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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The mapping class group of connect sums of $S^2 \times S^1$
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by Tara Brendle, Nathan Broaddus and Andrew Putman HTML | PDF
Trans. Amer. Math. Soc. 376 (2023), 2557-2572 Request permission


Let $M_n$ be the connect sum of $n$ copies of $S^2 \times S^1$. A classical theorem of Laudenbach says that the mapping class group $\operatorname {Mod}(M_n)$ is an extension of $\operatorname {Out}(F_n)$ by a group $(\mathbb {Z}/2)^n$ generated by sphere twists. We prove that this extension splits, so $\operatorname {Mod}(M_n)$ is the semidirect product of $\operatorname {Out}(F_n)$ by $(\mathbb {Z}/2)^n$, which $\operatorname {Out}(F_n)$ acts on via the dual of the natural surjection $\operatorname {Out}(F_n) \rightarrow GL_n(\mathbb {Z}/2)$. Our splitting takes $\operatorname {Out}(F_n)$ to the subgroup of $\operatorname {Mod}(M_n)$ consisting of mapping classes that fix the homotopy class of a trivialization of the tangent bundle of $M_n$. Our techniques also simplify various aspects of Laudenbach’s original proof, including the identification of the twist subgroup with $(\mathbb {Z}/2)^n$.
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Additional Information
  • Tara Brendle
  • Affiliation: School of Mathematics & Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, UK
  • MR Author ID: 683339
  • ORCID: 0000-0002-9594-8229
  • Email:
  • Nathan Broaddus
  • Affiliation: Department of Mathematics, Ohio State University, 231 W. 18th Ave., Columbus, Ohio 43210
  • ORCID: 0000-0003-2054-2627
  • Email:
  • Andrew Putman
  • Affiliation: Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, Indiana 46556
  • MR Author ID: 794071
  • Email:
  • Received by editor(s): January 12, 2022
  • Received by editor(s) in revised form: May 24, 2022
  • Published electronically: January 24, 2023
  • Additional Notes: Supported in part by NSF grant DMS-1811210
  • © Copyright 2023 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 2557-2572
  • MSC (2020): Primary 57S05, 20F34; Secondary 20E36
  • DOI:
  • MathSciNet review: 4557874