Projective bundles over small covers and the bundle triviality problem
Projective bundles over small covers and the bundle triviality problem
复制标题
小覆盖范围内的投影束和束平凡性问题
DOI：
10.1515/forum-2015-0007
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发表时间：
2016
影响因子：
0.8
通讯作者：
Lu Zhi
中科院分区：
文献类型：
--
作者：
Kuroki Shintaro;Lu Zhi
摘要 本文研究了小覆盖范围内的射影丛。我们首先给出实向量束在小覆盖上投影成为小覆盖的充要条件。然后结合矩角流形，通过引入简单凸多面体上的新特征函数，进一步研究了这种作为小覆盖的投影化的结构。作为一个应用，我们通过将纤维和运算解释为某种组合运算来表征二维小覆盖上的真实射影丛。接下来，我们确定实射影空间上同义反复线丛和切丛的惠特尼和的投影何时与两个实射影空间的乘积微分同胚。这回答了有关 Monster-Semple 塔光纤拓扑的悬而未决的问题。
Abstract The present paper investigates the projective bundles over small covers. We first give a necessary and sufficient condition for the projectivization of a real vector bundle over a small cover to be a small cover. Then associated with moment-angle manifolds, we further study the structure of such a projectivization as a small cover by introducing a new characteristic function on simple convex polytopes. As an application, we characterize the real projective bundles over 2-dimensional small covers by interpreting the fiber sum operation to some combinatorial operation. We next determine when the projectivization of Whitney sum of the tautological line bundle and the tangent bundle over real projective space is diffeomorphic to the product of two real projective spaces. This answers an open question regarding the topology of the fiber of the Monster-Semple tower.