Compactness Of Moduli Space For Morse Category

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.