Algebraic topology, Math 414b, Spring 2001

Algebraic topology is the study of topological spaces using tools of an algebraic nature, such as homology groups, cohomology groups and homotopy groups. It is one of the major accomplishments of twentieth century mathematics and has applications to many areas of mathematics and to other fields, such as physics, computer science, and economics. I encourage students from other departments to attend. This is a first course in algebraic topology which will introduce the invariants mentioned above, explain their basic properties and develop both methods of computation and geometric intuition.

Course outline:   Homotopy, fundamental group, Van Kampen's theorem, fundamental theorem of algebra, Jordan curve theorem, singular homology, homotopy invariance, long exact sequence of a pair, excision, Mayer-Vietoris sequence, Brouwer fixed point theorem, Jordan-Brouwer separation theorem, invariance of domain, Euler characteristic, cell complexes, projective spaces, Poincaré theorem.

Text:  The text for the course is Algebraic Topology: A first course, by Marvin J. Greenberg and John R. Harper. Published by Perseus Books. Revised edition. ISBN 0-8053-3557-9. Here is a list of other reading material: dvi, ps, or pdf. A few of these will be available at the bookstore, and most will be on reserve in the library.

New addition: Alan Hatcher has a very nice book on algebraic topology.

Homework:  Homework will be due every week or two, in class. Doing problems and talking about the material are both important ways to learn, 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 show your written work to others. Your solutions should be clear and carefully written and you should give credit to those who helped you and to any references you used. Late problem sets won't be accepted unless arranged in advance for a good reason.

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

Evaluation:   Evaluation will be based on homework and the presentations. There will be no tests or quizzes.

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