A localization formula for equivariant Lyusternik-Schnirelmann category

TL;DR

This paper introduces a localization formula for the equivariant LS-category of smooth manifolds with Lie group actions, enabling calculations based on minimal orbit-type strata, with applications to symplectic toric manifolds.

Contribution

It provides a new localization formula for equivariant LS-category in the presence of specific covers, applicable to symplectic toric manifolds and general Lie group actions.

Findings

The localization formula computes equivariant LS-category using minimal orbit-type strata.

Such covers exist on all symplectic toric manifolds.

The known LS-category equals the number of fixed points in symplectic toric manifolds.

Abstract

The LS-category of a topological space is a numerical homotopy invariant, introduced originally in a course on the global calculus of variations by Lyusternik and Schnirelmann, to estimate the number of critical points of a smooth function. When the topological space is a smooth manifold equipped with a proper action of a Lie group, we give a localization formula to calculate the equivariant analogue of this category in terms of the minimal orbit-type strata. The formula holds provided that the manifold admits a specific cover. We show that such a cover exists on every symplectic toric manifold. The known result stating that the LS-category of a symplectic toric manifold is equal to the number of fixed points of the torus action follows from our localization formula.