  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.
