The universal surface bundle over the Torelli space has no sections

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.