Math 742, Differential Topology, Spring 2007

There will be no graded exercises in the course.  Some useful books are:

• Bredon, Topology and Geometry
• Brocker and Janich, Introduction to Differential Topology
• Hirsch, Differential Topology
• Milnor, Topology from the Differentiable Viewpoint
• others to appear later

After covering the absolute basics, we will be free to choose topics to explore.  So if there is anything you would like to learn about, be sure and let me know and I'll see if it is feasible.  I expect the students in the course have widely varying backgrounds.  Be sure to ask questions if you don't understand something in class.  Rule number one is there are no stupid questions, so don't be shy.  (I reserve the right to rescind this rule if need be, but I doubt it will be necessary.)  I intend to keep a log of what we discuss each day.  For comparison here is the old log from a previous incarnation of this course.


I have had a couple requests for things to do eventually.  They may or may not be feasible but I'll list them anyway.  If I know a topic is feasible to do I will so indicate.  If you have told me a topic and I have forgotten to include it below, let me know.

• Exotic smooth structures-  Some spaces have many nondiffeomorphic smooth structures on them.  For example R⁴ has uncountably many nondiffeomorphic smooth structures on it.  Spheres of dimension 7 or more have finitely many, but more than one smooth structure.  I suspect this topic is not feasible to cover except very generally but I'll look into it.
• Ricci flow- I suspect this is not feasible.
• Differential forms and DeRham cohomology- This is feasible to do and can be done without previous knowledge of (co)homology, although those who know about ordinary singular cohomology will get more out of it.
• Symplectic topology and/or Kahler theory - we might be able to say a little here.  It would fit in with the discussion of differential forms above.
  
• Stuff useful for studying smooth dynamical systems - Some of this is stuff we will be doing anyway (Sard's Theorem, transversality), some I wanted to do (introduction to Morse theory), some which should fit in (Abraham transversality).