Section problems for configuration spaces of surfaces

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.

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