Closed 1-forms in topology and geometric group theory

M. Farber, R. Geoghegan, D. Schütz

Research output: Contribution to journalArticlepeer-review


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. Bibliography: 32 titles.

Original languageEnglish
Pages (from-to)143-172
Number of pages30
JournalRussian Mathematical Surveys
Issue number1
StatePublished - 2010
Externally publishedYes


  • Lusternik-Schnirelmann category
  • Movability of homology classes
  • Novikov ring
  • Sigma invariants


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