orbifold Cohomology

TL;DR

This paper extends Morse theory to orbifolds, establishing Morse inequalities relating orbifold Betti numbers to critical points, and demonstrates that generic functions on orbifolds are Morse, developing necessary differential geometric tools.

Contribution

It introduces Morse inequalities for orbifold cohomology and develops foundational differential geometric tools for differentiable Deligne-Mumford stacks.

Findings

Morse inequalities relate orbifold Betti numbers to critical points.

Generic functions on orbifolds are Morse.

Develops differential geometry tools for orbifold Morse theory.

Abstract

This paper begins the study of Morse theory for orbifolds, or more precisely for differentiable Deligne-Mumford stacks. The main result is an analogue of the Morse inequalities that relates the orbifold Betti numbers of an almost-complex orbifold to the critical points of a Morse function on the orbifold. We also show that a generic function on an orbifold is Morse. In obtaining these results we develop for differentiable Deligne-Mumford stacks those tools of differential geometry and topology -- flows of vector fields, the strong topology -- that are essential to the development of Morse theory on manifolds.