Compactness of Moduli Space for Morse Category

【作者】 李勇；

【导师】 杨明升； 赫海龙；

【作者基本信息】 南京师范大学 ， 基础数学， 2017， 硕士

【摘要】 本文给出了莫尔斯范畴的模空间的配边的一个分析证明,配边是在紧性和粘合的意义下.证明主要依据的是由Matthias Schwars给出的莫尔斯复形(Morse-Smale-Witten complex)中的紧性和粘合下的配边理论.关键点在于用分析的语言对紧性和粘合进行表达.紧性主要用到了阿尔泽拉-阿斯科利定理,粘合主要用到是弗雷德霍姆理论.所以本文从以下两个方面展开.第一部分,介绍一些基本概念和证明所需要的理论知识.第二部分,首先给出A无穷范畴的定义.其次是定义莫尔斯复形之间的乘法,这样就引出本文所讨论的模空间.最后给出模空间紧性的主要定理的证明,得到了满足A无穷范畴定义的条件,这样给出的A无穷范畴也叫莫尔斯范畴.更多还原

【Abstract】 In this thesis we prove cobordism theory of the moduli space of Morse category in the sense of compactness and gluing. The proof is based on the theory of compactness and gluing for the moduli space of Morse complex,that Matthias Schwars put forward. Its critical point aims at expressing the compactness and gluing with the analysis of the language. The compactness mainly uses the theorem of Arzela-Ascoli, and the gluing is applied to the Fredholm theory. And now, we will start this paper from two aspects.In the first part, we introduce some basic concepts and theoretical knowl-edge needed when proving theories.In the next part, firstly, we give the definition of A∞-category. Secondly, we define multiplication of Morse complex which leads to the concept of moduli space that we will discuss in this paper. At last, we show the proof of the major theorems of the compactness of moduli space. Thus, we obtain all the conditions of the definition of A∞-category, with the result A∞-category is also called Morse category.更多还原