Topological Complexity of H-Spaces

TL;DR

This paper establishes a precise relationship between higher topological complexity and Lusternik-Schnirelmann category for H-spaces, providing new insights into their topological properties and bounds.

Contribution

It proves that for H-spaces, the higher topological complexity equals the Lusternik-Schnirelmann category of their n-fold product, and extends this to a broader inequality involving spaces with actions.

Findings

TC_{n+1}(X) = cat(X^n) for H-spaces

Provides an upper bound for TC_{n+1}(X) in a generalized setting

Extends known relationships between topological complexity and category

Abstract

Let X be a (not-necessarily homotopy-associative) H-space. We show that TC_{n+1}(X) = cat(X^n), for n >= 1, where TC_{n+1}(-) denotes the so-called higher topological complexity introduced by Rudyak, and cat(-) denotes the Lusternik-Schnirelmann category. We also generalize this equality to an inequality, which gives an upper bound for TC_{n+1}(X), in the setting of a space Y acting on X.