Homotopy Theory I, Math 546a, Fall 2000

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, and general theory was developed to handle the machinery. The theory clarified the logical underpinnings, and has led to the expansion of homotopy theoretic thinking to many other areas of mathematics, most notably in algebra and algebraic geometry. Because of this, an understanding of homotopy theory is quite valuable. This course will focus on the original 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. The choice of topics and the pace will depend on the participants. We will cover the standard material, such as fibrations, cofibre sequences, Whitehead theorems, the Freudenthal suspension theorem, classifying spaces, 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 September 12.

Text: There is no text for the course, but there is a list of books you may like to refer to: dvi, ps, or pdf. A few of these will be available at the bookstore, and most will be on reserve in the library. You are not required to buy any books for this course.

[This paragraph updated Nov 2003.] Alan Hatcher has a nice book on algebraic topology which also includes much of the homotopy theory we will cover. And the first chapter of his book on spectral sequences treats the Serre spectral sequence, which will be the last topic of our course. Both of these books can be freely downloaded and printed, or can be purchased in bound form.

Presentations: In the second half of the semester, each student will give two presentations on topics 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 and the presentations. There will be no tests.

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