Revisiting Pontryagin's Proof of Stable Stems 1 and 2

2025/03/14 by Trishan Mondal, Mondal, Trishan
Mathematics · #55Q45(Primary)55-02 #Advanced Operator Algebra Research #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2503.11211

openalex publication_date 2025/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \citePont1 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 \citePont2 regarding π₂^S is incorrect.

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• Geometric approach towards stable homotopy groups of spheres. The Steenrod-Hopf invariant I