To support this question I'm going to use a theoretical example.
Suppose we have some Hamiltonian (H) which is a function of a continuous parameter, say $\alpha$. Also suppose that as a function of alpha this Hamiltonian moves from a space of trivial topology to a space of non-trivial topology (i.e sphere($S^2$) -> torus($T$)).
Clearly the Hamiltonian is continuous as I have stipulated, but the change in topology across the phase transition is certainly discontinuous in the sense that topology has been generated by something (usually the breaking of some continuous symmetry).
So my question more robustly stated is: Is there a good model for describing this change mathematically, smoothly, in terms of the underlying manifolds? And how well understood is this process?
