Closed 1-form on topological spaces, and Lusternik-Schnirelman type Category

TL;DR

This paper extends Lusternik-Schnirelmann theory to continuous closed 1-forms on CW-complexes, relating zero-points to topological category and dynamical systems, generalizing prior smooth form results.

Contribution

It introduces a generalized theory for continuous closed 1-forms, linking zero-points with cohomological category and dynamical systems, expanding on Farber's smooth form framework.

Findings

Established a relation between zero-points and cohomological category.

Generalized Lusternik-Schnirelmann theory to continuous closed 1-forms.

Connected zero-points to homoclinic cycles in dynamical systems.

Abstract

This article deals with a continuous closed 1-form defined on a CW-complex. In particular, we show Lusternik-Schnirelmann type theory on continuous closed 1-forms which is related to gradient-like flows. M.Farber defined a continuous closed 1-form and a category with a respect to a cohomology class and constructed a Lusternik-Schnirelman Theory for smooth closed 1-forms. We denote about a definition of a zero-point of continuous closed 1-form and generalize this theory to continuous closed 1-forms. It shows a relation between the number of a zero-point of continous closed 1-form and the category with a respect to a cohomology class, through a dynamical system on a original space, homoclinic cycle.