Algebraic and Geometric Topology 2 (2002), paper no. 9, pages 171-217.

Controlled connectivity of closed 1-forms

Dirk Schuetz

Abstract. We discuss controlled connectivity properties of closed 1-forms and their cohomology classes and relate them to the simple homotopy type of the Novikov complex. The degree of controlled connectivity of a closed 1-form depends only on positive multiples of its cohomology class and is related to the Bieri-Neumann-Strebel-Renz invariant. It is also related to the Morse theory of closed 1-forms. Given a controlled 0-connected cohomology class on a manifold M with n = dim M > 4 we can realize it by a closed 1-form which is Morse without critical points of index 0, 1, n-1 and n. If n = dim M > 5 and the cohomology class is controlled 1-connected we can approximately realize any chain complex D_* with the simple homotopy type of the Novikov complex and with D_i=0 for i < 2 and i > n-2 as the Novikov complex of a closed 1-form. This reduces the problem of finding a closed 1-form with a minimal number of critical points to a purely algebraic problem.

Keywords. Controlled connectivity, closed 1-forms, Novikov complex

AMS subject classification. Primary: 57R70. Secondary: 20J05, 57R19.

DOI: 10.2140/agt.2002.2.171

E-print: arXiv:math.DG/0203283

Submitted: 3 December 2001. Accepted: 8 March 2002. Published: 26 March 2002.

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Dirk Schuetz
Department of Mathematics, University College Dublin
Belfield, Dublin 4, Ireland

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