## Differential Topology [MA5122]

### Sommersemester 2017

### News

- There will be no lecture on July 26.
- Oral exams are being scheduled for July 28 and 29. Please check your inbox.
- Exercise classes will take place once every two weeks. The first class is on Wednesday May 3. The preliminary dates for the other classes are: May 17, May 31, June 7, June 21 , July 12, July 26.

### Contents

This course will provide an introduction to basic concepts of differential topology. We will discuss immersions, submersions and embeddings, critical points and Sard's theorem, Whitney's embedding theorem, as well as some mapping degree theory. Applications include fixed point theorems and the Borsuk-Ulam theorem. By the end of this course, students should be able to analyse topological problems from a differentiable viewpoint and to see differential problems from a topological perspective.

### Literature

In addition to the handwritten notes, some suggested references are

- V. Guillemin, A. Pollack, Differential Topology
- J.W. Milnor, Topology from the differentiable viewpoint; video recordings of a classic lecture by Milnor can be found here: part I
^{}, part II ^{}, part III ^{}
- Bröcker, Jänich, Einführung in die Differentialtopologie
- M.W. Hirsch, Differential Topology

### Handwritten notes

- 1. Topological spaces (updated on May 3)
- 2. Topological manifolds
- 3. Differentiable/smooth manifolds (updated on May 14)
- 4. Tangent space and differential, inverse function theorem, smooth invariance of domain, the tangent bundle, vector bundles (updated on May 26)
- 5. Rank of a map, immersions, submersions, constant rank theorem, regular values and points
- 6. Sard's theorem
- 7. Embeddings and Whitney's embedding theorem
- 8. Manifolds with boundary
- 9. Classification of smooth 1-manifolds, no retraction theorem, Brouwer's fixed point theorem
- 10. Mod-2-degree (updated on July 12)
- 11. Winding number mod 2, Borsuk-Ulam theorem
- 12. Oriented manifolds, Brouwer degree, Hopf theorem

### Exercises

### Other comments

You may be interested in the topology course taught concurrently by Prof. Ulrich Bauer, see

here .