I am mainly interested in the $3$-dimensional case. It is a well-known fact, following from the work of E. E. Moise and R. H. Bing in the 1950s, that every $3$-dimensional topological manifold (with or without boundary) admits a triangulation, i.e. its homeomorphic to (the geometric realization of) an abstract simplicial complex. Furthermore, it is a well known fact that a manifold is piecewise-linear if and only if it admits a combinatorial triangulation, i.e. a triangulation in which the link of each simplex is Pl-homeomorphic to a sphere, and that in $d\leq 4$, every triangulation of a manifold is combinatorial. In other words, every $3$-manifold admits a PL-structure.


I am interested in the other way round: Is there a bunch of properties an abstract simplicial complex has to have in order to define a topological manifold? Clearly, not all $3$-dimensional simplicial complexes which one can draw give rise to a manifold. The complex should be at least pure and non-branching, I guess. Is it maybe enough to assume that a complex is combinatorial?


In the literature, I also have found the notions of ''pseudo-manifolds'', which are abstract simplicial complexes, which are pure, non-branching and strongly-connected. How is this related to my question?


Any help is appreciated. If someone could provide some reference, I would be happy too.



Properties a triangulation must have in order to describe a manifold
From the comments:


1. Suppose that $T$ is a triangulation. If all vertex links are PL $(n-1)$-dimensional spheres then the realisation space of $T$ is a PL manifold and thus a topological manifold. (In the compact case this is equivalent to the usual definition.)
2. There are triangulations of topological manifolds (in fact, of $S^5$) that do not have this property. Examples come from the [double suspension theorem](https://en.wikipedia.org/wiki/Double_suspension_theorem) of Cannon and also Edwards.


Furthermore:


3. There are triangulations of topological manifolds that have no PL structure. This is (essentially) the failure of the [Hauptvermutung](https://en.wikipedia.org/wiki/Hauptvermutung).
4. There are closed topological manifolds that do not admit any triangulation. This is the failure of the Triangulation Conjecture, and is obtained by Casson in dimension four and to [Manolescu](https://arxiv.org/abs/1607.08163) in all higher dimensions.


