  We show that the main theorem of Morse theory holds for a large class of
functions on singular spaces. The function must satisfy certain conditions
extending the usual requirements on a manifold that Condition C holds and the
gradient flow around the critical sets is well-behaved, and the singular space
must satisfy a local deformation retract condition. We then show that these
conditions are satisfied when the function is the norm-square of a moment map
on an affine variety, and that the homotopy equivalence from this theorem is
equivariant with respect to the associated Hamiltonian group action. An
important special case of these results is that the main theorem of Morse
theory holds for the norm square of a moment map on the space of
representations of a finite quiver with relations.
