All Borel Group Extensions of Finite-Dimensional Real Space Are Trivial

2025/05/12 by Linus Richter, Richter, Linus
Computer Science · Mathematics · #03E40 (Secondary) #20J06 #54H05 (Primary) 20K35 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2505.07216

openalex publication_date 2025/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For N ≥ 2, we study the structure of definable abelian group extensions of the additive group (ℝ^N,+) by countable abelian (Borel) groups G. Given an extension H of (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 ≥ 2, there are no non-trivial Borel definable abelian cocycles coding group extensions of (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.

Citations

• Algorithmic information, plane Kakeya sets, and conditional dimension
• The Dimensions of Individual Strings and Sequences
• On the stability of the linear mapping in Banach spaces

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