拓扑学基础课程教学大纲-上海交通大学数学系

1、拓扑学基础课程教学大纲课程基本信息（ Course Information ）课程代码MA3133/* 学时* 学分（Course Code）MA307644（Credit Hours）（Credits）（中文）拓扑学基础* 课程名称（ Course Name）课程性质(Course Type)授课对象（ Audience）授课语言(Language of Instruction)* 开课院系（ School）先修课程（ Prerequisite ）授课教师（ Instructor ）（英文） Elementary Topology专业选修A 组本科生中文数学系数学分析 ,解析几何 ,高等代数

2、 ,抽象代数课程网址杨义虎(Course Webpage)这个课程分三个部分：第一部分是关于点集拓扑；第二部分是基本群和覆盖空间；* 课程简介 （Description）第三部分为闭曲面的拓扑分类，实际为前面的内容的应用。This course has three parts. The first part is on point-set topology; the second part is* 课程简介 （Description） fundamental group and covering spaces theory; the third part is (partly as appli

3、cations of the previous chapters) on the (topological) classification of closed surfaces.课程教学大纲（course syllabus）The first part (24hrs) is on set topology, basic language of this subject. Concretely, we will teach1. Definition of topological spaces and examples (open and closed sets, closure), contin

4、uous map and homeomorphism, product space and topological basis.2. Some important topological properties: separation axioms (in particular Hausdorff spaces) and countability axiom, compactness and sequence compactness (of a metric space), connectivity and path connectivity; Urysohn lemma and Tietze

5、extension.The second part (28 hrs) is on the fundamental group of a (path connected) topological space and covering theory of topological spaces. These should be one of basic language of modern geometry. Concretely, we will teach1.Definitionsof (relative) homotopyof maps and spaces, definitionof the

6、 fundamentalgroup of a (pathconnected) space, the induced homomophism of a continuous map, homotopical invariance of fundamental groups. computation of fundamental groups of some simple topological spaces, Von Kampen Theorem without proof, some simple applications.2. Definition of covering spaces an

7、d examples and basic properties, the map lifting theorem, covering transformations and regular covering spaces, universal covering and deck transformations groups,fundamental domains and some examples.The third part (12 hrs) is the (topological) classification of closed surfaces, which can be (partl

8、y) considered as an application of the previous parts.1. Quotient spaces and quotient maps, construction of some simple surfaces (by quotient space method), definition of topological manifolds and closed surface.2. general construction of (orientable and nonorientable) closed surfaces, polygon repre

9、sentation of a closedsurface, surface surgery and canonical polygon representations, computation of the fundamental group of a closed surface using Van Kampen theorem.* 学习目标 (Learning Outcomes)* 教学内容、进度安排及要求(Class Schedule&Requirements)基本地 , 纯数学有三大块 , 几何 , 分析 , 代数 . 大学本科数学系学生几何的学习始于解析几何，然后是微分几何 , 但这

10、些内容某种意义上是十分特殊的例子。为了考虑更复杂的几何问题，我们需要更精确地语言整体几何的语言。因此，该课程除了学习并掌握一些基本内容外，也是为学生将来在几何方面的进一步学习提供相应的语言。具体地，我们希望1 培养学生整体地思考几何问题的能力(A4,A5,B2,B3,B7)；2培养学生的几何直观，空间想象能力(A4,A5)；3 同时为将来进一步学习更高深的几何内容打下坚实的基础(A3)。教学内容学时教学方式作业及要求基本要求考查方式点集拓扑 :拓扑空间定义(开集 ,闭集 ,闭包 ,邻域等 ),积拓扑 ,可分性 ,10+4课堂教学分离公理 ,连通性 (连通与道路连通 ),紧性 (列紧和紧致 );

11、 习题课商空间 ,商拓扑 ,商映射 ;曲8+2课堂教学面的分类 ;同伦 ,同胚 ,基本群及其应16+4课堂教学用 ;习题课覆盖空间 ,映照提升定理 ,万有覆盖 ;习题课16+4课堂教学* 考核方式闭卷考试(Grading)Text book:Chongye You, Lecture on Basic Topology, Press of Peking University, 1996. (inChinese)*教材或参考资料Reference books(Textbooks & Other1.M. A. Armstrong, Basic Topology, McGraw-Hill, 1979.Materials)2.T. Lawson, Topology: a geometric approach,Oxford University Press, 2003.3.J. R. Munkres, Topology: a first course, Prentice-Hall, Inc., 19