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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

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 n5, for which m can one continuously add m points to a configuration of n points? For n 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.

spherical braid group, configuration space, section, canonical reduction system
Mathematical Subject Classification 2010
Primary: 20F36, 55S40
Received: 6 June 2019
Revised: 26 October 2019
Accepted: 24 November 2019
Published: 8 December 2020
Lei Chen
Department of Mathematics
Pasadena, CA
United States
Nick Salter
Department of Mathematics
Columbia University
New York, NY
United States