Chen, Lei
(2021)
*Section problems for configuration spaces of surfaces.*
Journal of Topology and Analysis, 13
(2).
pp. 469-497.
ISSN 1793-5253.
doi:10.1142/s1793525320500181.
https://resolver.caltech.edu/CaltechAUTHORS:20210729-213026874

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## Abstract

In this paper, we give a close-to-sharp answer to the basic questions: When is there a continuous way to add a point to a configuration of n ordered points on a surface S of finite type so that all the points are still distinct? When this is possible, what are all the ways to do it? More precisely, let PConf_n(S) be the space of ordered n-tuple of distinct points in S. Let f_n(S):PConf_(n+1)(S)→PConf_n(S) be the map given by f_n(x₀,x₁,…,x_n):=(x₁,…,x_n). We classify all continuous sections of f_n up to homotopy by proving the following: (1) If S=R² and n > 3, any section of f_n(S) is either “adding a point at infinity” or “adding a point near x_k”. (We define these two terms in Sec. 2.1; whether we can define “adding a point near x_k” or “adding a point at infinity” depends in a delicate way on properties of S.) (2) If S=S² a 2-sphere and n > 4, any section of f_n(S) is “adding a point near x_k”; if S=S² and n=2, the bundle f_n(S) does not have a section. (We define this term in Sec. 3.2). (3) If S=S_g a surface of genus g > 1and for n > 1, we give an easy proof of [D. L. Gonçalves and J. Guaschi, On the structure of surface pure braid groups, J. Pure Appl. Algebra 182 (2003) 33–64, Theorem 2] that the bundle f_n(S) does not have a section.

Item Type: | Article | |||||||||
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Additional Information: | © 2021 World Scientific Publishing Co Pte Ltd. Received 10 March 2019; Accepted 17 May 2019; Published: 10 July 2019. | |||||||||

Subject Keywords: | Braid groups; section problems | |||||||||

Issue or Number: | 2 | |||||||||

Classification Code: | AMSC: 20F36, 20F34, 57M27 | |||||||||

DOI: | 10.1142/s1793525320500181 | |||||||||

Record Number: | CaltechAUTHORS:20210729-213026874 | |||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20210729-213026874 | |||||||||

Official Citation: | Section problems for configuration spaces of surfaces. Lei Chen. Journal of Topology and Analysis 2021 13:02, 469-497; DOI: 10.1142/s1793525320500181 | |||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||

ID Code: | 110081 | |||||||||

Collection: | CaltechAUTHORS | |||||||||

Deposited By: | Tony Diaz | |||||||||

Deposited On: | 02 Aug 2021 18:44 | |||||||||

Last Modified: | 02 Aug 2021 18:44 |

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