Realising sets of integers as mapping degree sets
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866918124572377088 |
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| author | Neofytidis, Christoforos Wang, Shicheng Wang, Zhongzi |
| author_facet | Neofytidis, Christoforos Wang, Shicheng Wang, Zhongzi |
| contents | Given two closed oriented manifolds $M,N$ of the same dimension, we denote the set of degrees of maps from $M$ to $N$ by $D(M,N)$. The set $D(M,N)$ always contains zero. We show the following (non-)realisability results:
(i) There exists an infinite subset $A$ of $\mathbb Z$ containing $0$ which cannot be realised as $D(M,N)$, for any closed oriented $n$-manifolds $M,N$.
(ii) Every finite arithmetic progression of integers containing $0$ can be realised as $D(M,N)$, for some closed oriented $3$-manifolds $M,N$.
(iii) Together with $0$, every finite geometric progression of positive integers starting from $1$ can be realised as $D(M,N)$, for some closed oriented manifolds $M,N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_13790 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Realising sets of integers as mapping degree sets Neofytidis, Christoforos Wang, Shicheng Wang, Zhongzi Geometric Topology Algebraic Topology Number Theory Given two closed oriented manifolds $M,N$ of the same dimension, we denote the set of degrees of maps from $M$ to $N$ by $D(M,N)$. The set $D(M,N)$ always contains zero. We show the following (non-)realisability results: (i) There exists an infinite subset $A$ of $\mathbb Z$ containing $0$ which cannot be realised as $D(M,N)$, for any closed oriented $n$-manifolds $M,N$. (ii) Every finite arithmetic progression of integers containing $0$ can be realised as $D(M,N)$, for some closed oriented $3$-manifolds $M,N$. (iii) Together with $0$, every finite geometric progression of positive integers starting from $1$ can be realised as $D(M,N)$, for some closed oriented manifolds $M,N$. |
| title | Realising sets of integers as mapping degree sets |
| topic | Geometric Topology Algebraic Topology Number Theory |
| url | https://arxiv.org/abs/2109.13790 |