TMA4192 Differential Topology - Spring 2025
Welcome to Differential Topology!
- Our last meeting for this semester will be on Thursday, April 10, 12:15-13:00 in R50. We will discuss orientations and Hopf's theorem. Have a look at the semester plan for the relevant sections in the notes.
- This is the place where you find all information about the course. So please check this page regularly!
- The course will follow these lecture notes. Note that they will be updated regularly during the course. The notes are mainly based on the books by Guillemin and Pollack [GP] and Milnor [M].
- The teaching activities this year will consist of one weekly meeting where we discuss the main topics of the week. We will discuss in the first week which day and time suits best for everyone. Participants are expected to read the relevant sections of the lecture notes in advance and to prepare questions and comments for the discussion.
- This will be the last time the course TMA4192 will be taught.
| Schedule | Room | ||
|---|---|---|---|
| Lectures: | Tuesday | 14:15 – 16:00 | GL-RFB R51 |
| Thursday | 12:15 – 14:00 | GL-RFB R50 | |
| Office Hours: | upon request | ||
| Exam: | 03 June 2025 | 15:00 – 19:00 | Exam info |
| Lecturer | |||
| Gereon Quick | |||
| Office: | 1246 in Sentralbygg 2 | ||
| Email: | gereon [dot] quick [at] ntnu [dot] no | ||
What this course is about
The aim of the course is to introduce fundamental concepts and examples in differential topology and the idea of using algebraic invariants to study geometric objects. The course discusses smooth manifolds, tangent spaces and bundles, embeddings, submersions, regular vs critical points, transversality, homotopy, the Brouwer degree and much more. Important examples of smooth manifolds are spheres, the Möbius band, surfaces, and projective spaces. Applications presented in the course may range from Brouwer's fixed point theorem to vector fields on spheres. The methods and ideas we discuss have had fundamental influence on and are still used in many other parts of mathematics, but also in physics and other areas of science. You may also want to visit study handbook for further information.
What you need to know before this course
Ideally, you will have taken TMA4190 Introduction to Topology before or at the same time with this class. But I will not assume that you have done so. :) Hence my goal is to assume as little previous knowledge as possible, but you should have taken Calculus 1 and 2 andLinear Algebra or similar courses. For example, you should have seen what it means for a function \( f \colon \mathbb{R}^n \to \mathbb{R}^m\) to be differentiable and what it means for a subset \(U\subseteq \mathbb{R}^n\) to be open, closed,… :)
Semester Plan
This is our tentative lecture plan which we will update during the semester. We will most likely be a bit slower than scheduled. :) The numbers of the chapters, sections, and exercises refer to the lecture notes [Q].
| Week | Topics | Chapter | Exercises | Remarks |
|---|---|---|---|---|
| 2 | Introduction + Topology in \(\mathbb{R}^n\) and smooth maps | 2.1 – 2.2 | 2.1 – 2.2 | |
| 3 | Smooth manifolds, Tangent spaces and derivatives | 2.3 – 2.5 | 2.3 – 2.13 | |
| 4 | First examples and computations | 2.3 – 2.5 | ||
| 5 | The Inverse Function Theorem, Immersions and Embeddings | 3.1 –3.3 | 3.1 – 3.7 | |
| 6 | Submersions and Regular Values | 4.1 – 4.4 | 4.1 – 4.10 | |
| 7 | Transversality and Sard's Theorem | 6.1 – 7.1 | 6.1 – 6.6 | |
| 8 | Smooth Homotopies | 8.1 – 8.4 | 8.1 – 8.7 | |
| 9 | Abstract Manifolds and Examples | 9.1 – 9.5 | 9.1 – 9.4 | |
| 10 | Manifolds with Boundary | 10.1 – 10.4 | 10.1 – 10.5 | |
| 11 | One-manifolds and Brouwer Fixed Point Theorem | 11.1 – 11.7 | 11.1 – 11.2 | |
| 12 | Brouwer Degree modulo 2,Winding Numbers, Borsuk-Ulam Theorem | 12.1 – 12.3 | 12.1 – 12.10 | |
| 13 | Hopf invariant and Intersection Numbers modulo 2 | 12.4, 14.1 – 14.5 | 14.1 – 14.8 | |
| 14 | Orientation and Hopf's Theorem | 15 – 16 | ||
| 15 | Summary and outlook |
Reference Group
- Members:
Course material
Main sources:
- [Q] G. Quick, Lecture Notes on Differential Topology, online draft, 2024.
- [GP] V. Guillemin and A. Pollack, Differential Topology, Prentice Hall, 1974.
- [M] J. Milnor, Topology from the differentiable viewpoint, The University Press of Virginia, 1969.
A highly recommended companion to these books is:
- [T] L. W. Tu, An Introduction to Manifolds, Springer Verlag, 2007.
Some other interesting books are:
- [BM] J. Baez, J. Muniain, Gauge fields, knots and gravity, World Scientific Publishing, 1994.
- [D] B. Dundas, A Short Course in Differential Topology, Cambridge Mathematical Textbooks, 2018.
- [MM] J. Milnor, Morse Theory, Annals of Mathematic Studies No. 51, Princeton University Press, Princeton, N.J., 1963.
- [BJ] T. Bröcker, K.Jänich, Introduction to differential topology, Cambridge Univ. Press, 1982.
- [L] J.M. Lee, Introduction to Smooth manifolds, Springer-Verlag.
- [S] D. Spivak, Calculus on Manifolds, Addison-Wesley, 1965.
Some books on general topology:
- [J] K.Jänich, Topology, Springer-Verlag, 1984.
- [Mu] J.R. Munkres, Topology: a first course, Prentice-Hall, 1975.