# TMA4190 Introduction to Topology - Spring 2021

Dear all,

Find all information about the exam in the exam section. Due to the recent spike in the spreading of SARS-CoV-2 it seems as if NTNU will be forced to shut down its campus in Trondheim. Therefore, as a backup, I will create Zoom meetings for all of you having your exam in week 22 (possibly extended also to those of you having your exam in week 23 depending on what is decided and how long the new guidelines will be in effect).

Updated June 1, 10:30am: Oral exams may go ahead as planned. See here for further information. Notice the absolute requirement about NTNU Check-In. I ask that those of you wanting a Zoom exam notify me as soon as possible.

Best wishes,
Marius

Schedule Room
Lectures: Tuesday 14.15 - 16.00 S1
Thursday 10.15 - 12.00 S4
Exam: Practical information
Schedule
Marius Thaule
Office: 1248 Sentralbygg 2
Email: marius [dot] thaule [at] ntnu [dot] no

## What this course is about

This course is a first introduction to topology. It will start with basic concepts in point set topology (e.g. topological spaces, continuous maps, metric spaces, constructions of topological spaces, compactness, connectedness). Then it will give an introduction to algebraic topology, such as homotopy, fundamental group, and covering spaces. A short introduction to homology will give an outlook on computational methods.

## What you need to know before this course

You should have seen multivariate calculus and linear algebra. Ideally you should have taken TMA4150 Algebra and/or MA3201 Rings and Modules as well, but if you haven't don't worry!

If you have any questions, just contact me!

## Lecture Plan

The first lectures will be January 12 and 14.

Lecture Date Topic Notes
1.1 12.01 Introduction. Continuous maps Chapter 1 and 2
1.2 14.01 Continuous maps. Topological spaces Chapter 2 and 3
2.1 19.01 Topological spaces Chapter 3
2.2 21.01 Topological spaces Chapter 3
3.1 26.01 Exercises (from chapter 2) Chapter 2
3.2 28.01 Generating topologies Chapter 4
4.1 02.02 Generating topologies. Exercises (from chapter 3) Chapter 3 and 4
4.2 04.02 Constructing spaces Chapter 5
5.1 09.02 Constructing spaces Chapter 5
5.2 11.02 Constructing spaces Chapter 5
6.1 16.02 Constructing spaces. Exercises (from chapter 4) Chapter 4 and 5
6.2 18.02 Topological properties Chapter 6
7.1 23.02 Topological properties Chapter 6
7.2 25.02 Topological properties Chapter 6
8.1 02.03 Topological properties Chapter 6
8.2 04.03 Topological properties. Exercises (from chapter 5) Chapter 5 and 6
9.1 09.03 The fundamental group Chapter 7
9.2 11.03 The fundamental group Chapter 7
10.1 16.03 The fundamental group Chapter 7
10.2 18.03 The fundamental group Chapter 7
11.1 23.03 The fundamental group. Exercises (from chapter 6) Chapter 6 and 7
11.2 25.03 The fundamental group of the circle Chapter 8
12.1 06.04 Cancelled due to Easter break
12.2 08.04 The fundamental group of the circle Chapter 8
13.1 13.04 The fundamental group of the circle Chapter 8
13.2 15.04 The fundamental group of the circle. Exercises (from chapter 7) Chapter 7 and 8
14.1 20.04 Exercises (from chapter 7 and 8) Chapter 7 and 8
14.2 22.04 Exercises (from chapter 8). Summary Chapter 8
15.1 27.04 Summary. Information about the oral exam Practical information about the exam

## Course material

• Lecture notes

We will not follow any particular textbook.

Some interesting books:

• [A] M.A. Armstrong, Basic Topology, Springer-Verlag, 1983.
• [Croo] F.H. Croom, Basic Concepts of Algebraic Topology, Springer-Verlag, 1978.
• [Cros] M. Crossley, Essential Topology, Spring-Verlag, 2005.
• [H] A. Hatcher, Algebraic Topology, Cambridge University Press, 2000.
• [Ma] J.P. May, A Concise Course in Algebraic Topology, Chicago Lectures in Mathematics, 1999.

Some books on general topology:

• [J] K. Jänich, Topology, Springer, 1984.
• [Mu] J.R. Munkres, Topology: a first course, Prentice-Hall, 1975.

## Exam

Note that you must notify me no later than May 4 if you want your exam to be given via Zoom. I will organize a trial run for those students that want to have their exam via Zoom.