We have a characterization when we want $|X|$ to be a PL-manifold, in particular that the links of all the vertices are themselves (PL) spheres. If we are in the category of PL- spaces then this is a necessary and sufficient condition. If however, we leave the PL category, then we get simplicial complexes that are topologically spheres, but not PL spheres. For instance if we take the double suspension of a homology sphere, then we get a sphere. If our homology sphere has a triangulation (and is not itself a sphere), then the double suspension also has a triangulation. The links of the suspension points are not spheres, but these links do have the homotopy type of spheres. Furthermore, the links of these suspension points are also pure simplicial complexes (pure meaning that that all top dimensional simplicies have the same dimension).
I have drawn the following conclusion: If $X$ is a simplicial complex, |X| is a topological manifold of dimension $N$, then the links of all of the vertices are pure simplicial complexes of dimension $N-1$ and have the homotopy type of the $N-1$ sphere.
What is the converse to this conclusion? Namely what extra conditions must be put on the links of the vertices other than purity and being homotopy sphere that will be equivalent to $|X|$ is a manifold?
note: This question is similar to When are (finite) simplicial complexes (smooth) manifolds? . However, the I did not find the answer there. This is also a slightly different question.
Best Answer
In dimensions $\geq 5$, Theorem 1.5 of Galewski and Stern implies that if the links of the vertices of a homology manifold of dimension $\geq 5$ are simply-connected, then it is a manifold.
I'm not sure if this is equivalent to your condition. Certainly your condition implies that the links of vertices are simply-connected and are homology spheres. But to be a homology $n$-manifold, the links of the barycentric subdivision must be $n-1$-homology spheres as well, and I'm not sure whether that is implied by your conditions.
In $1, 2, 3$ and (I believe) $4$ dimensions, I think it's necessary and sufficient to have links be $n-1$-spheres.