Classification of $T^2$-bundles over $T^2$
Classification of $T^2$-bundles over $T^2$
复制标题
$T^2$ 上的 $T^2$ 捆绑的分类
DOI：
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发表时间：
1983
期刊：
影响因子：
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通讯作者：
S. Fukuhara
中科院分区：
文献类型：
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作者：
Koichi Sakamoto;S. Fukuhara
通过$T^{2}$，我们表示二维环面$R^{2}/Z^{2}$。本文的目的是对以$T^{2}$为纤维和基空间的纤维丛进行分类。（我们简单地称它们为$T ^{2}上的$T ^{2}$-丛。从丛同构的观点出发，得到定理4，给出了两个丛同构的充要条件。根据这个定理，人们甚至可以用计算机确定两个丛是否同构。根据全空间的同胚类型，我们得到了定理5：全空间是同胚的当且仅当它们的基本群是同构的。在定理3中，我们证明了$T ^{2}$上的任何$T^{2}$-丛同构于某些标准类型的丛之一。
By $T^{2}$ , we mean a two dimensional torus $R^{2}/Z^{2}$ . The purpose of this paper is to classify fiber bundles which have $T^{2}$ as fibers and base spaces. (Simply we call them $T^{2}$-bundles over $T^{2}.$ ) In view of bundle isomorphisms, we obtain Theorem 4 which gives necessary and sufficient condition that two bundles are bundle isomorphic. By this theorem, one might determine even by computer whether two bundles are isomorphic or not. In view of homeomorphism types of total spaces, we obtain Theorem 5 which says that total spaces are homeomorphic if and only if their fundamental groups are isomorphic. In Theorem 3, we show that any $T^{2}$-bundle over $T^{2}$ is isomorphic to one of some standard types of bundles.