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.
Notes on file formats
Dirk Schuetz
Department of Mathematics, University College Dublin
Belfield, Dublin 4, Ireland
Email: dirk.schuetz@ucd.ie
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