Applications of Borel-definable homological algebra to locally compact groups

[2509.25474] Applications of Borel-definable homological algebra to locally compact groups Skip to main content Search Submit Donate Log in Mathematics > Group Theory arXiv:2509.25474 (math) [Submitted on 29 Sep 2025 ( v1 ), last revised 27 Aug 2026 (this version, v3)] Title: Applications of Borel-definable homological algebra to locally compact groups Authors: Martino Lupini View PDF HTML (experimental) Abstract: We apply the description of the left heart of the category $\mathbf{LCPAb}$ of locally compact Polish abelian groups in terms of groups with a Polish cover and Borel-definable group homomorphisms to study injective and projective objects in the left heart of $\mathbf{LCPAb}$, as well as in the left hearts of its thick subcategories. In particular, we prove that the left hearts of the following categories have no nonzero injective objects: locally compact Polish abelian groups, compactly generated locally compact Polish abelian groups, locally compact Polish abelian groups of finite ranks, abelian Lie groups, totally disconnected locally compact Polish abelian groups, topological torsion locally compact Polish abelian groups, locally compact Polish abelian topological $p$-groups for any prime $p$. Comments: 94 pages, minor edits, changed formatting, Proposition 3.9 strengthened Subjects: Group Theory (math.GR) ; Category Theory (math.CT); General Topology (math.GN); Logic (math.LO) MSC classes: 54H05, 20K45, 18F60 (Primary) 26E30, 18G10, 46M15 (Secondary) Cite as: arXiv:2509.25474 [math.GR] (or arXiv:2509.25474v3 [math.GR] for this version) https://doi.org/10.48550/arXiv.2509.25474 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Martino Lupini [ view email ] [v1] Mon, 29 Sep 2025 20:27:04 UTC (879 KB) [v2] Thu, 6 Aug 2026 07:01:30 UTC (67 KB) [v3] Thu, 27 Aug 2026 19:05:49 UTC (929 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.GR < prev | next > new | recent | 2025-09 Change to browse by: math math.CT math.GN math.LO References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

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