The universal surface bundle over the Torelli space has no sections

TL;DR

This paper proves that the universal surface bundle over the Torelli space does not admit sections for genus greater than 3, clarifying errors in previous claims and establishing the uniqueness of sections in certain cases.

Contribution

It provides two proofs that the Birman exact sequence for the Torelli group does not split, correcting a prior claim and analyzing the section properties of universal surface bundles.

Findings

The Birman exact sequence for the Torelli group does not split for genus > 3.

The universal surface bundle over Torelli space has no sections when fixing more than one point.

The bundle admits exactly n sections up to homotopy for n points.

Abstract

For $g>3$, we give two proofs of the fact that the \emph{Birman exact sequence} for the Torelli group \[ 1\to \pi_1(S_g)\to {\cal I}_{g,1}\to {\cal I}_g\to 1 \] does not split. This result was claimed by G. Mess in \cite{mess1990unit}, but his proof has a critical and unrepairable error which will be discussed in the introduction. Let ${\cal UI}_{g,n}\xrightarrow{Tu'_{g,n}} {\cal BI}_{g,n}$ (resp. ${\cal UPI}_{g,n}\xrightarrow{Tu_{g,n}}{\cal BPI}_{g,n}$) denote the universal surface bundle over the Torelli space fixing $n$ points as a set (resp. pointwise). We also deduce that $Tu'_{g,n}$ has no sections when $n>1$ and that $Tu_{g,n}$ has precisely $n$ distinct sections for $n\ge 0$ up to homotopy.