Covers of groups definable in o-minimal structures

Abstract

We develop in this paper the theory of covers for Hausdorff properly $\bigvee$-definable manifolds with definable choice in an o-minimal structure $\mathbb{N}$. In particular, we show that given an $\mathbb{N}$-definably connected $\mathbb{N}$-definable group $G$ we have $1\to {\pi }_1(G)\to \tilde{G}\stackrel{p}{\to }G\to 1$ in the category of strictly properly $\bigvee$-definable groups with strictly properly $\bigvee$-definable homomorphisms, where ${\pi }_1(G)$ is the o-minimal fundamental group of $G$.

69 pages, submited