Spaces of Topological Complexity One

TL;DR

This paper characterizes spaces with minimal topological complexity, showing that a space with TC=1 is homotopy equivalent to an odd-dimensional sphere, and explores similar properties for higher topological complexities.

Contribution

It establishes a homotopy equivalence between spaces with TC=1 and odd-dimensional spheres, and investigates analogous results for higher topological complexities.

Findings

Spaces with TC=1 are homotopy equivalent to odd-dimensional spheres.

Spaces with higher topological complexity TC_n(X) = n-1 exhibit similar properties.

Provides partial results extending the characterization to higher topological complexities.

Abstract

We prove that a space whose topological complexity equals 1 is homotopy equivalent to some odd-dimensional sphere. We prove a similar result, although not in complete generality, for spaces X whose higher topological complexity TC_n(X) is as low as possible, namely n-1.