Given a triangulation (as in, simplicial complex) of a compact closed orientable surface, is it always possible to extend this triangulation to the “bounded” handlebody? If yes, is there any control on the complexity of the extended triangulation in terms of the original one?
The only reference I found for such a question is this paper by Jaco and Rubinstein, but they mention only extensions of 1-vertex triangulations (not strictly simplicial but that's OK) and don't seem to analyze complexity.
I'd be grateful for some kind of overview or survey article!