The universal surface bundle over the Torelli space has no sections

Abstract

For $$g>3$$ , we give two proofs of the fact that the Birman exact sequence for the Torelli group $$\begin{aligned} 1\rightarrow \pi _1(S_g)\rightarrow {\mathcal I}_{g,1}\rightarrow {\mathcal I}_g\rightarrow 1 \end{aligned}$$ does not split. This result was claimed by Mess (Unit tangent bundle subgroups of the mapping class groups, MSRI Pre-print, 1990), but his proof has a critical and unrepairable error which will be discussed in the introduction. Let $${\mathcal U\mathcal I}_{g,n}\xrightarrow {Tu'_{g,n}} {\mathcal B} {\mathcal I}_{g,n}$$ (resp. $${\mathcal U}{\mathcal P}{\mathcal I}_{g,n}\xrightarrow {Tu_{g,n}}{\mathcal B}{\mathcal P}{\mathcal I}_{g,n}$$ ) 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.