Some Aspects of Equivariant LS-category

TL;DR

This paper investigates bounds for LS-category and equivariant LS-category, computes these invariants for torus manifolds, and discusses their behavior under products, providing counterexamples to previous claims.

Contribution

It provides new bounds, explicit computations for torus manifolds, and counterexamples regarding the behavior of equivariant LS-category under products.

Findings

Computed LS-category and equivariant LS-category for torus manifolds.

Identified sharpness of bounds through examples.

Presented counterexamples to previous results on product spaces.

Abstract

We study the lower bounds and upper bounds for LS-category and equivariant LS-category. In particular we compute both invariants for torus manifolds. There are some examples to show the sharpness of conditions in the theorems. Moreover the equivariant LS-category of the product space is discussed and counterexamples of some previous results are given.