Can (deformation classes of local) gapped Hamiltonians be moved about from one manifold to another?

I’ll explain how a generalization of Kirby’s torus trick from "Stable homeomorphisms and the annulus conjecture", Annals of Mathematics (1969) can be used to relocate a certain phases of matter from one manifold to another --- restrictions on both the phase and the manifolds may apply.

(This talk is a part of the International Physics Summer School on Topological Matter. Note the location of the talk, in the Science Teaching Building.)