During the Spring, I will be teaching a class on differential topology. Lecture Notes will not be posted on this blog since I will be explicitly using several books. The course will mainly be organized in two parts.
Part 1. Introduction to differential topology
In this part, to simplify the presentation, all manifolds are taken to be infinitely differentiable and to be explicitly embedded in euclidean space. A small amount of point-set topology and of real variable theory is taken for granted. We shall follow the classical reference by John Milnor: Topology from the differentiable viewpoint.
- Smooth manifolds and smooth maps
- Sard-Brown theorem
- Smooth homotopy and smooth isotopy
- Brouwer degree
- Vector fields and Euler number
Part 2. Riemannian geometry and differential topology
In this part we plan to show how Riemannian geometry can be used to study topological properties of a manifold. Depending on the pace of the class, all of the following topics may not be covered.
- Abstract manifolds
- Whitney embedding theorem
- Differential forms on manifolds
- De Rham cohomology
- Riemannian metrics
- The Laplacian on forms and Hodge theory
- Connections and curvature
- Weitzenbock formula
- The Chern-Gauss-Bonnet formula
- Introduction to index theory and characteristic classes
General references for this part of the course will be
- Riemannian Geometry, by Gallot-Hulin-Lafontaine
- The Laplacian on a Riemannian manifold, by Rosenberg