First Cohomology Groups of Minimal Flows

TL;DR

This paper investigates the structure of group extensions of minimal flows with compact abelian groups, focusing on their first cohomology groups and algebraic properties, revealing new insights into their dynamics and existence.

Contribution

It determines the first cohomology groups for flows with simply connected groups and topologically free flows, and demonstrates the existence of rich algebraic structures in minimal extensions.

Findings

First cohomology groups characterized for specific flows

Existence of minimal extensions with complex algebraic structures

Connections established between dynamics and algebraic invariants

Abstract

Our interest in this work is in group extensions of minimal flows with compact abelian groups in the fibres. We study their structure from categorical and algebraic points of view, and describe relations of their dynamics to the one-dimensional algebraic-topological invariants. We determine the first cohomology groups of flows with simply connected acting groups and those of topologically free flows possessing a free cycle. As an application we show that minimal extensions of these flows not only do exist, but they have a rich algebraic structure.