Controlled connectivity of closed 1–forms

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 $n=dimM\ge 5$ we can realize it by a closed 1–form which is Morse without critical points of index 0, 1, $n-1$ and $n$. If $n=dimM\ge 6$ and the cohomology class is controlled 1–connected we can approximately realize any chain complex $D_{\ast }$ with the simple homotopy type of the Novikov complex and with $D_i=0$ for $i\le 1$ and $i\ge n-1$ 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.