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Topological complexity of unordered configuration spaces of surfaces

Andrea Bianchi and David Recio-Mitter

Algebraic & Geometric Topology 19 (2019) 1359–1384

We determine the topological complexity of unordered configuration spaces on almost all punctured surfaces (both orientable and nonorientable). 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.

topological complexity, braid groups, robotics, surfaces, algebraic topology
Mathematical Subject Classification 2010
Primary: 55M99, 55P20
Secondary: 20J06, 55M30, 68T40
Received: 23 February 2018
Revised: 21 October 2018
Accepted: 1 November 2018
Published: 21 May 2019
Andrea Bianchi
Mathematics Institute
University of Bonn
David Recio-Mitter
Institute of Mathematics
University of Aberdeen
United Kingdom
Department of Mathematics
Lehigh University
Bethlehem, PA
United States