The Birman exact sequence does not virtually split

This paper answers a basic question about the Birman exact sequence in the theory of mapping class groups. We prove that the Birman exact sequence does not admit a section over any subgroup Γ contained in the Torelli group with finite index. A fortiori this implies that there is no multi-section for the universal surface bundle with Torelli monodromy. This theorem was announced in a 1990 preprint of G. Mess, but an error was uncovered and described in a recent paper of the first author