The universal $n$-pointed surface bundle only has $n$ sections

TL;DR

This paper proves that the only sections of the universal surface bundle with fixed points are the obvious ones, confirming a conjecture and revealing the structure of sections in these bundles.

Contribution

It proves Hain's conjecture that all sections are homotopic to fixed point sections and characterizes homomorphisms from configuration space groups to surface groups.

Findings

Any section of the universal bundle is homotopic to a fixed point section.

Homomorphisms from configuration space fundamental groups to surface groups are essentially forgetful maps.

The universal surface bundle fixing points as a set has no sections.

Abstract

The classifying space BDiff$(S_{g,n})$ of the orientation-preserving diffeomorphism group of the surface $S_{g,n}$ of genus $g>1$ with $n$ ordered marked points has a universal bundle \[ S_g \to \text{UDiff}(S_{g,n})\xrightarrow{\pi}\text{BDiff}(S_{g,n}). \] The fixed $n$ points provide $n$ sections $s_i$ of $\pi$. In this paper we prove a conjecture of R. Hain that any section of $\pi$ is homotopic to some $s_i$. Let $\text{PConf}_n(S_g)$ be the ordered $n$-tuples of distinct points on $S_g$. As part of the proof, we prove a result of independent interest: any surjective homomorphism $\pi_1(\text{PConf}_n(S_g))\to \pi_1(S_g)$ is equal to one of the forgetful maps $\{p_i:\pi_1(\text{PConf}_n(S_g))\to \pi_1(S_g)\}$, possibly post-composed with an automorphism of $\pi_1(S_g)$. Using similar arguments, we then show that the universal surface bundle that fixes $n$ points as a set does not have any section.