Cappell-Shaneson polynomials
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912482961915904 |
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| author | Endo, Hisaaki Iwaki, Kazunori Pajitnov, Andrei |
| author_facet | Endo, Hisaaki Iwaki, Kazunori Pajitnov, Andrei |
| contents | S. Cappell and J. Shaneson constructed a pair of inequivalent embeddings of $(n-1)$-spheres in homotopy $(n+1)$-spheres for every square matrix of order $n$ with special properties (a Cappell-Shaneson matrix). A Cappell-Shaneson polynomial is the characteristic polynomial of a Cappell-Shaneson matrix. In this paper, we interpret part of the definition of Cappell-Shaneson polynomial as algebraic conditions on polynomials in terms of signed reciprocal polynomial and reduction modulo primes, and give complete lists of all Cappell-Shaneson polynomials of degrees $4$ and $5$. We construct several infinite series of Cappell-Shaneson polynomials of degree $6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10885 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cappell-Shaneson polynomials Endo, Hisaaki Iwaki, Kazunori Pajitnov, Andrei Geometric Topology Algebraic Topology Number Theory 57K45, 12E05, 11C20, 11D72, 11Yxx S. Cappell and J. Shaneson constructed a pair of inequivalent embeddings of $(n-1)$-spheres in homotopy $(n+1)$-spheres for every square matrix of order $n$ with special properties (a Cappell-Shaneson matrix). A Cappell-Shaneson polynomial is the characteristic polynomial of a Cappell-Shaneson matrix. In this paper, we interpret part of the definition of Cappell-Shaneson polynomial as algebraic conditions on polynomials in terms of signed reciprocal polynomial and reduction modulo primes, and give complete lists of all Cappell-Shaneson polynomials of degrees $4$ and $5$. We construct several infinite series of Cappell-Shaneson polynomials of degree $6$. |
| title | Cappell-Shaneson polynomials |
| topic | Geometric Topology Algebraic Topology Number Theory 57K45, 12E05, 11C20, 11D72, 11Yxx |
| url | https://arxiv.org/abs/2507.10885 |