论文标题
$ s^2 \ times s^1 $的映射类连接总和
The mapping class group of connect sums of $S^2 \times S^1$
论文作者
论文摘要
令$ m_n $为$ s^2 \ times s^1 $的$ n $副本的连接总和。 Laudenbach的经典定理说，映射类组$ \ text {mod}（m_n）$是$ \ text {out}（f_n）$的扩展，该组$（\ mathbb {z}/2）^n $ feed twists twists twists twists。我们证明了此扩展的拆分，因此$ \ text {mod}（m_n）$是$ \ text {out}（out}（f_n）$ by $（\ mathbb {z}/2）^n $的半领产品，$ \ \ \ \ text {out text}（out}（f_n）$ py n of the of text of text $ \ right text { \ text {gl} _n（\ mathbb {z}/2）$。我们的拆分将$ \ text {out}（f_n）$带到$ \ text {mod}（m_n）$的子组中，由映射类组成，这些类固定了$ m_n $的切线束的琐事类别的同型类别。我们的技术还简化了Laudenbach原始证明的各个方面，包括用$（\ Mathbb {z}/2）^n $识别Twist子组。
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 $(\mathbb{Z}/2)^n$.
