MA3403 Algebraic Topology - Fall 2021

Schedule Room
Lectures: Monday 10:15-12:00 MA24
Wednesday 14:15-16:00 H1
Exam: Oral exams TBC
Lecturer
Drew Heard
Office: 1258 Sentralbygg 2
Email: drew [dot] k [dot] heard [at] ntnu [dot] no

What this course is about

Studying geometric objects by associating algebraic invariants to them is a powerful idea that has influenced many areas of mathematics. For example, determining the existence of a map between spaces (often a difficult task) may be translated into deciding whether an algebraic equation has a solution (sometimes a piece of cake). One of the birthplaces of this idea is Algebraic Topology. The goal of the course is to introduce some of the most important examples of such invariants, namely singular homology and cohomology groups. Along the way we are going to calculate many examples and see some applications.

See the study handbook for more information.

What you need to know before this course

The key idea of algebraic topology is to apply algebra to study topological spaces, so to follow the course you should have some background in both algebra and topology. Ideally this means you have already taken the courses "Introduction to topology" (TMA4190) and "Rings and modules" (MA3201); the course "Differential topology" (TMA4192) also covers much of the required background in topology. That said, if you are motivated it is still possible to follow the course even if you haven't taken these other courses yet, as the amount of material that we will actually use is fairly limited. In this case it is a good idea to gain some familiarity with topological spaces, for example by looking at the books by Jänich [J] or Munkres [Mu2].

In addition, it is recommended to take the course "Homological algebra" (MA3204) this semester (but it is not required as we will cover the small amount of homological algebra we need).

If you have any questions, feel free to contact me!

Lecture Plan

Lecture Date Topic Notes Exercises Video
1 23/08 Introduction, review [Q, Lecture 1]
2 25/08 Review of topological spaces, products and coproducts [H, 2.1-2.4] 2.1,2.3,2.7
3 30/08 Homotopies, path-components, singular chains [H,2.5-2.7,3.1] 2.16, 2.17,2.20
4 01/09 Singular chains, singular homology [H,3.1- 3.4] Link
5 06/09 Functoriality and H_0 [H,3.6 - 3.8], [Q, Lecture 4] 3.2,3.3,3.4.
6 08/09 Relative homology and exact sequences [H,4,1.4,2], [Q,Lecture 5] 4.1-4.3 Link Sorry, I am still having technical problems getting it to record from the camera in the room.
7 13/09 SES of complexes, Eilenberg-Steenrod axioms [H,4.3, 4.4] 4.4,4.5
8 15/09 Quotients and excision, homology of spheres [H,4.5,4.6] 4.7 Success!

Here is a link to an exercise sheet from a previous version of this course. Here are the solutions.

Reference group

The reference group consists of Even Aslaksen, Bjørnar Hem, and Chileshe Mwamba. We will meet for the first time on Wednesday 15/09. Please pass any feedback on to them.

Course material

We will not follow any particular textbook. The previous lecture notes will give you a good idea of the content of the course:

References

Other good lecture notes:

Some interesting books:

  • [H] A. Hatcher, Algebraic Topology, Cambridge University Press, 2000.
  • [Mu] J.R. Munkres, Elements of Algebraic Topology, Westview Press, 1996.
  • [F] W. Fulton, Algebraic Topology - A First Course, Springer-Verlag, 1995.
  • [V] J.W. Vick, Homology Theory - An Introduction to Algebraic Topology, Springer, 1994.
  • [Ma] J.P. May, A Concise Course in Algebraic Topology, Chicago Lectures in Mathematics, 1999.
  • [MS] J. Milnor, J. Stasheff, Characteristic Classes, Princeton University Press, 1974.

Some books on general topology:

  • [J] K. Jänich, Topology, Springer, 1984.
  • [Mu2] J.R. Munkres, Topology: a first course, Prentice-Hall, 1975.
2021-09-15, Drew Kenneth Heard