Morse inequalities for orbifold Borel homology

Abstract

Using a notion of Morse function on classifying spaces for finite groups, we define Morse numbers relating the critical point data of an orbifold Morse function to the homology of an associated Borel construction. In particular, we establish Morse inequalities relating our numbers to integral equivariant homology, generalizing the inequalities for manifold Morse functions due to E. Pitcher. To showcase the sharpness of these new inequalities, we will provide a complete classification of all closed, orientable, two-dimensional orbifolds for which equality may be achieved. For three-dimensional orbifolds with singular circles, we prove partial results in this direction, with an emphasis on singular knots in orbifold three-spheres.