## The mapping class group of connect sums of $S^2 \times S^1$

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Tara Brendle, Nathan Broaddus and Andrew Putman
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## Abstract:

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 $(\mathbb {Z}/2)^n$.## References

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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: tara.brendle@glasgow.ac.uk
**Nathan Broaddus**- Affiliation: Department of Mathematics, Ohio State University, 231 W. 18th Ave., Columbus, Ohio 43210
- ORCID: 0000-0003-2054-2627
- Email: broaddus.9@osu.edu
**Andrew Putman**- Affiliation: Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, Indiana 46556
- MR Author ID: 794071
- Email: andyp@nd.edu
- 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: https://doi.org/10.1090/tran/8758
- MathSciNet review: 4557874