Topological complexity of motion planning and Massey products

TL;DR

This paper uses Massey products to improve lower bounds on the topological complexity of spaces, providing new examples where the complexity exceeds previous estimates based on cup-length.

Contribution

It introduces a novel application of Massey products to derive sharper lower bounds for the Schwarz genus and topological complexity of non-formal spaces.

Findings

Massey products yield better lower bounds for topological complexity.

Examples of non-formal spaces with higher topological complexity than zero-divisors cup-length.

Demonstration of the limitations of previous bounds in certain spaces.

Abstract

We employ Massey products to find sharper lower bounds for the Schwarz genus of a fibration than those previously known. In particular we give examples of non-formal spaces $X$ for which the topological complexity $\TC(X)$ (defined to be the genus of the free path fibration on $X$) is greater than the zero-divisors cup-length plus one.