Higher (equivariant) topological complexity of Milnor manifolds

TL;DR

This paper investigates the higher topological complexity of Milnor manifolds, providing bounds and exact values, and enhances understanding of their equivariant topological complexity under group actions.

Contribution

It establishes precise bounds and exact values for the higher topological complexity of Milnor manifolds and improves bounds on their equivariant topological complexity.

Findings

Derived exact values for the higher topological complexity of many Milnor manifolds.

Established sharper bounds on the equivariant topological complexity with group actions.

Provided improved upper bounds on the equivariant topological complexity of Milnor manifolds.

Abstract

J. Milnor introduced a specific class of codimension-$1$ submanifolds in the product of projective spaces, known as Milnor manifolds. This paper establishes precise bounds on the higher topological complexity of these manifolds and provides exact values for this invariant for numerous Milnor manifolds. Furthermore, we improve the upper bounds on the higher equivariant topological complexity. As an application, we obtain sharper bounds on the higher equivariant topological complexity of Milnor manifolds with free $\mathbb{Z}_2$ and $S^1$-actions.