标题： 可定向环面在闭可定向曲面上的分类 显示英文标题

标题： Classification of orientable torus bundles over closed orientable surfaces

摘要： 设$g$为一个非负整数，$\Sigma _g$为一个亏格为$g$的闭合可定向曲面，$\mathcal{M}_g$为其映射类群。 我们根据$\mathcal{M}_g$在$\pi _1(\Sigma _g)$上的作用，对所有的群同态$\pi _1(\Sigma _g)\to G$进行分类，在以下情况下；(1)$G=PSL(2;\mathbb{Z})$，(2)$G=SL(2;\mathbb{Z})$。作为情况(2)的应用，我们完全根据丛同构对闭合定向曲面上的定向$T^2$-丛进行分类。 特别地，我们证明任何在$\Sigma _g$上的$T^2$-丛，若具有$g\geq 1$，则同构于$g$个在$T^2$上的$T^2$-丛的纤维连通和。 此外，情况(1)中的分类结果可以推广到$G$是有限个有限循环群的自由积的情况。 我们还将它应用于从闭曲面到透镜空间的连通和的映射的扩展问题。 显示英文摘要

摘要： Let $g$ be a non-negative integer, $\Sigma _g$ a closed orientable surface of genus $g$, and $\mathcal{M}_g$ its mapping class group. We classify all the group homomorphisms $\pi _1(\Sigma _g)\to G$ up to the action of $\mathcal{M}_g$ on $\pi _1(\Sigma _g)$ in the following cases; (1) $G=PSL(2;\mathbb{Z})$, (2) $G=SL(2;\mathbb{Z})$. As an application of the case (2), we completely classify orientable $T^2$-bundles over closed orientable surfaces up to bundle isomorphisms. In particular, we show that any orientable $T^2$-bundle over $\Sigma _g$ with $g\geq 1$ is isomorphic to the fiber connected sum of $g$ pieces of $T^2$-bundles over $T^2$. Moreover, the classification result in the case (1) can be generalized into the case where $G$ is the free product of finite number of finite cyclic groups. We also apply it to an extension problem of maps from a closed surface to the connected sum of lens spaces.