I'm having trouble with solving the following question:
Let $T_1, T_2$ be two finite triangulations of a compact surface. Show that if $E_{T_1}\cap E_{T_2}$ is a finite set of points, where $E_{T_i}$ is the set of edges of $T_i$, then one can get from $T_1$ to $T_2$ by a finite sequence of moves of type (and their inverses):
- Dividing an edge by inserting a vertex.
- Dividing a polygon by inserting an edge and the corresponding vertices.
- Adding a vertex in the interior of a polygon and the corresponding connecting edge.
My general idea is to take two arbitrary triangulations and express them with the corresponding simplicial complexes. It seems plausible that you can somehow embed one simplicial complex into the other, such that one of the complexes will be a subdivision of the other. In such a case, I could simply remove vertices and edges from the more refined simplicial complex, until I end up with the other complex (I.E - get from one to the other in a finite sequence of the permitted steps).
My main two problems are:
- I'm not entirely sure that you can always perform the embedding I suggested.
- Even if you can, I'm not entirely sure how to describe the process in a precise way. How can I show that one complex can be embedded into the other and then transformed into the other?
Any hint would be of great help. If you could refer me to relevant reading materials, that would also be very helpful (if I'm way off and there's possibly a simpler way than what I've suggested- I'd be happy to hear).
Thank you.