Principal bundle structure of the space of metric measure spaces
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866915025399054336 |
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| author | Kazukawa, Daisuke Nakajima, Hiroki Shioya, Takashi |
| author_facet | Kazukawa, Daisuke Nakajima, Hiroki Shioya, Takashi |
| contents | We study the topological structure of the space $\mathcal{X}$ of isomorphism classes of metric measure spaces equipped with the box or concentration topologies. We consider the scale-change action of the multiplicative group $\mathbb{R}_+$ of positive real numbers on $\mathcal{X}$, which has a one-point metric measure space, say $*$, as only one fixed-point. We prove that the $\mathbb{R}_+$-action on $\mathcal{X}_* := \mathcal{X} \setminus \{*\}$ admits the structure of nontrivial and locally trivial principal $\mathbb{R}_+$-bundle over the quotient space. Our bundle $\mathbb{R}_+ \to \mathcal{X}_* \to \mathcal{X}_*/\mathbb{R}_+$ is a curious example of a nontrivial principal fiber bundle with contractible fiber. A similar statement is obtained for the pyramidal compactification of $\mathcal{X}$, where we completely determine the structure of the fixed-point set of the $\mathbb{R}_+$-action on the compactification. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_06880 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Principal bundle structure of the space of metric measure spaces Kazukawa, Daisuke Nakajima, Hiroki Shioya, Takashi Metric Geometry 53C23, 57R22 We study the topological structure of the space $\mathcal{X}$ of isomorphism classes of metric measure spaces equipped with the box or concentration topologies. We consider the scale-change action of the multiplicative group $\mathbb{R}_+$ of positive real numbers on $\mathcal{X}$, which has a one-point metric measure space, say $*$, as only one fixed-point. We prove that the $\mathbb{R}_+$-action on $\mathcal{X}_* := \mathcal{X} \setminus \{*\}$ admits the structure of nontrivial and locally trivial principal $\mathbb{R}_+$-bundle over the quotient space. Our bundle $\mathbb{R}_+ \to \mathcal{X}_* \to \mathcal{X}_*/\mathbb{R}_+$ is a curious example of a nontrivial principal fiber bundle with contractible fiber. A similar statement is obtained for the pyramidal compactification of $\mathcal{X}$, where we completely determine the structure of the fixed-point set of the $\mathbb{R}_+$-action on the compactification. |
| title | Principal bundle structure of the space of metric measure spaces |
| topic | Metric Geometry 53C23, 57R22 |
| url | https://arxiv.org/abs/2304.06880 |