Every compact topological manifold has the homotopy type of a finite CW-complex. So, in the compact case, there is no such invariant of the type you are looking for (or, at least, such an invariant cannot also be a homotopical invariant).
This is a celebrated result by Kirby and Siebenmann, see
Kirby, R. C.; Siebenmann, L. C., On the triangulation of manifolds and the Hauptvermutung, Bull. Am. Math. Soc. 75, 742-749 (1969). ZBL0189.54701." Bull AMS 75 (1969).
More details can be found in the answers to MO36838.