The mapping class group of connect sums of S2 x S1

Brendle, T. , Broaddus, N. and Putman, A. (2022) The mapping class group of connect sums of S2 x S1. Transactions of the American Mathematical Society, (Accepted for Publication)

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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:Accepted for Publication
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
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