Cup products and the higher topological complexity of configuration spaces of the circle with two anchored points

TL;DR

This paper develops a method using discrete Morse theory to compute cup products in the anchored configuration space of a circle with two points, enabling sharper bounds for higher topological complexity.

Contribution

It introduces a novel approach to compute cup products in specific configuration spaces, improving bounds for topological complexity.

Findings

Computed cup products in the configuration space of the circle with two anchored points.

Provided sharp bounds for higher topological complexity $TC_s$ for large $s$.

Demonstrated the effectiveness of discrete Morse theory in this context.

Abstract

In this paper we show how to compute cup products in the anchored configuration space of the circle with two anchored points using discrete Morse theory. Knowing how to compute cup products allows us to obtain bounds for the (higher) topological complexity $TC_s$, which are sharp for a sufficiently large value of $s$.