All Borel Group Extensions of Finite-Dimensional Real Space Are Trivial
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916732427304960 |
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| author | Richter, Linus |
| author_facet | Richter, Linus |
| contents | For $N \geq 2$, we study the structure of definable abelian group extensions of the additive group $(\mathbb{R}^N,+)$ by countable abelian (Borel) groups $G$. Given an extension $H$ of $(\mathbb{R}^N,+)$ by $G$, we measure the definability of $H$ by investigating its complexity as a Borel set. We do this by combining homological algebra and descriptive set theory, and hence study the Borel complexity of those functions inducing $H$, the abelian cocycles. We prove that, for every $N \geq 2$, there are no non-trivial Borel definable abelian cocycles coding group extensions of $(\mathbb{R}^N,+)$ by a countable abelian group $G$, and hence show that no non-trivial such group extensions exist. This completes the picture first investigated by Kanovei and Reeken in 2000, who proved the case $N = 1$, and whose techniques we adapt in this work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07216 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | All Borel Group Extensions of Finite-Dimensional Real Space Are Trivial Richter, Linus Logic 54H05 (Primary) 20K35, 20J06, 03E40 (Secondary) For $N \geq 2$, we study the structure of definable abelian group extensions of the additive group $(\mathbb{R}^N,+)$ by countable abelian (Borel) groups $G$. Given an extension $H$ of $(\mathbb{R}^N,+)$ by $G$, we measure the definability of $H$ by investigating its complexity as a Borel set. We do this by combining homological algebra and descriptive set theory, and hence study the Borel complexity of those functions inducing $H$, the abelian cocycles. We prove that, for every $N \geq 2$, there are no non-trivial Borel definable abelian cocycles coding group extensions of $(\mathbb{R}^N,+)$ by a countable abelian group $G$, and hence show that no non-trivial such group extensions exist. This completes the picture first investigated by Kanovei and Reeken in 2000, who proved the case $N = 1$, and whose techniques we adapt in this work. |
| title | All Borel Group Extensions of Finite-Dimensional Real Space Are Trivial |
| topic | Logic 54H05 (Primary) 20K35, 20J06, 03E40 (Secondary) |
| url | https://arxiv.org/abs/2505.07216 |