Volume 20, issue 6 (2020)

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Section problems for configurations of points on the Riemann sphere

Lei Chen and Nick Salter

Algebraic & Geometric Topology 20 (2020) 3047–3082
DOI: 10.2140/agt.2020.20.3047
Abstract

We prove a suite of results concerning the problem of adding $m$ distinct new points to a configuration of $n$ distinct points on the Riemann sphere, such that the new points depend continuously on the old. Altogether, these results provide a complete answer to the following question: given $n\ne 5$, for which $m$ can one continuously add $m$ points to a configuration of $n$ points? For $n\ge 6$, we find that $m$ must be divisible by $n\left(n-1\right)\left(n-2\right)$, and we provide a construction based on the idea of cabling of braids. For $n=3,4$, we give some exceptional constructions based on the theory of elliptic curves.

Keywords
spherical braid group, configuration space, section, canonical reduction system
Mathematical Subject Classification 2010
Primary: 20F36, 55S40
Publication
Revised: 26 October 2019
Accepted: 24 November 2019
Published: 8 December 2020
Authors
 Lei Chen Department of Mathematics Caltech Pasadena, CA United States Nick Salter Department of Mathematics Columbia University New York, NY United States