  Ren\'e Thom's remarkable and far-reaching concept of transversality has found
numerous powerful applications. Most importantly, it allowed Thom to develop
cobordism theory, which led to a piercing insight into the topology of smooth
manifolds, when he generalized Lev Pontryagin's earlier concept of framed
cobordism. For his profound findings, in 1958 he was awarded the Fields Medal.
The present paper provides a down-to-earth approach to transversality theory
which only assumes basic knowledge about smooth manifolds. Following an idea of
Andrew Putman, we then use transversality to give a novel and detailed proof of
an isomorphism theorem of Pontryagin about homotopy groups of spheres whose
original proof is generally believed to be rather hard to access.
