TMA4192 Differential Topology - Spring 2021

Welcome to Differential Topology!

  • This is the place where you will find all messages concerning the course.
  • Here is an updated version of the Lecture Notes. The course will follow these notes which are mainly based on the book by Guillemin and Pollack [GP].
  • The exam form has been changed from written to oral exam in order to have a safe and predictable exam. If you don't like oral exams, don't worry, it'll be OK. :)
  • Delta students created a Maths Discord. It has a channel for our class as well. Here are the coordinates to sign up.
  • For the time being, the lectures will take place on campus as initially scheduled (Friday starts at 08.30) in the timetable. The lectures will also be streamed via zoom as usual. Please remember to use the Check-in options when you are on campus!
  • Here is a suggestion for a schedule for the final exam. Please fill in your full name in preferred time slot as soon as possible. Please also contact me if you have any question, time conflict, etc.
Schedule Room
Lectures: Monday 12.15-14.00 online via zoom and in B1 Berg
Friday 08.30-10.00 online via zoom and in MA24 Grønnbygget
Office Hour: upon request online via zoom
Exam: oral exams
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 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 lecture plan. Note that we deviated slightly at the end. Here are the collected Lecture Notes.

Week Topic Reference Exercises Notes
2.1 Introduction Lecture 1
2.2 Topology in \(\mathbb{R}^n\) and smooth maps GP 1.1 Lecture 2
3.1 Smooth manifolds GP 1.1 Set 1 + Soln Lecture 3
3.2 Lecture is cancelled
4.1 Tangent spaces and derivatives GP 1.2 Set 2 + Soln Lecture 4
4.2 The Inverse Function Theorem and Immersions GP 1.3 Lecture 5
5.1 Immersions and Embeddings GP 1.3 Set 3 + Soln Lecture 6
5.2 Submersions and Regular Values GP 1.4 Lecture 7
6.1 Milnor's proof of the Fundamental Theorem of Algebra M 1.3 + GP 1.4 Set 4 + Soln Lecture 8
6.2 Transversality GP 1.5 Lecture 9
7.1 Transversality of submanifolds GP 1.5 Set 5 + Soln Lecture 10
7.2 Homotopy and Stability GP 1.6 Set 6 + Soln Lecture 11
8.1 Sard's Theorem and Morse functions GP 1.7 Lecture 12
8.2 Tangent Bundles and Embedding Manifolds in Euclidean Space GP 1.8 Lecture 13
9.1 Abstract Manifolds and Real Projective Space Set 7 + Soln Lecture 14
9.2 Stiefel Manifolds and Grassmannians Lecture 15
10.1 Partition of unity and Whitney's Embedding Theorem GP 1.8 Lecture 16
10.2 Manifolds with Boundary GP 2.1 Set 8 + Soln Lecture 17
11.1 Brouwer Fixed Point Theorem and One-Manifolds GP 2.2 + M 2 Lecture 18
11.2 Transversality is generic GP 2.3 Set 9 + Soln Lecture 19
12.1 Intersection Numbers and Degrees modulo 2 GP 2.4 + M 4 Set 10 + Soln Lecture 20
12.2 Intersection Numbers: Examples and Exercises GP 2.4 + M 4 L20 and Set 10
13.1 Easter Break
13.2 Easter Break
14.1 Easter Break
14.2 Winding Numbers and the Borsuk-Ulam Theorem GP 2.5 + 2.6 Set 11 + Soln Lecture 21
15.1 Orientations GP 3.2 + M 5 Set 12 + Soln Lecture 22
15.2 Intersection Theory, Euler characteristic and Degrees GP 3.3 + M 5 Set 13 + Soln Lecture 23
16.1 Lefschetz Fixed-Point Theory GP 3.4 Lecture 24
16.2 Vector bundles and the Poincare-Hopf Theorem GP 3.5 Lecture 25
17.1 Hopf Degree Theorem GP 3.6 + M 7 Lecture 26

Reference Group

Johanna Aigner, Leona Rodenkirchen, Robin Fissum

  • 1st reference group meeting: Friday, 5 February 2021, at 10.00.
  • 2nd reference group meeting: Friday, 19 March 2021, at 10.00.
  • 3rd reference group meeting: Monday, 7 June 2021, at 10.00

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:

  • [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.
  • [G] R. Ghrist, Elementary Applied Topology.
  • [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.

2022-03-31, Gereon Quick