Algebraic Topology -1


Prerequisites:

The only formal requirements are some basic knowledge in point-set topology and group theory. However, the more familiarity you have with algebra and topology, the better.


Syllabus:

Algebraic topology seeks to capture the essence of a topological space in terms of various algebraic and combinatorial objects. We will construct three such gadgets: the fundamental group, covering spaces and homology groups. We will apply these to prove various classical results such as the classification of surfaces, the Brouwer fixed point theorem, the Jordan curve theorem, the Lefschetz fixed point theorem, and more. An important topic related to algebraic topology is the study of manifolds. I will emphasize the topology of manifolds, in order to provide more intuition and applications. In addition, knot theory would also haunt our class throughout the term.


Textbooks amd References:

No single textbook does all the things that I want to do in this course. The official textbook is 《Algebraic Topology》by A. Hatcher which is free. This is a very nice book, although it does not say much about differential topology. In addition, J. Rotman's《Introduction to Algebraic Topology》 may be suitable for students who like more homological algebraic approach toward topology. Most of the elementary materials of algebraic topology can be found in the two classical textbooks 《Basic Topology》 by A. Armstrong and 《Elements of Algebraic Topology》 by J. Munkres. Another very nice and comprehenstive textbook is 《Topology and Geometry》by G. Bredon which also contains some materials in differential topology (e.g. de Rham cohomology and characteristic classes).

As for students who have difficulty to read English books, I recommend the following textbooks in Chinese:


Time Schedule (17 weeks):




Updated in April 18, 2022