Differential Topology
Differential Topology is the study of differentiable manifolds, spaces that locally look like an Euclidean space, but as a whole might look much different, such as for instance a sphere. Central questions we will address in this course include:
- When can a manifold be embedded into another manifold, for instance, Euclidean space?
- What can we say, globally, about the space of smooth maps from one manifold to another?
- How do submanifolds in a larger manifold intersect one another?
- How can we decompose a smooth manifold into smaller parts and obtain a more combinatorial description?
Prerequisites
No background in algebraic topology is required. Some background in point-set topology is expected. Knowledge about the basic definitions of manifolds and their tangent spaces is useful, but can be acquired during the course.
Literature
We will largely be following the following textbook:
Victor Guillemin, Alan Pollack, Differential Topology, AMS Chelsea publishing, 2010 (reprint)
I will provide or point to additional material.
Examination
The examination is based on weekly homework and presentations of problems in the exercise classes. There will be no final exam.
There will be 12 homework sets, each worth 10 points. At the end of the course, the two worst-scoring homework sets will be disregarded (this includes missed homework), for a maximum of 100 points. The grade boundaries are:
| F | Fx | E | D | C | B | A |
| 0 | 45 | 50 | 60 | 70 | 80 | 90 |
In addition, you are required to present solutions to homework problems in the exercise sessions, three times in total during the courses duration. This is a pass-fail criterion. You may not choose which exercise you want to present, but you have some control over it, as follows:
On every exercise sheet you turn in, you clearly mark the problems you would be willing to present in the following exercise session with an asterisk (*). The assistant may or may not call on you to present this problem. To pass the course, you have to mark at least 50% of the problems with an asterisk. If you mark any problems on an exercise sheet with an asterisk, you have to go to the corresponding exercise session, otherwise the asterisks are not counted.
Times and dates
All lectures will be online only. There will be lectures and exercise sessions, always on Tuesdays and Fridays. First lecture: January 19, 1pm. The exercise sessions are led by Eric Ahlqvist.
The zoom link for all lectures and exercises is https://kth-se.zoom.us/j/69184586345 Links to an external site.
| T, Jan 19 | lecture | Introduction; definition of smooth submanifolds of Rn and abstract smooth manifolds; smooth maps between manifolds |
Guillemin-Pollack ([GP]): I.1 |
| F, Jan 22 | lecture | Tangent vectors and derivatives of maps; immersions, submersions, and embeddings |
[GP]: I.2-3 |
| T, Jan 26 | exercises | Examples to understand the definitions of manifolds, immersions etc. Jonathan, Adam, and Ludvig presented. |
Exercises 1 |
| F, Jan 29 | lecture | Standard form theorems for immersions and submersions; implicit function theorem; proper injective immersions; transversality; statement of Sard's Theorem; proof of Fubini's measure-zero Theorem |
[GP]: I.3-5, Appendix I Lecture Notes Download Lecture Notes (including half of next lecture) |
| T, Feb 2 | lecture | Proof of Sard's Theorem; stability; the "easy" Whitney embedding theorem. |
[GP]: I.6-7, Appendix I Lecture Notes Download Lecture Notes (including parts of next lecture) |
| F, Feb 5 | exercises | Examples of regular/critical values and a proof of the fundamental theorem of algebra. Daniel, Jonathan, and Lina presented. | Exercises 2 |
| T, Feb 9 | lecture 10am instead of 1pm! | Existence of smooth partitions of unity; the Whitney embedding theorem (into R2n+1) for compact and noncompact manifolds |
[MT]: Appendix A, Chapter 8 Lecture Notes Download Lecture Notes (including parts of next lecture) |
| F, Feb 12 | exercises | Discussed previous homework, the structure of the tangent bundle, and Lie groups. Daniel, Jonathan and Lina presented. | Exercises 3 |
| T, Feb 16 | lecture | Homotopy and Stability; Manifolds with boundary |
[GP] I.6, II.1 Lecture Notes Download Lecture Notes (including next lecture) |
| F, Feb 19 | exercises | Discussed the previous homework in detail. Daniel and Ludvig presented. | Exercises 4 |
| T, Feb 23 | lecture | Classification of 1-dimensional manifolds, Brouwer's fixed point theorem | [GP] II.2, Appendix 2 |
| F, Feb 26 | exercises | Discussed solutions to previous homework and looked at oriantations. Lina and Ludvig presented. | Exercises 5 |
| T, Mar 2 | lecture | Transversality and transversality homotopy theorems; smooth approximation |
[GP] II.3 |
| F, Mar 5 | exercises | Solutions to Homework 6. Compared definitions of orientations and solved exercise about deformations. | Exercises 6 |
| Exam weeks | |||
| T, Mar 23 | lecture cancelled | ||
| F, Mar 26 | exercises cancelled | ||
| T, Mar 30 | lecture | Unoriented intersection theory, mapping degrees, winding numbers, Jordan-Brouwer separation theorem, Borsuk-Ulam theorem |
[GP] II.4-6 |
| W, Mar 31 | extra exercise session at 1pm-3pm | Solved exercises 1-4 of the previous homework. | Exercises 7 |
| Easter | |||
| T, Apr 13 | lecture | proof of the Borsuk-Ulam theorem; orientations; oriented intersection theory |
[GP] III.1-3 |
| F, Apr 16 | exercises | Solved previous homework and compared the new definition of orientation with the old one. | Exercises 8 |
| T, Apr 20 | lecture | Lefschetz fixed point theory |
[GP] III.4 |
| F, Apr 23 | exercises | Discussed orientations and solved exercises from homework 9. | Exercises 9 |
| T, Apr 27 | lecture | Vector fields and indices; triangulations and Euler characteristics of simplicial complexes; Poincaré-Hopf theorem; Statement and preparations for the Hopf degree theorem. |
[GP] III.5-7 Lecture notes Download Lecture notes, containing parts of next lecture |
| Walpurgis night | |||
| T, May 4 | lecture | proof of the Hopf degree theorem, introduction to Lie groups |
[GP] III.7 Adams, J.F.: Lectures on Lie groups [A], Chapter 1 |
| F, May 7 | exercises | Solved homework 10 and discussed triangulations. Seuri presented problem 2. |
Exercises 10 |
| T, May 11 | lecture | The exponential map for Lie groups, classification of abelian Lie groups, Cartans theorem on closed subgroups of Lie groups. |
[A] Chapter 2 |
| Kristi klämdag | |||
| T, May 18 | lecture | The quotient manifold theorem and applications |
Lee, J. M: Introduction to smooth manifolds, Ch. 7 Lecture notes |
| F, May 21 | exercises | Solved homework 11 and 12. Seuri presented problem 1 and 2 on HW12. | Exercises 11-12 |