Algebraic Topology II, Math 9152b, Winter 2011

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 Monday, January 3, 2011.

Text: We will use Allen Hatcher's book on algebraic topology. 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.

Homework:  Homework will be due every two weeks, in class. Doing problems and talking about the material are both essential for learning the material in this course, so you are encouraged to discuss the problems with classmates and with me. But you must write up the solutions on your own and must not look at other students' written solutions nor should you attempt to find solutions to problems online or in textbooks. Your solutions should be clear and carefully written and you should give credit to those who helped you and to any references you used. Homework will be graded based on both correctness and clarity. Late problem sets will not be accepted unless arranged in advance for a good reason.

Presentations: In the second half of the semester, each student will give one 45-55 minute presentation on a topic related to the course.

Exam:   There will be a final exam at the end of the course that we will schedule later.

Evaluation: Evaluation will be based upon homework (35%), the presentation (35%) and the final exam (30%).

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