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

2025-06-10, Gereon Quick