复Bott流形的上同调刚性问题

项目名称： 复Bott流形的上同调刚性问题

项目编号： No.11201170

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 李方

作者单位： 吉林大学

项目金额： 22万元

中文摘要： 2006年在大阪举行的环面拓扑会议上，Mikiya Masuda 和Dong Youp Suh 提出了环面流形的上同调刚性问题：如果两个环面流形的上同调环作为分次环是同构的，那么它们是否是同胚甚至是微分同胚的？在过去五年来出现的许多结果都倾向于这个回答是肯定的，它们都是针对Bott流形这一类环面流形的。 2009年Kamishima和Masuda证明了实Bott流形可以由它们的模2上同调环来微分同胚分类。而对于复Bott流形, 目前只证明了几类特殊情形的上同调刚性问题是肯定的如：维数小于等于4或只有一个扭的复Bott流形以及有理平凡化的复Bott流形。在本课题中，主要研究复Bott流形的上同调刚性问题，首先计算上同调环之间的同态（同构），接下来考虑这些同态（同构）能否由流形间的映射诱导得到。通过回答这个问题来说明复Bott流形能否由它们的上同调环来分类。

中文关键词： 复Bott流形；复Bott塔；上同调刚性；同态；

英文摘要： In the proceedings of 2006 Osaka conference on toric topology, Mikiya Masuda and Dong Youp Suh proposed the cohomological rigidity problem for toric manifolds asking whether two toric manifolds are homeomorphic or even diffeomorphic if their integral cohomology rings are isomorphic as graded rings. The affirmative solution to the problem seems implausible at first glance. Instead, during the last five years, many results supporting the affirmative solution to the problem have appeared. So far, all affirmative results are on (generalized) Bott manifolds. The class of real Bott manifolds up to diffeomorphism is determined by their mod 2 cohomology rings. This is shown by Kamishima and Masuda in 2009. However, for the complex Bott manifolds, we just know some affirmtive results to the cohomological rigidity problem such as n-stage(n<=4) or Q-trivial complex Bott manifolds and complex Bott manifolds with one twist. In this project, we focus on the complex Bott manifolds. Firstly, we want to express the homomorphism(isomorphism) between the cohomology rings of complex Bott manifolds. Whether the homomorphism (isomorphism) can be induced by the map between the manifolds, this is the following problem we will study. With the answer for this problem we can know whether the complex Bott manifolds can be classified by

英文关键词： Bott manifold；Bott tower；cohomological rigidity；homomorphism；