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