TMA4192 Differential Topology - Spring 2022

Welcome to Differential Topology!

Classes start on Thursday, January 13. The first two weeks (at least) will be only online on zoom. Further info will follow soon, both here and via Blackboard. So do sign up and check this page regularly! :)
The course will follow these Lecture Notes. However, they will be updated regularly during the course. The notes are mainly based on the book by Guillemin and Pollack [GP].
You can use this course to fulfil the requirements for the Master degree to take classes in Geometry & Topology. For more details contact Gereon in his capacity as the academic study adviser for the Master program.
Please make sure you chose your spot in the exam schedule and contact Gereon as soon as possible if there are any problems or questions.

	Schedule		Room
Lectures:	Thursday	12.15-14.00	zoom and H3 Hovedbygg
	Friday	12.15-14.00	zoom and R4 Realfagsbygg
Office Hours:	upon request
Exam:	oral exams		schedule
	Lecturers
	Gereon Quick	Melvin Vaupel
Office:	1246 in Sentralbygg 2
Email:	gereon [dot] quick [at] ntnu [dot] no	melvin [dot] vaupel [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 the inverse function theorem, differentiable structures and smooth manifolds, tangent bundles, embeddings, submersions and regular/critical points. Important examples of spaces are surfaces, spheres, and projective spaces. Other key concepts are homotopy, the fundamental group and covering spaces. Applications presented in the course may range from Brouwer's fixed point theorem to vector fields on spheres. These methods and ideas have been influential to and are used in many other parts of mathematics, but also in physics and other areas of application. See the study handbook for more 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, 2 and 3 or similar courses. For example, you should have seen what it means for a function from \( f \colon \mathbb{R}^n \to \mathbb{R}^m\) to be differentiable and what it means for a subset \(U\subseteq \mathbb{R}^n\) to be open, closed, compact,… :)

Lecture Plan

This is our tentative lecture plan. We will most likely be a bit slower than scheduled. Here are the collected Lecture Notes.

Week	Topic	Reference	Exercises	Notes
2.1	Introduction			Lecture 01
2.2	Topology in \(\mathbb{R}^n\) and smooth maps	GP 1.1		Lecture 02
3.1	Smooth manifolds	GP 1.1	Set 01 + Soln	Lecture 02 + Lecture 03 + Notes 03
3.2	Examples of smooth manifolds	GP 1.1		Lecture 03 + Set 01
4.1	Tangent spaces and derivatives	GP 1.2	Set 02 + Soln	Lecture 04
4.2	The Inverse Function Theorem and Immersions	GP 1.3		Lecture 05
5.1	Immersions and Embeddings	GP 1.3	Set 03 + Soln	Lecture 06
5.2	Submersions and Regular Values	GP 1.4	Set 04 + Soln	Lecture 07
6.1	More on regular values and submanifolds	GP 1.4		Lecture 07 + Lecture 08
6.2	Lie groups		Set 05 + Soln	Lecture 08
7.1	Milnor's proof of the Fundamental Theorem of Algebra	M 1.3 + GP 1.4		Lecture 09
7.2	Transversality	GP 1.5	Set 06 + Soln	Lecture 10
8.1	Transversality of submanifolds	GP 1.5		Lecture 11
8.2	The Theorem of Sard and Brown	GP 1.7		Lecture 12
9.1	Smooth Homotopies	GP 1.6	Set 07 + Soln	Lecture 13
9.2	Tangent Bundles and Embedding Manifolds in Euclidean Space	GP 1.8		Lecture 14
10.1	Abstract Manifolds and Real Projective Space		Set 08 + Soln	Lecture 15
10.2	Stiefel Manifolds and Grassmannians			Lecture 16
11.1	Partition of unity and Whitney's Embedding Theorem	GP 1.8		Lecture 17
11.2	Manifolds with Boundary: Motivation	GP 2.1	Set 09 + Soln	Lecture 18
12.1	Manifolds with Boundary and one-manifolds	GP 2.2 + M 2		Lecture 18 + Lecture 19
12.2	Brouwer Fixed Point Theorem	GP 2.2 + M 2	Set 10 + Soln	Lecture 19
13.1	Intersection Numbers modulo 2	GP 2.4 + M 4	Set 11 + Soln	Lecture 21
13.2	Degrees modulo 2	GP 2.4 + M 4		Lecture 21
14.1	Intersection Numbers: Examples and Exercises	GP 2.4 + M 4		Lecture 21 + Set 11
14.2	Winding Numbers and the Borsuk-Ulam Theorem	GP 2.5 + 2.6	Set 12 + Soln	Lecture 22
15.1	Easter Break
15.2	Easter Break
16.1	Outlook: Orientations	GP 3.2 + M 5
16.2	Outlook: Intersection Theory revisited	GP 3.3 + M 5

Reference Group

Members: Naglius Kirvaitis Brandal, Eivind Xu Djurhuus, Mikkel Elkjær Kårstein

1st reference group meeting: 09 February 2022 at 13.15
2nd reference group meeting:
3rd reference group meeting:

Course material

Main sources:

[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.

Some other interesting books are:

[BM] J. Baez, J. Muniain, Gauge fields, knots and gravity, World Scientific Publishing, 1994.
[D] B. Dundas, Differential Topology, Version January 2013.
[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.

Jeff Weeks's geometry games.