Section problems for configurations of points on the Riemann sphere

Chen, Lei; Salter, Nick

doi:10.2140/agt.2020.20.3047

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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 $n \neq 5$ , for which $m$ can one continuously add $m$ points to a configuration of $n$ points? For $n \geq 6$ , we find that $m$ must be divisible by $n (n - 1) (n - 2)$ , 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

References

Publication

Received: 6 June 2019

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

Volume 20, issue 6 (2020)