When is simplicial complex a manifold?

Question

I'm looking for necessary and sufficient conditions for an abstract simplicial complex so that it has a geometric realization that is homeomorphic to a manifold. I can't seem to find a single reference for this online. Any help would be greatly appreciated.

Answer 1

One very important result of geometric topology is the double suspension theorem of Bob Edwards, in which he started with a certain triangulated 3-manifold $M$ having the same homology groups as the $3$-sphere $S^3$, and proved that its double suspension $\Sigma^2 M$ is homeomorphic to $S^5$. Double suspension is a very simple construction that one can carry out on any simplicial complex $X$, producing a new simplicial complex $\Sigma^2 X$ whose dimension is $2$ plus the dimension of $X$. Roughly speaking you form one simplicial cone on $X$, and another simplicial cone on $X$, and you glue the bases of the two cones together in the obvious manner, and you get the single suspension $\Sigma X$. Then you do it again to $\Sigma X$ to get $\Sigma^2 X$.

The reason this is relevant to your question is that the hard part of Edwards' proof is to show that $\Sigma^2 M$ is a $5$-dimensional manifold. There were no techniques available to do this, Edwards proof is just shear hard work.

So if there were simple and straightforward necessary and sufficient conditions for the geometric realization of an abstract simplicial complex to be a manifold, we could apply those conditions to $\Sigma^2 M$ to get a cheapo proof of Edwards' theorem.

But no such cheapo proof has shown itself. Edwards's Theorem is a hard theorem.

Steve Ferry's lecture notes also give some historical insight into the significance of the double suspension theorem.

Answer 2

Disclaimer: I know nothing of how to prove all this, I just know the results. This is the simplicial complex recognition problem. It can be phrased so as to make it algorithmic, because a simplicial complex can just be thought of as a set of subsets of some underlying set. The set of such collections is recursive. From that perspective, it turns out that this is undecidable: no algorithm can take a simplicial complex, and tell you whether or not it's homeomorphic to a manifold. So the best that can be hoped for is a partial result.