Revisiting Pontryagin's Proof of Stable Stems 1 and 2

Abstract

In this paper, we introduce fundamental notions of homotopy theory, including homotopy excision and the Freudenthal suspension theorem. We then explore framed cobordism and its connection to stable homotopy groups of spheres through the Pontryagin-Thom construction. Using this framework, we compute the stable stems in dimensions $0$, $1$, and $2$. This work is primarily expository, revisiting proofs from \cite{Pont1} with slight modifications incorporating modern notation. Furthermore, in the final section, we discuss 2-dimensional framed manifolds with Arf invariant one and examine why the result of \cite{Pont2} regarding $\pi_2^S$ is incorrect.