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 [Formula: see text] ordered points on a surface [Formula: see text] 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[Formula: see text] be the space of ordered [Formula: see text]-tuple of distinct points in [Formula: see text]. Let [Formula: see text] be the map given by [Formula: see text]. We classify all continuous sections of [Formula: see text] up to homotopy by proving the following: (1) If [Formula: see text] and [Formula: see text], any section of [Formula: see text] is either “adding a point at infinity” or “adding a point near [Formula: see text]”. (We define these two terms in Sec. 2.1; whether we can define “adding a point near [Formula: see text]” or “adding a point at infinity” depends in a delicate way on properties of [Formula: see text].) (2) If [Formula: see text] a [Formula: see text]-sphere and [Formula: see text], any section of [Formula: see text] is “adding a point near [Formula: see text]”; if [Formula: see text] and [Formula: see text], the bundle [Formula: see text] does not have a section. (We define this term in Sec. 3.2). (3) If [Formula: see text] a surface of genus [Formula: see text] and for [Formula: see text], 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 [Formula: see text] does not have a section.