>>11729899I don't know, because I'm not a low-dimensional topologist. not at all, lol. but look:
classification only tells you that S is homeomorphic to a certain surface based on its fundamental group or something. how many homeomorphisms up to homotopy are there ? what about up to isotopy ? is the homeomorphism homotopic to a diffeomorphism ? what about if it's supposed to preserve some group action ? then you can study fibre bundles over surfaces or whatever. there's a lot things, classification is really just a start.
