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

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