Closed 1-forms in topology and dynamics

TL;DR

This survey explores how closed 1-forms are used in topology and dynamics to analyze flow properties and manifold invariants through homotopical and cohomological methods.

Contribution

It introduces a Lusternik-Schnirelmann type theory for closed one-forms and discusses recent advances in cohomological invariants related to flows.

Findings

Development of a Lusternik-Schnirelmann type theory for closed 1-forms

Analysis of the focusing effect for flows using closed 1-forms

Explicit computation of cohomological invariants in specific examples

Abstract

This article surveys recent progress of results in topology and dynamics based on techniques of closed one-forms. Our approach allows us to draw conclusions about properties of flows by studying homotopical and cohomological features of manifolds. More specifically we describe a Lusternik - Schnirelmann type theory for closed one-forms, the focusing effect for flows and the theory of Lyapunov one-forms. We also discuss recent results about cohomological treatment of the invariants cat(X, \xi) and cat^1(X, \xi) and their explicit computation in certain examples.