Revisiting Pontryagin's Proof of Stable Stems 1 and 2
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909537247690752 |
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| author | Mondal, Trishan |
| author_facet | Mondal, Trishan |
| contents | 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 $π_2^S$ is incorrect. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_11211 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Revisiting Pontryagin's Proof of Stable Stems 1 and 2 Mondal, Trishan Algebraic Topology Geometric Topology 55Q45(Primary)55-02 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 $π_2^S$ is incorrect. |
| title | Revisiting Pontryagin's Proof of Stable Stems 1 and 2 |
| topic | Algebraic Topology Geometric Topology 55Q45(Primary)55-02 |
| url | https://arxiv.org/abs/2503.11211 |