The multiple points of maps from sphere to Euclidean space
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866911328348667904 |
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| author | Wang, Jun Zhao, Xuezhi |
| author_facet | Wang, Jun Zhao, Xuezhi |
| contents | In this paper, we obtain some sufficient conditions to guarantee the existence of multiple points of maps from $S^m$ to $\mathbb{R}^d$. Our main tool is the ideal-valued index of $G$-space defined by E. Fadell and S. Husseini. We obtain more detailed relative positional relationship of multiple points. It is proved that for a continuous real value function $f: S^m\rightarrow \mathbb{R}$ such that $f(-p)=-f(p)$, if $m+1$ is a power of $2$, then there are $m+1$ points $p_1, \ldots, p_{m+1}$ in $S^m$ such that $f(p_1)=\cdots=f(p_{m+1})$, where $p_1, \ldots, p_{m+1}$ are linearly dependent and any $m$ points of $p_1, \ldots, p_{m+1}$ are linearly independent. As a generalization of Hopf's theorem, we also prove that for any continuous map $f: S^m\rightarrow \mathbb{R}^d$, if $m> d$, then there exists a pair of mutually orthogonal points having the same image in addition to the antipodal points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_11575 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The multiple points of maps from sphere to Euclidean space Wang, Jun Zhao, Xuezhi Algebraic Topology Geometric Topology Primary 57M99, Secondary 55M35, 55N91 In this paper, we obtain some sufficient conditions to guarantee the existence of multiple points of maps from $S^m$ to $\mathbb{R}^d$. Our main tool is the ideal-valued index of $G$-space defined by E. Fadell and S. Husseini. We obtain more detailed relative positional relationship of multiple points. It is proved that for a continuous real value function $f: S^m\rightarrow \mathbb{R}$ such that $f(-p)=-f(p)$, if $m+1$ is a power of $2$, then there are $m+1$ points $p_1, \ldots, p_{m+1}$ in $S^m$ such that $f(p_1)=\cdots=f(p_{m+1})$, where $p_1, \ldots, p_{m+1}$ are linearly dependent and any $m$ points of $p_1, \ldots, p_{m+1}$ are linearly independent. As a generalization of Hopf's theorem, we also prove that for any continuous map $f: S^m\rightarrow \mathbb{R}^d$, if $m> d$, then there exists a pair of mutually orthogonal points having the same image in addition to the antipodal points. |
| title | The multiple points of maps from sphere to Euclidean space |
| topic | Algebraic Topology Geometric Topology Primary 57M99, Secondary 55M35, 55N91 |
| url | https://arxiv.org/abs/2109.11575 |