Introductory Lectures On Manifold Topology | |
Student No.： | 50 |
Time： | Wed 15:20-17:00/Fri 13:20-15:00 |
Instructor： | Thomas Farrell [Tsinghua University] |
Place： | Conference Room 1, Floor 1, Jin Chun Yuan West Building |
Starting Date： | 2013-3-1 |
Ending Date： | 2013-7-3 |
Description:
A basic problem in topology is to classify n-dimensional manifold up to homeomorphism (diffeomorphism); i.e. classify spaces which are locally homeommorphic (diffeomorphic) to n-dimensional Euclideean space. Until 1956 it was implicitly assumed that there would be no difference in the two classifications. Then Milnor exhibited a manifold homeomorphic but not diffeomorphic to the 7-dimmmensional sphere.
And this was followed four years later by Kervaire's example of a topologically simple ten dimensional manifold which doesn't support any differential structure. These two examples led to methods (called surgery) for attacking the classification problem in general. We will carefully explain the work of Kervaire and Milnor on this problem and show how it led to the general method of surgery. There are important unsolved problems in this area which we hope to introduce to people who might be interested in working on them.
Prerequisite:
Algebraic topology through cup products and differential topology through the Whitney embedding theorem. Some knowledge of characteristic classes (as in the text by Milnor and Stasheff) will be very useful and will be reviewed as needed.
Reference:
THE TOPOLOGY OF FIBRE BUNDLES by Norman Steenrod
CHARACTERISTIC CLASSES by J.W. Milnor and J.D. Stasheff