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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 CWcomplexes 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 LusternikSchnirelmann category of a closed 1-form and to the existence of a nonsingular closed 1-form in a given cohomology class on a high-dimensional closed manifold. To S.P. Novikov on the occasion of his 70-th birthday

Замкнутые 1-формы в топологии и геометрической теории групп