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.

In the comments, it was asked to provide some references to the claims made in the first paragraph, so here we go:

• "A manifold if piecewise-linear iff it admits a combinatorial triangulation" (see Hudson: Piecewise Linear Topology, WA Benjamin Inc., 1969. starting from page 24)
• "Every 2- and 3-dimensional manifold admits a triangulation" (This is a classical result of Rado (1924) and Moise (1952), Bing (1959))
• "Every triangulation of a $d\leq 4$ dimensional manifold is combinatorial" (For $d=1,2,3$ this follows from the Theorems of Bing-Moise and Rado. For $d=4$ this is highly non-trivial and a consequence of the Poincare conjecture (in fact it is equivalent to it), which has been shown by Perelman in 2003 (cf. Millennium problems))

This are all the claims made in my first paragraph. Some other interesting results in this direction:

• "Every smooth manifold (in fact $C^{1}$ is enough) of every dimension admits a combinatorial triangulation and hence a PL-structure" (classical result of Cairns (1935) and Whitehead (1940))
• "For $d\geq 4$, there are topological manifolds that do not admit triangulations at all" (Example: $E^{8}$, result of Freedman 1982 and Akbulut-McCarthy 1990)
• General non-existence result: For every $d\geq 5$, there is a manifold that does not admit a PL-structure (Kirby-Siebenmann 1969). In fact, for every $d\geq 5$, there exists a manifold that does not even admit a triangulation (Manolescu 2016). Note that there are also manifolds that admit a triangulation, but not a PL structure, like $E_{8}\times T^{k}$ where $T^{k}$ denotes the $k$-torus.
• Last but not least, above I have mentioned that every smooth manifold has a natural PL structure, however, there is in general not a one-to-one correspondence between these two things. For instance, Kervaire (1960) constructed a $10$-dimensional topological manifold that admits a PL-structure but not a smooth structure. However, this is only true in sufficient high dimensions, since for $d\leq 7$, it has been shown that every Pl-manifold admits a combatible smooth structure (cf. Hirsch-Mazur obstruction theory).