The universal n-pointed surface bundle only has n sections

Abstract

The classifying space BDiff[Formula: see text] of the orientation-preserving diffeomorphism group of a surface [Formula: see text] of genus [Formula: see text] fixing [Formula: see text] points pointwise has a universal bundle [Formula: see text] The [Formula: see text] fixed points provide [Formula: see text] sections [Formula: see text] of [Formula: see text]. In this paper we prove a conjecture of R. Hain that any section of [Formula: see text] is homotopic to some [Formula: see text]. Let [Formula: see text] be the space of ordered [Formula: see text]-tuple of distinct points on [Formula: see text]. As part of the proof of Hain’s conjecture, we prove a result of independent interest: any surjective homomorphism [Formula: see text] is equal to one of the forgetful homomorphisms [Formula: see text], possibly post-composed with an automorphism of [Formula: see text]. We also classify sections of the universal hyperelliptic surface bundle.