Suppose I have two simplicial two-manifolds ("triangle meshes") $M_1$ and $M_2$.
I want to compute a surjective simplicial map between $M_1$ and $M_2$, i.e. a surjective function $\phi$ between the vertices of $M_1$ and $M_2$ so that every triangle of $M_1$ maps to either a triangle, edge, or vertex of $M_2$.
Alternatively I want to know if $M_2$ can be constructed from $M_1$ via some sequence of elementary edge collapses (and if so, find that sequence).
Clearly the problem is in NP, and a naive exponential-time algorithm for finding a map is to enumerate all possible vertex maps $\phi$ and check if they are a valid simplicial map. Is there a faster algorithm? I know that for the related graph isomorphism problem, it helps to assume that the graphs are of bounded genus; would that assumption also help here?