Transversality and framed cobordism

TL;DR

This paper offers an accessible introduction to transversality theory and employs it to provide a new, detailed proof of Pontryagin's isomorphism theorem concerning homotopy groups of spheres, simplifying the original complex proof.

Contribution

It introduces a straightforward approach to transversality and applies it to give a clearer proof of a fundamental theorem in topology, enhancing understanding of homotopy groups.

Findings

Simplified proof of Pontryagin's isomorphism theorem

New application of transversality in homotopy theory

Enhanced accessibility to complex topological proofs

Abstract

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.