Topological complexity of unordered configuration spaces of surfaces

TL;DR

This paper calculates the topological complexity of unordered configuration spaces on various punctured surfaces, providing new bounds and applying advanced methods to improve understanding of these aspherical spaces.

Contribution

It determines the topological complexity for most punctured surfaces and refines bounds for closed aspherical surfaces using novel applications of existing methods.

Findings

Topological complexity of most punctured surfaces is determined.

Bounds for closed aspherical surfaces are significantly improved.

Methods from 2015 are effectively applied to new classes of spaces.

Abstract

We determine the topological complexity of unordered configuration spaces on almost all punctured surfaces (both orientable and non-orientable). We also give improved bounds for the topological complexity of unordered configuration spaces on all aspherical closed surfaces, reducing it to three possible values. The main methods used in the proofs were developed in 2015 by Grant, Lupton and Oprea to give bounds for the topological complexity of aspherical spaces. As such this paper is also part of the current effort to study the topological complexity of aspherical spaces and it presents many further examples where these methods strongly improve upon the lower bounds given by zero-divisor cup-length.