复Bott流形的上同调刚性问题

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 their cohomology rings.

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。

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？在本课题中，主要研究复Bott流形这种环面流形的上同调刚性问题，完成了以下创新工作：1.每一个严格上三角整矩阵都定义一个复Bott塔，我们在这类矩阵中定义了一种等价关系，使得等价矩阵对应的复Bott塔同构，即利用矩阵给出了复Bott塔的同构分类。同时给出了高为2,3的复Bott塔的矩阵分类代表元。2. 通过计算得到两个n阶复Bott流形上同调环之间的同态对应了一个n阶整矩阵（同构对应了可逆矩阵），并且给出了这个矩阵满足的n个矩阵方程，同时满足这些方程的n阶整矩阵也对应一个上同调环之间的同态。随后主要研究了这些矩阵方程的求解问题。对同态的具体形式，以及随后的实现问题的研究是解决复Bott流形这种环面流形的上同调刚性问题的重要途径。