# TMA4190 Introduction to Topology - Spring 2018

Welcome to Introduction to Topology!

• This is the place where you will find all messages concerning the course.
• Update on Exercise Set 5: Exercise 1c) turned out to be more difficult than intended. Suggestions for additional assumptions are now added. Suggestions for solutions are available. Apologies for the late notice!
• There will be a Q&A-session on Tuesday, May 22, 14:00-15:30 in Seminarroom 734 in Sentralbygg 2. Stop by if you have any questions about the exam. Note the change of day!
• The Lecture Notes are now avaialable as one file: Lecture Notes
Schedule Room
Lectures: Monday 12.15-14.00 R92 Realfagbygget
Thursday 10.15-12.00 R54 Realfagbygget
Exercises: Thursday 16.15-17.00 R21 Realfagbygget
Exam: see here
Lecturer
Gereon Quick
Office: 1246 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

I will 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

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 Lecture 3
3.2 Tangent spaces and derivatives [GP] § 1.2 Set 2 Lecture 4
4.1 The Inverse Function Theorm and Immersions [GP] § 1.3 Lecture 5
4.2 Immersions and Embeddings [GP] § 1.3 Set 3 Lecture 6
5.1 Submersions and Regular Values [GP] § 1.4 Lecture 7
5.2 Milnor's proof of the Fundamental Theorem of Algebra [M] § 1.3 + [GP] § 1.4 Set 4 Lecture 8
6.1 A brief excursion to Lie groups - Part 1 [L] § 7 + [T] § 15 Lecture 9
6.2 A brief excursion to Lie groups - Part 2 [L] § 7 + [T] § 15 Set 5 Lecture 10
7.1 Transversality [GP] § 1.5 Lecture 11
7.2 Transversality [GP] § 1.5 Set 6 Lecture 12
8.1 Lecture is cancelled
8.2 Homotopy and Stability [GP] § 1.6 Set 7 Lecture 13
9.1 Sard's Theorem and Morse functions [GP] § 1.7 Lecture 14
9.2 Embedding Manifolds in Euclidean Space and Abstract Manifolds [GP] § 1.8 Set 8 Lecture 15
10.1 Embedding Abstract Manifolds in Euclidean Space [GP] § 1.8 Lecture 16
10.2 Manifolds with Boundary [GP] § 2.1 Set 9 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 10 Lecture 19
12.1 Intersection Numbers and Degrees modulo 2 [GP] § 2.4 + [M] § 4 Lecture 20
12.2 Winding Numbers and the Borsuk-Ulam Theorem [GP] § 2.5 + 2.6 Set 11 Lecture 21
13.1 Easter Break
13.2 Easter Break
14.1 Easter Break
14.2 Orientation [GP] § 3.2 + [M] § 5 Set 12 Lecture 22
15.1 Intersection Numbers and the Brouwer degree [GP] § 3.3 + [M] § 5 Lecture 23
15.2 Euler characteristic and Lefschetz fixed points [GP] § 3.3 + 3.4 Lecture 24
16.1 Euler characteristic and surfaces Lecture 25
16.2 Two dimensional Quantum Field Theories Lecture 26
17.1 Hopf Degree Theorem [GP] § 3.6 + [M] § 7 Lecture 27

## Reference Group

• 1st reference group meeting: Thursday, February 8, 2018.
• 2nd reference group meeting: Thursday, March 15, 2018.
• 3rd reference group meeting: Tuesday, May 22, 2018.

## Course material

Some interesting books are:

• [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.
• [MM] J. Milnor, Morse Theory, Annals of Mathematic Studies No. 51, Princeton University Press, Princeton, N.J., 1963.
• [D] B. Dundas, Differential Topology. Version January 2013
• [BJ] T.Bröcker, K.Jänich, Introduction to differential topology, Cambridge Univ. Press, 1982.
• [T] L. W. Tu, An Introduction to Manifolds, Springer Verlag.
• [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.