## 1MA259 Differential Topology, Spring 2017

Lecturer: Dmitry Tonkonog.

 Homework 1. Worth 20 points, Released on 10 Feb, Due by 24 Feb, 01:00pm Homework 2. Worth 20 points, Released on 8 March, Due by 29 March, 01:00pm Homework 3. Worth 30 points, Released on 13 April, Due by 4 May, 01:00pm

### Lecture notes

 [L01] [L02] [L03] [L04] [ExS] [L06] [L07] [L08] [L09] [ExS] [L11] [L12] [L13] [L14] [ExS] [L16] [L17] [L18] [L19] [L20] [ExS] [L22] [L23]

NB: the notes have typos.

### Exercises and problem sessions

These exercises are ungraded, and are discussed during problem sessions:

 [Ex 01-02] [Ex 03-04] [Ex 06-07] [Ex 08-09] [Ex 11-12] [Ex 13-14] [Ex 16-18] [Ex 19-20]

Schedule of problem sessions:

 Lec 5, 31 Jan Lec 10, 17 Feb Lec 15, 8 March Lec 21, 13 April

### Lecture plan

1. Excursion: The pendulum. Smooth maps and diffeomorphisms.

2. Manifolds, tangent planes, differential of a smooth map.

3. More on the differential. Inverse function theorem. Submersions.

4. Proof that SO(n) is a manifold. Immersions. Irrational winding on the torus and other pathological examples.

5. Exercise session.

6. Maps transverse to submanifolds. Preimage theorem. Transversally intersecting submanifolds. Sard's theorem: statement. Measure 0 sets.

7. Proof of Sard's theorem.

8. Morse functions. Excursion: Morse homology.

9. Existence of Morse functions. Quick excursion: Vector bundles. Whitney embedding theorem.

10. Exercise session.

11. Manifolds with boundary. Preimage theorem for maps from manifolds with boundary. Non-existence of retractions to the boundary.

12. Brouwer fixed point theorem. Degree modulo 2.

13. Degree and the fundamental theorem of algebra. Transversality theorem. Epsilon-neighbourhood theorem and the normal bundle.

14. Transversality theorems continued. Intersection theory mod 2. Examples for surfaces.

15. Exercise session.

16. Sketch of the definition & invariance of the integral degree. Orientations.

17. Orientations, continued. Oriented intersection theory.

18. Euler characteristic. Lefschetz fixed point theory.

19. Vector fields and the Poincare-Hopf theorem.

20. Homotopy groups. Hopf degree theorem. Pontryagin theory.

21. Exercise session.

22. Pontryagin theory, continued. Fundamental groups of surfaces, Dehn's solution of the word problem.

23. Excursion: mapping class groups, Nielsen-Thurston classification, sketch proof of the Nielsen realisation theorem.