All Borel Group Extensions of Finite-Dimensional Real Space Are Trivial
Linus Richter
TL;DR
This work classifies Borel definable abelian group extensions of the additive groups $(\mathbb{R}^N,+)$ by countable abelian groups $G$. By blending group cohomology with descriptive set theory and employing Cohen-forcing-type arguments, the authors show that every Borel 2-cocycle is a coboundary for all $N\ge 2$, so $H^2_{Bor}(\mathbb{R}^N,G)=0$ and all such definable extensions are trivial. The core analysis begins with a detailed $N=2$ case that introduces novel geometric lemmas and a staged trivialisation, then extends the results to general $N$ via a higher-dimensional analogue $T(N)$, with a consistent forcing-based strategy ensuring coboundarity. The results extend the Kanovei–Reeken theorem from $N=1$ to all $N\ge 2$, and the paper discusses open questions for uncountable $G$ and other Polish groups, outlining future directions at the intersection of logic, descriptive set theory, and homological algebra.
Abstract
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.