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

For N 2, we study the structure of definable abelian group extensions of the additive group (RN, +) by countable abelian (Borel) groups G. Given an extension H of (RN, +) 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 (RN, +) 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.