When I have some (abstract) $3$-dimensional simplicial complex, is it possible to "read of" which topology it describes. A little bit more precisely: Suppose I have a simplicial compelx and that I know that it really describes a (compact, oriented) manifold (and not a pseudo-manifold) possibly with boundary. Is it then possible to tell to which $3$-dimensional manifold the corresponding geometric realization is homeomorphic to, or in other words, which topology describes?
For $2$-dimensional complexes this is quite simple, because I can always read off the Euler characteristic $\chi=V-E+T$ and since compact orientable surfaces are classified by their genera we are done. However, as I have learnt from a previous question, $3$-dimensional (compact, oriented) manifolds are not classified by their genera. In particular, one can show that every closed odd-dimensional manifold has Euler characteristic zero. In the case of manifolds with boundary, this can also be understood as a special case of the statement that every compact and orientable $3$-manifold with boundary has half the Euler characteristic of its boundary.
Any ideas?