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

Abstract

For $N\ge 2$, 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 $N\ge 2$, 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.

18 pages