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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2501.12888 |
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| _version_ | 1866916849413783552 |
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| author | Meadows, Nicholas |
| author_facet | Meadows, Nicholas |
| contents | A series of recent papers by Bergfalk, Lupini and Panagiotopoulus developed the foundations of a field known as `definable algebraic topology,' in which classical cohomological invariants are enriched by viewing them as groups with a Polish cover. This allows one to apply techniques from descriptive set theory to the study of cohomology theories. In this paper, we will establish a `definable' version of a classical theorem from obstruction theory, and use this to study the potential complexity of the homotopy relation on the space of continuous maps $C(X, |K|)$, where $X$ is a locally compact Polish space, and K is a locally finite countable simplicial complex. We will also characterize the Solecki Groups of the Cech cohomology of X, which are the canonical chain of subgroups with a Polish cover that are least among those of a given complexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12888 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Definable Obstruction Theory Meadows, Nicholas Logic 55S35, 03E15 (Primary), 18G80, 18B99 (Secondary) A series of recent papers by Bergfalk, Lupini and Panagiotopoulus developed the foundations of a field known as `definable algebraic topology,' in which classical cohomological invariants are enriched by viewing them as groups with a Polish cover. This allows one to apply techniques from descriptive set theory to the study of cohomology theories. In this paper, we will establish a `definable' version of a classical theorem from obstruction theory, and use this to study the potential complexity of the homotopy relation on the space of continuous maps $C(X, |K|)$, where $X$ is a locally compact Polish space, and K is a locally finite countable simplicial complex. We will also characterize the Solecki Groups of the Cech cohomology of X, which are the canonical chain of subgroups with a Polish cover that are least among those of a given complexity. |
| title | Definable Obstruction Theory |
| topic | Logic 55S35, 03E15 (Primary), 18G80, 18B99 (Secondary) |
| url | https://arxiv.org/abs/2501.12888 |