Every compact topological manifold $M$ has the homotopy type of 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).  

Moreover, if $M$ is closed and $\dim(M) \geq 6$, then $M$ is homeomorphic to a CW-complex.

This is a result by Kirby and Siebenmann, see

<cite authors="Kirby, R. C.; Siebenmann, L. C.">_Kirby, R. C.; Siebenmann, L. C._, [**On the triangulation of manifolds and the Hauptvermutung**](https://doi.org/10.1090/S0002-9904-1969-12271-8), Bull. Am. Math. Soc. 75, 742-749 (1969). [ZBL0189.54701](https://zbmath.org/?q=an:0189.54701).</cite>" Bull AMS 75 (1969).

More details can be found in the answers to [MO36838][1].


  [1]: https://mathoverflow.net/questions/36838/are-non-pl-manifolds-cw-complexes/36841#36841