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

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.

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.

References.

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.

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.

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

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.