# TMA4192 Differential Topology - Spring 2024

Welcome to Differential Topology!

• Here are suggestions for solutions for the exam on 07 May and the grading guidelines.
• 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].
• You can use this course to fulfil the requirements for the Master degree to take classes in Geometry & Topology. For more details contact Gereon.
Schedule Room
Lectures: Thursday 12.15-14.00 656 Simastuen in Sentralbygg 2
Friday 08.15-10.00 GL-RFB R92 Realfagsbygg
Office Hours: upon request
Exam: 07 May 2024 09.00-13.00 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,… :)

## Lecture 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 Topic Chapter Exercises Remarks
2.1 Introduction
2.2 Topology in $\mathbb{R}^n$ and smooth maps 2.1 – 2.2 2.1 – 2.2
3.1 Smooth manifolds 2.3 2.3 – 2.8
3.2 Tangent spaces and derivatives 2.4 – 2.5 2.9 – 2.13
4.1 Examples and computations 2.3 – 2.5
4.2 The Inverse Function Theorem and Immersions 3.1 –3.2 3.1 – 3.2
5.1 Immersions and Embeddings 3.2 –3.3 3.3 – 3.7
5.2 Submersions and Regular Values 4.1 – 4.2 4.1 – 4.3
6.1 More on regular values 4.2 – 4.4 4.4 – 4.10
6.2 Milnor's proof of the Fundamental Theorem of Algebra 4.4
7.1 Transversality 6.1 – 6.2 6.1 – 6.6
7.2 Transversality and Sard's theorem 6.2 – 7.1
8.1 Smooth Homotopies 8.1 – 8.4 8.1 – 8.7
8.2 Abstract Manifolds and Real Projective Space 9.1 – 9.2
9.1 Examples: Stiefel and Grassmann manifolds 9.3 – 9.4 9.1 – 9.4
9.2 Tangent Bundles and Embedding Manifolds in Euclidean Space 2.5 + 9.5
10.1 Manifolds with Boundary: Motivation 10.1 – 10.4 10.1 – 10.5
10.2 One-manifolds and Brouwer Fixed Point Theorem 11.1 – 11.7 11.1 – 11.2
11.1 Brouwer Degree modulo 2 12.1 – 12.2 12.1 – 12.4
11.2 Winding Numbers, Borsuk-Ulam Theorem 12.3 12.5 – 12.10
12.1 Hopf invariant and Intersection Numbers modulo 2 12.4, 14.1 – 14.5 14.1 – 14.8
12.2 Tubular Neighborhoods 13.1 – 13.4
13.1 Easter Break
13.2 Easter Break
14.1 Intersection Numbers modulo 2 14.1 – 14.5 14.1 – 14.8
14.2 Orientation 15.1 – 15.4 15.1 – 15.8
15.1 Orientation and Brouwer Degree 15.5 – 15.8, 16.1 16.4 – 16.12
15.2 Hopf invariant and Degree Theorem 16.2 – 16.3 16.1 – 16.12
16.1 Vector Fields and the Poincaré-Hopf Theorem I
16.2 Vector Fields and the Poincaré-Hopf Theorem II

## Reference Group

• Members: Kasper Rettedal Eikeland, Tore Bjerkestrand Braathen, Ulrik Sørgaard Djupvik.

## Course material

Main sources:

• [Q] G. Quick, Lecture Notes on Differential Topology, online draft, 2023.
• [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.