TMA4190 Introduction to Topology - Spring 2025
| Schedule | Room | ||
|---|---|---|---|
| Lectures: | Thursday | 10.15 - 12.00 | F3 Gamle fysikk |
| Friday | 12.15 - 14.00 | G21 Geologi | |
| Instructor: | Fernando Abellán | ||
|---|---|---|---|
| Webpage: | https://www.ntnu.no/ansatte/fernando.a.garcia | ||
| Office: | 1204 Sentralbygg 2 | ||
| Email: | fernando [dot] a [dot] garcia [at] ntnu [dot] no | ||
About this course
Topology its a vast field of study in mathematics: Informally speaking, topology studies the general notion of "space" or "shape" and gives a foundational framework for doing geometric constructions. Topology can be seen everywhere: From algebra to analysis and even in applied mathematics and physics.
This course is intented a first introduction to topology. We will define the fundamental notions of study such as topological spaces and continuous maps among them. We will spend some time learning which properties a topological space can satisfy (compactness, connectedness,countability and many more!) and what are the main constructions that can be perfomed with topological spaces (products, quotients,etc).
By the end of the course we will make a short introduction to algebraic topology: We will define the fundamental group of a topological space and perfom elementary computations.
What you need to know before this course
You should have seen multivariate calculus and linear algebra. Some abstract algebra knowledge would be ideal (TMA4150 Algebra and/or MA3201 Rings and Modules) for the last part of this course. However, this is not mandatory and I will take care to define the necessary algebraic notions when the time comes.
If you have any questions, feel free to send me an email!
Lecture Plan
The first lectures will be January 9 and 10.
| Lecture | Date | |
|---|---|---|
| 0.1 | 09.01 | Introduction. |
| 0.2 | 10.01 | Metric spaces, continuous functions |
| 1.1 | 16.01 | Topological spaces: First definitionsand examples |
| Ex 0 | 17.01 | Solutions to exercise sheet 0 |
| 1.2 | 23.01 | Topological spaces: Continuous maps, homeomorphisms, closure, interior |
| 1.3 | 24.01 | Basis of a topology, subspace topology |
| 2.1 | 30.01 | Subspace topology, product topology, universal properties |
| Ex 1 | 31.01 | Solutions to exercise sheet 1 |
| 2.2 | 06.02 | Universal property of the product topology, quotient topology |
| 2.3 | 07.02 | Quotients, open maps, universal property of the quotient topology |
| 2.4 | 13.02 | Connected spaces, path connectedness |
| 2.5 | 14.02 | Connected spaces, Hausdorff spaces, compact spaces. |
| 2.6 | 20.02 | Compact spaces, product of compact spaces is compact. |
| Ex 2 | 21.02 | |
| xx | 27.02 | NO CLASS |
| xx | 28.02 | NO CLASS |
| 3.1 | 06.03 | Homotopy between maps, homotopy as equivalence relation, path homotopy |
| 3.2 | 07.03 | Concatenation of paths, associativity , unitality |
| 3.3 | 13.03 | Fundamental group, fundamental group of a product of spaces |
| Ex.3 | 14.03 | Solutions to Exercise sheet 3 |
| 3.4 | 20.03 | Homotopy equivalences and the fundamental group |
| Ex.4 | 21.03 | Solutions to exercise sheet 4 |
| 4.1 | 27.03 | Covering spaces |
| 4.2 | 28.03 | Homotopy lifting property covering spaces |
| 4.3 | 3.04 | Fundamental group of the circle and applications |
| Ex. 5 | 4.04 | Solutions to Exercise sheet 5 |
| Mock | 10.04 | Mock Exam |
THE END
Exercise sheets
Exercise sheet 0 | Metric spaces|
Exercise sheet 1 | Topological spaces|
Exercise sheet 2 | Constructing topological spaces |
Exercise sheet 3| Topological properties: Compact spaces|
Exercise sheet 4 | Connected and path connected topological spaces|
Exercise sheet 5 | Homotopy theory|
Mock Exam | Solutions to the Mock Exam|
Solutions to the exam
Reference group
Jørgen Risanger Sønstabø: jorgen.r.sonstabo@ntnu.no
Daniel Rustøen: danieru@stud.ntnu.no
David Persson: david.persson@ntnu.no
References
We will not follow any particular textbook.
Please follow the references listed below if you wish to do the exam but you are not planning to attend the lectures!
I do not have notes to share
Some books on general topology:
- [J] K. Jänich, Topology, Springer, 1984.
- [Mu] J.R. Munkres, Topology: a first course, Prentice-Hall, 1975.
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.