This is a soft question -- in Guillemin & Pollack's well-known book Differential Topology, they forego tools from algebraic topology like homotopy and homology, opting for a more concrete approach. They carry out a discussion of transversality, "intersection theory mod 2", and "integral intersection theory".
I was wondering whether there was a more natural/succinct/power way to state or rephrase these ideas in the language of algebraic topology, possibly using homology and notions like Poincare duality for smooth manifolds. Are such perspectives present anywhere? Thank you.