Title: 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 $n$-tuples of distinct points in $S$. Let
$f_n(S): \text{PConf}_{n+1}(S) \to \text{PConf}_n(S)$ be the map given by
$f_n(x_0,x_1,\ldots,x_n):=(x_1,\ldots,x_n)$. We will classify all continuous
sections of $f_n$ by proving:
1. If $S=\mathbb{R}^2$ 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 Section 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 $S=S^2$ a $2$-sphere and $n>4$, any section of $f_{n}(S)$ is "adding a
point near $x_k$", if $S=S^2$ and $n=2$, the bundle $f_n(S)$ does not have a
section. (We define this term in Section 3.2)
3. If $S=S_g$ a surface of genus $g>1$ and for $n>1$, the bundle $f_{n}(S)$
does not have a section.