Controlled connectivity of closed 1-forms

Details Controlled connectivity of closed 1-forms Controlled connectivity of closed 1-forms

2002-03-27

arxiv.org/abs/math/0203283v1

Algebr. Geom. Topol. 2 (2002) 171-217

Mathematics

Differential Geometry

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-9.abs.html

Scientific paper

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.

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