  In this article we describe relations of the topology of closed 1-forms to
the group theoretic invariants of Bieri-Neumann-Strebel-Renz. Starting with a
survey, we extend these Sigma invariants to finite CW- complexes and show that
many properties of the group theoretic version have analogous statements. In
particular we show the relation between Sigma invariants and finiteness
properties of certain infinite covering spaces. We also discuss applications of
these invariants to the Lusternik- Schnirelmann category of a closed 1-form and
to the existence of a non- singular closed 1-form in a given cohomology class
on a high-dimensional closed manifold.
