1MA259 Differential Topology, Spring 2017

Lecturer: Dmitry Tonkonog.

Links: University course homepage, brief syllabus.

Graded homework assignments

[Homework Assignment 3]

[Homework Assignment 2]

[Homework Assignment 1]

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.

Grading scheme

The final grade is determined by three compulsory graded homework assignments. The total number of points one can earn from the three homework assignments is 70, and the final grades are determined according to the table:

Grade 5: 56-70 points
Grade 4: 46-55 points
Grade 3: 32-45 points

Additionally, you must earn at least 25% of points for each of the three homeworks in order to get Grade 3, 4 or 5.

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