Math 215a: Algebraic topology

UC Berkeley, Fall 2003


Michael Hutchings
Office phone: 510-642-4329.
Office: 923 Evans.
Tentative office hours: Tuesday 3 - 5 (primarily for Math 53), Thursday 10-11 (primarily for Math 215a).

About the course

Algebraic topology is a beautiful subject developing algebraic invariants (homology, homotopy groups, etc.) which extract the "essence" of a topological space and provide powerful machinery for proving many deep theorems (Brouwer fixed point theorem, Jordan curve theorem, many others like this, classification results, etc.).

In the first semester we will begin to study the essential "core" material of algebraic topology: homotopy groups, homology, and cohomology, and interesting examples and applications thereof.

In the second semester we will finish whatever is left over from the first semester, and then study a selection of more advanced, but still very important topics. Possible topics include but are not limited to bundles and characteristic classes, Morse theory, and spectral sequences.


The textbook for the core material that we will be doing in the first semester and some of the second semester is "Algebraic topology" by Allen Hatcher; this is available in paperback from Cambridge University Press, and can also be downloaded from the author's website, although printing out the whole book might cost more than buying it. A little bit in the first semester, and a lot in the second semester, we will discuss some topics and perspectives that are not in this book. I will post additional references here later.


Basic knowledge of abstract algebra, linear algebra, and point-set topology, together with sufficient independence and mathematical maturity to handle a graduate course, are essential. Undergraduate algebraic or differential topology would be helpful but is not required.

Lecture summaries and plans