Extensions of definable local homomorphisms in o-minimal structures and semialgebraic groups

TL;DR

This paper investigates conditions under which definable local homomorphisms in o-minimal structures can be extended, and explores the structure of universal coverings of semialgebraic groups, providing new insights into their algebraic and topological properties.

Contribution

It establishes criteria for extending local homomorphisms in simply connected groups and characterizes universal coverings of semialgebraic groups within o-minimal structures.

Findings

Unique extension of local homomorphisms in simply connected groups

Universal covering groups are open subgroups of algebraic group coverings

Descriptions of universal covers as extensions of algebraic group subgroups

Abstract

We state conditions for which a definable local homomorphism between two locally definable groups $\mathcal{G}$, $\mathcal{G^{\prime}}$ can be uniquely extended when $\mathcal{G}$ is simply connected (Theorem 2.1). As an application of this result we obtain an easy proof of [3, Thm. 9.1] (see Corollary 2.2). We also prove that Theorem 10.2 in [3] also holds for any definably connected definably compact semialgebraic group $G$ not necessarily abelian over a sufficiently saturated real closed field $R$; namely, that the o-minimal universal covering group $\widetilde{G}$ of $G$ is an open locally definable subgroup of $\widetilde{H\left(R\right)^{0}}$ for some $R$-algebraic group $H$ (Thm. 3.3). Finally, for an abelian definably connected semialgebraic group $G$ over $R$, we describe $\widetilde{G}$ as a locally definable extension of subgroups of the o-minimal universal covering groups of commutative $R$-algebraic groups (Theorem 3.4)