TMA4190 Introduction to Topology - Spring 2022

Dear all,

Best of luck with your future studies in topology!

Best wishes,
Marius

Schedule Room
Lectures: Monday 12.15 - 14.00 S6
Tuesday 08.15 - 10.00 EL6
Exam: Oral exams
Marius Thaule
Office: 1248 Sentralbygg 2
Email: marius [dot] thaule [at] ntnu [dot] no

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.

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 10 and 11.

Lecture Date Topic Notes
1.1 10.01 Introduction. Continuous maps Chapter 1 and 2
1.2 11.01 Continuous maps. Topological spaces Chapter 2 and 3
2.1 17.01 Topological spaces Chapter 3
2.2 18.01 Topological spaces. Exercises Chapter 3
3.1 24.01 Exercises. Generating topologies Chapter 2 and 4
3.2 25.01 Generating topologies Chapter 4
4.1 31.01 Generating topologies. Exercises Chapter 3 and 4
4.2 01.02 Constructing spaces Chapter 5
5.1 07.02 Constructing spaces Chapter 5
5.2 08.02 Constructing spaces Chapter 5
6.1 14.02 Constructing spaces. Exercises. Topological properties Chapter 4, 5 and 6
6.2 15.02 Topological properties Chapter 6
7.1 21.02 Topological properties Chapter 6
7.2 22.02 Topological properties Chapter 6
8.1 28.02 Topological properties Chapter 6
8.2 01.03 Topological properties. Exercises Chapter 5 and 6
9.1 07.03 Exercises. The fundamental group Chapter 5 and 7
9.2 08.03 The fundamental group Chapter 7
10.1 14.03 The fundamental group Chapter 7
10.2 15.03 The fundamental group Chapter 7
11.1 21.03 The fundamental group. Exercises. The fundamental group of the circle Chapter 6, 7 and 8
11.2 22.03 The fundamental group of the circle Chapter 8
12.1 28.03 The fundamental group of the circle Chapter 8
12.2 29.03 The fundamental group of the circle Chapter 8
13.1 04.04 The fundamental group of the circle Chapter 8
13.2 05.04 Exercises. Summary Chapter 1 through 7
14.1 18.04 Cancelled due to Easter break
14.2 19.04 Cancelled due to Easter break
15.1 25.04 Summary/Exercises Chapter 1 through 8
15.2 26.04 Exercises Chapter 8

Course material

• Lecture notes

References

We will not follow any particular textbook.

Some interesting books:

• [A] M.A. Armstrong, Basic Topology, Springer-Verlag, 1983.
• [AF] C. Adams and R. Franzosa, Introduction to Topology. Pure and Applied, Pearson Prentice Hall, 2008.
• [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.
• [J] K. Jänich, Topology, Springer, 1984.
• [Ma] J.P. May, A Concise Course in Algebraic Topology, Chicago Lectures in Mathematics, 1999.
• [Mu] J.R. Munkres, Topology: a first course, Second edition, Prentice-Hall, 2000.
• [S] T.B. Singh, Introduction to Topology, Springer-Verlag, 2019.

Oral exams.