Topology of definable abelian groups in o-minimal structures

TL;DR

This paper proves that in o-minimal structures, definably connected, compact abelian groups are topologically equivalent to tori, with specific dimension restrictions, extending known results in the semialgebraic case.

Contribution

It establishes a topological classification of abelian definable groups in o-minimal structures, generalizing previous semialgebraic results to broader contexts.

Findings

Definably connected, compact abelian groups are homeomorphic to tori in most cases.

The result holds for all dimensions in the semialgebraic setting.

Dimension 4 cases are excluded in the general o-minimal setting.

Abstract

In this note we show that every definably connected, definably compact abelian definable group in an o-minimal expansion of a real closed field of dimension not 4 is definably homeomorphic to a torus of the same dimension. Moreover, in the semialgebraic case the result holds for all dimensions.