Low dimensional topology is currently a very active area of research. This course will introduce the basic ideas and theorems in the study of 4-dimensional manifolds. Surprisingly manifolds of other dimensions are much better understood, but there is not even a conjectural picture of the structure of 4-manifolds. Despite this one can say a lot about 4-manifolds. Some likely topics for the course are:
- algebraic topology of 4-manifolds
- Wall's theorem
- complex surfaces
- "exotic" 4-manifolds
- Kirby calculus
- Seiberg-Witten invariants for Heegaard Floer invariants
and Thursday 9:35 in Skiles 257.
Office: Skiles 106
e-mail: etnyre "at" math .gatech.edu
While I will assign homework problems in class, these are purely for your benefit. I hope you think about them and talk to me about them, but they will not be graded. Your grade will be based on class participation (mainly attendance) and one talk in the Friday working topology seminar. The topic of this talk is completely up to you. While it would be nice if it had some relevance to the class, it is not necessary that it does. Any topological or geometric talk will be fine. The level of the talk is also completely up to you. It can be a fairly basic talk that covers something tangential to the class or it can be a talk on recent research. Please come talk to me early in the semester to figure out what you will present and schedule a time for your talk.
I am assuming you have had a basic point set topology course, but here are some basic notes on quotient spaces that might help refresh this important idea (we will be using this to "glue" spaces together all the time).
There are no official text book for this course, but the following books are quite useful and more or less take the same perspective on the theory we will in this class.
- 4-manifolds and Kirby calculus by Gompf and Stipsicz
This book is a very thorough introduction to Kirby calculus (the art of drawing 4-manifolds) and countless examples of 4-manifolds. It also does a quick and dirty run through of Seiberg-Witten theory and a bit of symplectic geometry.
- The topology of 4-manifolds by Kirby
This book is much shorter than the other and only covers a little Kirby calculus, but it covers many of the classic results in 4-manifolds and gives an overview of Freedman's work.