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

TL;DR

This paper proves that the mapping class group of connected sums of $S^2 imes S^1$ splits as a semidirect product, clarifying its structure and simplifying previous proofs.

Contribution

It establishes that the extension of the mapping class group by sphere twists splits, providing an explicit semidirect product structure and simplifying Laudenbach's original proof.

Findings

The extension of $ ext{Mod}(M_n)$ splits as a semidirect product.

$ ext{Out}(F_n)$ acts on the sphere twist subgroup via the dual of a natural surjection.

The techniques simplify the identification of the twist subgroup with $( ext{Z}/2)^n$.

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 $\text{Mod}(M_n)$ is an extension of $\text{Out}(F_n)$ by a group $(\mathbb{Z}/2)^n$ generated by sphere twists. We prove that this extension splits, so $\text{Mod}(M_n)$ is the semidirect product of $\text{Out}(F_n)$ by $(\mathbb{Z}/2)^n$, which $\text{Out}(F_n)$ acts on via the dual of the natural surjection $\text{Out}(F_n) \rightarrow \text{GL}_n(\mathbb{Z}/2)$. Our splitting takes $\text{Out}(F_n)$ to the subgroup of $\text{Mod}(M_n)$ 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 $(\mathbb{Z}/2)^n$.