Manifold pathologies and Baire-1 functions as cohomotopy groups
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2024
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| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| contents | A slight extension of a construction due to Calabi-Rosenlicht (and later Gabard, Baillif and others) produces a typically non-metrizable $n$-manifold $\mathbb{P}$ by gluing two copies of the open upper half-space $\mathbb{H}_{++}$ in $\mathbb{R}^n$ along the disjoint union of the spaces of rays within $\mathbb{H}_{++}$ originating at points ranging over a subset $S\subseteq \mathbb{R}^{n-1}$ of the boundary $\mathbb{R}^{n-1}=\partial\overline{\mathbb{H}_{++}}$. The fundamental group $π_1(\mathbb{P})$ is free on the complement $S^{\times}$ of any singleton in $S\ne\emptyset$, and the main result below is that the first cohomotopy group $π^1(\mathbb{P})$, regarded as a space of functions $S^{\times}\to \mathbb{Z}$, is precisely the additive group of integer-valued Baire-1 functions on $S^{\times}$.
This occasions a detour on characterizations (perhaps of independent interest) of Baire-1 real-valued functions on a metric space $(B,d)$ as various types of non-tangential boundary limits of continuous functions on $B\times \mathbb{R}_{>0}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_05276 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Manifold pathologies and Baire-1 functions as cohomotopy groups Chirvasitu, Alexandru Geometric Topology Algebraic Topology General Topology Metric Geometry 26A21, 55Q55, 57N65, 54E52, 54C35, 26A16, 55Q05, 55Q52 A slight extension of a construction due to Calabi-Rosenlicht (and later Gabard, Baillif and others) produces a typically non-metrizable $n$-manifold $\mathbb{P}$ by gluing two copies of the open upper half-space $\mathbb{H}_{++}$ in $\mathbb{R}^n$ along the disjoint union of the spaces of rays within $\mathbb{H}_{++}$ originating at points ranging over a subset $S\subseteq \mathbb{R}^{n-1}$ of the boundary $\mathbb{R}^{n-1}=\partial\overline{\mathbb{H}_{++}}$. The fundamental group $π_1(\mathbb{P})$ is free on the complement $S^{\times}$ of any singleton in $S\ne\emptyset$, and the main result below is that the first cohomotopy group $π^1(\mathbb{P})$, regarded as a space of functions $S^{\times}\to \mathbb{Z}$, is precisely the additive group of integer-valued Baire-1 functions on $S^{\times}$. This occasions a detour on characterizations (perhaps of independent interest) of Baire-1 real-valued functions on a metric space $(B,d)$ as various types of non-tangential boundary limits of continuous functions on $B\times \mathbb{R}_{>0}$. |
| title | Manifold pathologies and Baire-1 functions as cohomotopy groups |
| topic | Geometric Topology Algebraic Topology General Topology Metric Geometry 26A21, 55Q55, 57N65, 54E52, 54C35, 26A16, 55Q05, 55Q52 |
| url | https://arxiv.org/abs/2405.05276 |