Controlled connectivity of closed 1–forms
Dirk Schütz · Algebraic & Geometric Topology · 2002
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 ≥ 5 we can realize it by a closed 1form which is Morse without critical points of index 0, 1, n -1 and n.If n = dim M ≥ 6 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 ≤ 1 and i ≥ n -1 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.