As some guy on the street hinted, any "elementary" inductive approach is a spectral sequence in disguise. You have not indicated what topological invariants you have in mind. If Betti numbers suffices for your needs, then in some instances the spectral sequences become relatively simple.
One such instance comes from the stratification of Grassmannians (or more generally flag manifolds) by Schubert cells. In this case the $k$-th Betti number of the corresponding f;ag space is equal to the number of Schubert cells of dimension $k$.
More generally, suppose that you have a Whitney stratification of a compact space $X$ with the following properties.
- All the strata are diffeomorphic to open balls.
- There exists no pair of strata $(S,S')$ such that $|\dim S-\dim S'|=1$.
Then the $k$-th Betti number of $X$ is equal to the number of strata of dimension $k$.