The mapping class group of connect sums of S2 x S1

Brendle, T. , Broaddus, N. and Putman, A. (2023) The mapping class group of connect sums of S2 x S1. Transactions of the American Mathematical Society, 376(4), pp. 2557-2572. (doi: 10.1090/tran/8758)

Abstract

Let Mn be the connect sum of n copies of S2×S1. A classical theorem of Laudenbach says that the mapping class group Mod(Mn) is an extension of Out(Fn) by a group (ℤ/2)n generated by sphere twists. We prove that this extension splits, so Mod(Mn) is the semidirect product of Out(Fn) by (ℤ/2)n, which Out(Fn) acts on via the dual of the natural surjection Out(Fn)→GLn(ℤ/2). Our splitting takes Out(Fn) to the subgroup of Mod(Mn) consisting of mapping classes that fix the homotopy class of a trivialization of the tangent bundle of Mn. Our techniques also simplify various aspects of Laudenbach's original proof, including the identification of the twist subgroup with (ℤ/2)n.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Brendle, Professor Tara
• Authors: Brendle, T., Broaddus, N., and Putman, A.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Transactions of the American Mathematical Society
• Publisher: American Mathematical Society
• ISSN: 0002-9947
• ISSN (Online): 1088-6850
• Published Online: 24 January 2023
• Copyright Holders: Copyright © 2023 American Mathematical Society
• First Published: First published in Transactions of the American Mathematical Society 376(4):2557-2572
• Publisher Policy: Reproduced in accordance with the publisher copyright policy

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