Math 215a: Algebraic topology

UC Berkeley, Fall 2007


Instructor: Michael Hutchings, 923 Evans, [my last name with the last letter deleted] Tentative office hours (may be rescheduled some weeks): Wednesday 9-12. Outside of office hours, the best way to reach me is to send me an email. I generally check email once per day.

GSI: Qin Li. Office hours: Friday 3-5pm, room 935 Evans.

Lectures: MWF 1-2, room 3 Evans.

Prerequisites: 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 easier this course will be. I think that all the point-set topology we will need (and a lot more) is reviewed in Bredon, Chapter I, Sections 1-13.

Syllabus: Algebraic topology seeks to capture key information about 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.


Homework is essential for learning the material, and will form the basis for your grade. Assignments will be given every 1.5-2 weeks and will be posted here.

Tentative course outline (There is a lot of material here, so we might not get to all of it, and I will refer to the textbooks for details of some of the proofs. My goal will be to explain the important ideas.)

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