Algebraic topology, Math 414b/501b, Winter 2008

The final exam is Wednesday, April 23, 2-5 pm, MC107.

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. This is a first course in algebraic topology which will introduce the invariants mentioned above, explain their basic properties and develop geometric intuition and methods of computation.

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, by Allen Hatcher. Published by Cambridge University Press. ISBN 0-521-79540-0. The seventh printing, from 2006, contains many corrections and improvements. The book will be available at the campus bookstore, and is also available online. The book's webpage also contains a list of errata for the printed copy.

Here is a list of other reading material. None of these are required, but you might find them interesting. A few are on reserve at the library.

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 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. Homework will be graded based on both correctness and clarity. 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. See the presentations page for more details.

Exam:   There will be a final exam at the end of the course: Wednesday, April 23, 2-5 pm, MC107.

Evaluation:   Evaluation will be based on homework, presentations and the final exam, with equal weight. Graduate students will have extra work, which will be determined once I see the enrollment.

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