LS-category and topological complexity of several families of fibre bundles

TL;DR

This paper investigates upper bounds and exact values of topological complexity and LS-category for various fibre bundles, including Klein bottles and generalized projective spaces, using cohomology and equivariant methods.

Contribution

It provides new tight bounds and exact calculations for topological complexity and LS-category of specific fibre bundles and their families, expanding understanding of these invariants.

Findings

Exact topological complexity of 3D Klein bottle computed.

Upper bounds for topological complexity of n-dimensional Klein bottles established.

LS-category and topological complexity of generalized projective product spaces determined.

Abstract

In this paper, we study upper bounds for the topological complexity of the total spaces of some classes of fibre bundles. We calculate a tight upper bound for the topological complexity of an $n$-dimensional Klein bottle. We also compute the exact value of the topological complexity of $3$-dimensional Klein bottle. We describe the cohomology rings of several classes of generalized projective product spaces with $\mathbb{Z}_2$-coefficients. Then we study the LS-category and topological complexity of infinite families of generalized projective product spaces. We reckon the exact value of these invariants in many specific cases. We calculate the equivariant LS-category and equivariant topological complexity of several product spaces equipped with $\mathbb{Z}_2$-action.