Mathematical Research Letters
Volume 28 (2021)
The Birman exact sequence does not virtually split
Pages: 383 – 413
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 $\Gamma$ 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.
N.S. is supported by the National Science Foundation under Award No. DMS-1703181.
Received 8 January 2019
Accepted 24 February 2020
Published 13 May 2021