How can I force a topological manifold $M$ to be homeomorphic to a finite simplicial complex? Obviously $M$ needs to be compact, but this is far from sufficient.
In particular is there any "purely topological" way to do it? (Without introducing more structures on $M$ such as smooth, ect?)
I am translating the question "How can I force a topological manifold $M$ to be..." as "what are sufficient conditions for a topological manifold $M$ to be..."
Classically (i.e. work of Rado and Moise), it is know that every topological manifold of dimension $\le 3$ admits a triangulation (i.e. is homeomorphic to a simplicial complex). In low dimensions, the existence of a triangulation is equivalent to the existence of a PL structure, but this breaks down in higher dimensions. The question of triangulability belongs to the circle of problems called "Hauptvermutung." You can find some history and references in this Wikipedia page on Hauptvermutung. In particular, if your manifold $M$ has dimension $\ge 5$ and $^4(,{\mathbb Z}/2)=0$, then $M$ admits a PL structure, in particular, a triangulation. Ditto if $M$ is an open 4-manifold. What happens (in the case of closed manifolds) in dimension 4 is anybody's guess, although in the simply-connected case many sufficient conditions follow from Freedman's and Donaldson's work.