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MATH 661: Geometric Topology (Fall
2007)
Instructor: Allen Hatcher
Meeting Time & Room
Here are two possible topics for the course:
- Basic topology of 3-manifolds.
Perelman's recent
work verifies a conjectural picture of the classification of 3-manifolds
that arose around 25 years ago, and the primary aim of the course would
be to present this picture. The main prerequisite for the course would
be basic algebraic topology as in 651 (fundamental group, covering
spaces, homology). Some prior exposure to differentiable manifolds
would also be helpful, although what's needed here
is minimal and could be picked up as one goes along.
- Bott Periodicity and its
applications such as the nonexistence of real division algebras in
dimensions other than 1, 2, 4, and 8. The framework here is vector
bundles and topological K-theory. Prerequisite: basic algebraic topology
as in 651. The source for the fiirst part of the course would be the
set of notes on my webpage, and a goal for the latter part of the course
would be to extend these notes to cover Bott periodicity for the real
case as well as the easier complex case. This involves the very nice
topic of Clifford algebras, generalizing quaternions.
The
choice between the two topics will depend on the interests of the audience.
Last modified:April 3, 2007
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