Let's restrict to triangulations $T$ of compact and closed smooth manifolds $M$ with $\dim M=2,3$. Such a triangulation is a PL manifold homeomorphic to $M$ which geometric realization is a simplicial complex.
Like a triangle, such a complex is completely (geometrically) fixed by its edges (or equivalently by its vertices and their connection matrix). This is basically the $1$-skeleton $\operatorname{Skel}_1$ of the complex, which is given by the union of all simplices of dimension $> 1$.
- Is it possible to reconstruct $T$ from $\operatorname{Skel}_1$, with the use of the properties of $T$ (compact and closed)? i.e. reconstruct the $2$-simplices, $3$-simplices
Or equivalently:
- Is it possible to construct two non-homeomorphic compact and closed triangulations $T_1,T_2$ having isomorphic $1$-skeleta $\operatorname{Skel}_1(T_1)$ and $\operatorname{Skel}_1(T_2)$?
If the answer is negative:
- What extra condition or information on the triangulation would be sufficient to maintain the the possibility of reconstruction?