On the relative Lusternik-Schnirelmann category with respect to a closed 1-form

TL;DR

This paper introduces a homotopy invariant generalizing Lusternik-Schnirelmann category for pairs of CW complexes with respect to a closed 1-form, linking topology with critical points and homoclinic cycles.

Contribution

It extends the Lusternik-Schnirelmann category to a new setting involving closed 1-forms, establishing connections with critical points and providing lower bounds.

Findings

Established the connection with the original Lusternik-Schnirelmann category.

Derived analogous results on critical points and homoclinic cycles.

Provided a cuplength lower bound for the new invariant.

Abstract

In this article we study a homotopy invariant cat(X,B,\xi) on a pair of finite CW complexes with respect to a continuous closed 1-form. This is a generalisation of a Lusternik-Schnirelmann category developed by Farber, studying the topology of a closed 1-form. The article establishes the connection with the original notion and obtains analogous results on critical points and homoclinic cycles. We also provide a similar cuplength lower bound for cat(X,B,\xi).