Algebraic Topology II, Math 9152a, Fall 2008

Homotopy theory originated as a branch of topology in which one studied the ways in which geometrical shapes can be deformed. As it progressed the methods became quite sophisticated. One of the milestones was the realization that homotopy theoretic methods are useful in many other areas of mathematics, most notably in algebra and algebraic geometry. Because of this, an understanding of homotopy theory is quite valuable for mathematicians coming from different backgrounds. This course will focus on the original topological ideas of homotopy theory, and will prepare the student for further work, either within topology, or in other fields.

Course outline: This course is an introduction to homotopy theory, which starts right at the beginning. We will cover the standard material, such as homotopy groups, relative homotopy groups, fibrations, cofibre sequences, Whitehead theorems, the Freudenthal suspension theorem, Eilenberg-Mac Lane spaces, Postnikov towers, etc., and will end up talking about the Serre spectral sequence which will allow us to do some computations of the homotopy groups of spheres. An effort has been made to organize the material in a way which emphasizes the geometrical ideas behind the results, rather than the most efficient proofs or the most generality.

The course begins on Tuesday, September 8.

Text: We will use Allen Hatcher's book on algebraic topology. And the first chapter of his book on spectral sequences treats the Serre spectral sequence, which will be the last topic of our course, but I don't know if I will follow his presentation. Both of these books can be freely downloaded and printed, and the first one can be purchased in bound form.

There is also a list of books you may like to refer to: dvi or pdf. Most of these are available in the library. I haven't put them on reserve, so share with other students.

Presentations: In the second half of the semester, each student will give one 40-50 minute presentation on a topic related to the course. The scheduling will be worked out later.

Evaluation: Homework will be due roughly every two weeks. Evaluation will be based upon homework (65%) and the presentations (35%). There will be no tests.

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