I have studied a basic course in differential geometry and algebraic,differential topology.I have clearly understood the differences between them which is"Differential geometry typically studies Riemannian metrics on manifolds, and properties of them whereas algebraic topology studies algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence",but, I cannot understand similarities between them.I have always wondered how the notion of homotopy,fundamental groups,higher homotopy gps,homology gps etc. work in the setup of riemannian geometry and extrinsic differential geometry which depend completely on notion of riemannian matrices as first one is concerned with classifying manifolds while the other is only concerned with certain local and global properties of manifold.
any help would be appericiated...