Homology, Homotopy and Applications

Volume 26 (2024)

Number 2

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

Pages: 273 – 295

DOI: https://dx.doi.org/10.4310/HHA.2024.v26.n2.a14

Authors

Navnath Daundkar (Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, Powai, India)

Soumen Sarkar (Department of Mathematics, Indian Institute of Technology Madras, Chennai, Tamil Nadu, India)

Abstract

In this paper, we study the 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 a $3$-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.

Keywords

LS-category, topological complexity, fibre bundle, projective product space

2010 Mathematics Subject Classification

55M30, 55N10, 55R10

Received 11 July 2023

Accepted 7 November 2023

Published 9 October 2024