Mathematical Research Letters

Volume 28 (2021)

Number 2

The Birman exact sequence does not virtually split

Pages: 383 – 413



Lei Chen (Department of Mathematics, California Institute of Technology, Pasadena, Cal., U.S.A.)

Nick Salter (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)


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