论文标题
句柄小组和第二个约翰逊同态的图像
The handlebody group and the images of the second Johnson homomorphism
论文作者
论文摘要
给定一个方向界定手柄机构的表面，我们研究其映射类组的子组定义为手柄组的相交和Johnson过滤的第二项：$ \ MATHCAL {A} \ CAP J_2 $。我们介绍了两个类似痕量的操作员，灵感来自莫里塔（Morita）的痕迹，并表明他们的内核与第二个约翰逊同构型$τ_2$ $ j_2 $和$ \ MATHCAL {A} \ CAP J_2 $的图像相吻合。特别是，我们对莱文询问的问题的否定回答是关于$τ_2的代数描述（\ Mathcal {a} \ cap j_2）$。通过相同的技术，对于$ s^3 $中的Heegaard表面，我们还通过$τ_2$计算了Goeritz Group $ \ MATHCAL {G} $与$ J_2 $的相交的$τ_2$。
Given an oriented surface bounding a handlebody, we study the subgroup of its mapping class group defined as the intersection of the handlebody group and the second term of the Johnson filtration: $\mathcal{A} \cap J_2$. We introduce two trace-like operators, inspired by Morita's trace, and show that their kernels coincide with the images by the second Johnson homomorphism $τ_2$ of $J_2$ and $\mathcal{A} \cap J_2$, respectively. In particular, we answer by the negative to a question asked by Levine about an algebraic description of $τ_2(\mathcal{A} \cap J_2)$. By the same techniques, and for a Heegaard surface in $S^3$, we also compute the image by $τ_2$ of the intersection of the Goeritz group $\mathcal{G}$ with $J_2$.
