Morse inequalities for orbifold cohomology

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.