Closed 1-Forms in Topology and Geometric Group Theory

TL;DR

This paper explores the relationship between closed 1-forms in topology and geometric group theory, extending Sigma invariants to CW-complexes and examining their applications to manifold topology and group properties.

Contribution

It extends Sigma invariants to finite CW-complexes and establishes their connections with topological and group-theoretic properties, including applications to manifold invariants.

Findings

Sigma invariants relate to finiteness properties of covering spaces

Extensions of Sigma invariants to CW-complexes are established

Applications to Lusternik-Schnirelmann category and closed 1-forms

Abstract

In this article we describe relations of the topology of closed 1-forms to the group theoretic invariants of Bieri-Neumann-Strebel-Renz. Starting with a survey, we extend these Sigma invariants to finite CW- complexes and show that many properties of the group theoretic version have analogous statements. In particular we show the relation between Sigma invariants and finiteness properties of certain infinite covering spaces. We also discuss applications of these invariants to the Lusternik- Schnirelmann category of a closed 1-form and to the existence of a non- singular closed 1-form in a given cohomology class on a high-dimensional closed manifold.