  In the presence of compact dimensions massive solutions of General Relativity
may take one of several forms including the black-hole and the black-string,
the simplest relevant background being R^{3+1} * S^1. It is shown how Morse
theory places constraints on the qualitative features of the phase diagram, and
a minimalistic diagram is suggested which describes a first order transition
whose only stable phases are the uniform string and the black-hole. The diagram
calls for a topology changing ``merger'' transition in which the black-hole
evolves continuously into an unstable black-string phase. As evidence a local
model for the transition is presented in which the cone over S^2 * S^2 plays a
central role. Horizon cusps do not appear as precursors to black hole merger. A
generalization to higher dimensions finds that whereas the cone has a tachyon
function for d=5, its stability depends interestingly on the dimension - it is
unstable for d<10, and stable for d>10.
