Differential Topology [MA5122]
Wintersemester 2018/19
News
Content
We will follow a direct approach to differential topology and to many of its applications without requiring and exploiting the abstract machinery of algebraic topology. The course will cover immersion, submersions and embeddings of manifolds in Euclidean space (including the basic results by Sard and Whitney), a discussion of the Euler number and winding numbers, fixed point theorems, the Borsuk-Ulam theorem and respective applications. At the end of the course, students should be able to analyse topological problems from a differentiable viewpoint and to see differential problems from a topological perspective.
| Date |
Content |
Black board notes |
Further reading |
| Oct 15 |
Intro, Topological spaces, subspace-, product- and quotient-topologies |
lec 1 |
A. Hatcher's notes  with more details and proofs |
| Oct 22 |
Compactness, homeomorphisms, topological manifolds and their embeddings into Rn, differential calculus |
lec 2 |
|
| Oct 29 |
Classification of low-dim. manifolds, manifolds with boundary, constant rank thm., smooth structures |
lec 3 |
|
| Nov 5 |
Smooth maps and diffeomorphisms, smooth invariance of domain and its consequences |
lec 4 |
|
| Nov 12 |
Embeddings, immerions, submersions, submanifolds, preimage theorems |
lec 5 |
|
| Nov 19 |
Tangent-vectors, -spaces and -bundles |
lec 6 |
|
| Nov 26 |
Differentials, measure zero sets, Sard's theorem with and without boundary |
lec 7 |
|
| Dec 3 |
Density of Morse functions, Morse's Lemma, no-retraction theorem, Brouwer's fixed point theorem |
lec 8 |
|
| Dec 10 |
Invariance of domain and dimension, Whitney's embedding and immersion theorem |
lec 9 |
|
Exercises
- Exercise sessions will be held every two weeks starting in the week from 22nd Oct - 26th Oct
- Date, time and room are: Wed 10:15-11:45 in room 03.12.020A and Thu 08:30-10:00 in room 03.12.020A
- Exercise sheets are to prepared at home and will be discussed during exercise sessions
| File |
Content |
Week |
Solution |
| Ex 1 |
Basic notions of topologies |
Oct. 22nd - Oct. 26th |
Sol 1 |
| Ex 2 |
Topological and smooth manifolds |
Nov. 5th - Nov. 9th |
Sol 2 |
| Ex 3 |
Lie groups, embeddings, immersions, submanifolds |
Nov. 19th - Nov. 23th |
Sol 3 |
| Ex 4 |
tangent space, more Lie groups |
Dec. 3th - Dec. 7th |
Sol 4 |
| Ex 5 |
Brouwer's fixed point theorem, classifiaction of 1-dim compact manifolds |
Dec. 17th - Dec. 21th |
|
Additional literature
- M.W. Hirsch, Differential Topology
- 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
- Munkres, Topology
