All Questions

What methods are there to approximate an arbitrary homotopy type by an algebraic-geometric object in a concrete (read: computable) way? I know this is an ambitious question, so maybe I should ...

Is there a (non-Abelian) homology theory that realizes the following: Let $X,Y$ be manifolds with complexes $C(X),C(Y)$. Then $X$ and $Y$ are homotopy-equivalent if and only if $C(X)$ and $C(Y)$ ...