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

TL;DR

This paper proves that all Borel definable abelian group extensions of finite-dimensional real spaces by countable abelian groups are trivial for dimensions two and higher, completing a classification initiated in earlier work.

Contribution

The authors extend previous results by showing that no non-trivial Borel definable abelian extensions exist for dimensions two and above, using homological algebra and descriptive set theory.

Findings

No non-trivial Borel definable abelian cocycles for N ≥ 2

Complete classification of such extensions for higher dimensions

Extension of techniques from N=1 case to N≥2

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.