Math 215a: Algebraic topology

UC Berkeley, Fall 2005


Michael Hutchings
Email: [my last name minus the last letter]
Office phone: 510-642-4329.
Office: 923 Evans.
Tentative office hours: Tu 2-5.

Spacetime coordinates

TuTh, 9:30-11:00. As of 9/8, we are officially moving to 9 Evans.


The only formal requirements are some basic algebra, point-set topology, and "mathematical maturity". However, the more familiarity you have with algebra and topology, the better.


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, homology groups, and the cohomology ring. 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 differential topology, i.e. the study of smooth manifolds. In fact, I don't think it really makes sense to study one without the other. So without making differential topology a prerequisite, I will emphasize the topology of manifolds, in order to provide more intuition and applications.

No single textbook does all the things that I want to do in this course. The official textbook is Algebraic Topology by Hatcher. This is a very nice book, although it does not say much about differential topology. Another very nice algebraic topology text which also covers some differential topology is Geometry and Topology by Bredon. Some further recommended books are listed below.


Doing homework exercises is essential for learning the material, and is an official course requirement. A reasonable effort on homework will result in a good grade. Homework grading policy. Since there are approximately fifty students in the class, it is not possible for me to read all of the homework carefully, and unfortunately I was not able to get a grader. So we will try the following, which might even be more educational than the traditional approach. Each homework assignment will be due at the beginning of class on some particular day. At the end of that lecture, everyone who handed in an assignment will be given a random assignment from someone else to grade. (I will keep a record of who gets which assignment.) After class, I will post solutions online to help with grading (although of course these solutions are not unique). If you can't make it to class on the day that the homework is due, please slide your homework under my office door before class, together with a note explaining how I can give you another assignment to grade.

To grade the assignment, please take a separate sheet of paper, and label it with both your name and the name of the person whose assignment you are grading. Write at most one page of commentary on the assignment. If an answer is good, say so; if you are not convinced by a proof, say why. Then give the whole assignment a grade of 3 (excellent), 2 (good), or 1 (needs improvement). The grades should not be taken too seriously; the important thing is to communicate and critique mathematical arguments. (At the end of the course, I will play some statistical games to adjust for the fact that different people grade more generously than others.)

Finally, staple your commentary to the assignment, and hand in the package to me in or before class one week after the assignment was due. I will then look over all of the graded assignments and return them in a subsequent class.

What I plan to do

What we actually did.


The following are some related books which are sitting on my shelf, along with my opinions about them. Some of these may be useful now, while others will be more useful after you have taken 215a.

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