The universal surface bundle over the Torelli space has no sections

For $g>3$, we give two proofs of the fact that the \emph{Birman exact sequence} for the Torelli group $$1\to {\pi }_1(S_g)\to {\mathcal{I}}_{g,1}\to {\mathcal{I}}_g\to 1$$ 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\mathrm{.}${\cal UPI}{g,n}\xrightarrow{Tu_{g,n}}{\cal BPI}_{g,n}$) denote the universal surface bundle over the Torelli space fixing $n$ points as a set (resp. pointwise). We also deduce that $T{u}_{g,n}^{\mathrm{\prime }}$ has no sections when $n>1$ and that $T{u}_{g,n}$ has precisely $n$ distinct sections for $n\ge 0$ up to homotopy.