This paper begins the study of Morse theory for orbifolds, or equivalently 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.
"Morse inequalities for orbifold cohomology." Algebr. Geom. Topol. 9 (2) 1105 - 1175, 2009. https://doi.org/10.2140/agt.2009.9.1105