代數拓樸

Algebraic topology has wide applications in many areas of mathematics. This introductory course covers some essential elements of algebraic topology, and briefly introduces some applications in differential equations and geometry. This is intended for graduate and advanced undergraduate students who have taken standard undergraduate courses for topology, differential geometry, differential equations, and algebra. Topics to be covered in the fall semester include: Compact surfaces, Fundamental group, Covering spaces, Homology theory, Cohomology theory 課程大綱 (Syllabus) Course keywords: Compact surfaces, Fundamental group, Covering spaces, Homology theory, Cohomology theory, 緊緻曲面，基本群，覆疊空間，上同調，下同調 一、課程說明(Course Description) Algebraic topology has wide applications in many areas of mathematics. This introductory course covers some essential elements of algebraic topology, and briefly introduces some applications in differential equations and geometry. In particular, the course includes brief introduction to homotopy theory, homology theory, and cohomology theory. This course is intended for graduate and advanced undergraduate students who have taken standard undergraduate courses for topology, differential geometry, differential equations, and algebra. * Students without background in basic group theory, point set topology, should NOT take this course. 二、指定用書(Text Books) William S. Massy: A Basic Course in Algebraic Topology, Springer, GTM volume 127, 1991. 三、參考書籍(References) James R. Munkres: Elements of Algebraic Topology, Addison-Wesley, 1984. 四、教學方式(Teaching Method) Standard 五、教學進度(Syllabus) Except for the introductory chapter, a rough schedule is as follows: 1. Compact surfaces -- 2 weeks 2. Fundamental group -- 3 weeks 3. Covering spaces -- 2 weeks 4. Homology theory -- 3 weeks 5. Cohomology theory -- 3 weeks 6. Some applications -- 1 week 六、成績考核(Evaluation) Homework (30%), Two Exams (70%) 七、可連結之網頁位址 http://www.math.nthu.edu.tw/~kchen/teaching/5525F26.html

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