MA3408 Algebraic Topology 2: Homotopy Theory

Web Page

The web page for this course will be set up soon, either here or in Blackboard.
To get Blackboard access, you need to

  • have an active NTNU user account,
  • have paid the semester fees for spring 2019, and
  • have registered for the course (check at StudentWeb).

Course Content

The course introduces the basic homotopy theory of spaces (fibrations and cofibrations, homotopy groups) and covers further classical topics in algebraic topology, such as: spectral sequences (in particular the Serre spectral sequence), vector bundles and characteristic classes, and cohomology operations.


We'll build on the course MA3403 Algebraic Topology 1: Homology and Cohomology. This means that I'll assume that you know about topological spaces and that you are somewhat familiar with homology and cohomology. It's also good if you have some familiarity with fundamental groups and covering spaces, but I will recall the definitions in places where I need (to generalize) them.

Class Hours

Classes are Thursdays 12-14 and Fridays 12-14 in 734, Sentralbygg II.

The first class is on 9 Jan, the last class is on 30 Apr.


There will be oral exams with a character grade. If you would like to take the exam, please send my an email with your date and time preferences before the end of April.

Lecture Plan

Lecture Date Topic References
02.1 Jan 09 01. Introduction and overview notes
02.2 Jan 10 Discussion and exercises [Ghrist] Chapter 8.9
03.1 Jan 16 02. Coverings and fiber bundles notes
03.2 Jan 17 03. Lifting homotopies and fibrations notes
04.1 Jan 23 Discussion and exercises
04.2 Jan 24 Class cancelled
05.1 Jan 30 04. Monodromy notes
05.2 Jan 31 Discussion and exercises
06.1 Feb 06 05. Higher homotopy groups notes
06.2 Feb 07 06. The long exact sequence notes
07.1 Feb 13 07. Examples notes
07.2 Feb 14 08. Pushouts notes
08.1 Feb 20 09. Cofibrations notes
08.2 Feb 21 10. Pearls and CW-complexes notes
09.1 Feb 27 Discussion and exercises
09.2 Feb 28 11. The homology of CW complexes notes
10.1 Mar 05 Equality and pedagogical training
10.2 Mar 06 12. The cohomology of a filtered space notes
11.1 Mar 12 13. Spectral sequences: construction notes
11.2 Mar 13 14. Spectral sequences: convergence notes
12.1 Mar 19 15. The spectral sequence of a fibration notes
12.2 Mar 20 16. Examples notes
13.1 Mar 26 17. The Hurewicz theorem notes
13.2 Mar 27 18. The Freudenthal theorem notes
14.1 Apr 02 19. Eilenberg-Mac Lane spaces notes
14.2 Apr 03 20. Killing homotopy groups notes
15.1 Apr 09 Easter break
15.2 Apr 10 Easter break
16.1 Apr 16 21. Representation of cohomology notes
16.2 Apr 17 22. Bockstein notes
17.1 Apr 23 23. Unitary groups and Stiefel manifolds notes
17.2 Apr 24 24. Grassmann manifolds and vector bundles notes
18.1 Apr 30 25. Characteristic classes notes

Textbook and References

You do not have to buy a textbook to follow this course. I will prepare notes for some parts of the class, and there will be specfic references for the others. Everyone should read Chapter 8 of [Ghrist] during the term.

[Ghrist] R. Ghrist, "Elementary Applied Topology", ed. 1.0, Createspace, 2014.

[Bott-Tu] R. Bott, L.W. Tu. Differential forms in algebraic topology. Springer-Verlag, New York-Berlin, 1982.

[Bredon] G.E. Bredon. Topology and geometry. Springer-Verlag, New York, 1993.

[Davis-Kirk] J.F. Davis, P. Kirk. Lecture notes in algebraic topology. American Mathematical Society, Providence, RI, 2001.

[tom Dieck] T. tom Dieck. Algebraic topology. European Mathematical Society (EMS), Zürich, 2008.

[May] J.P. May. A concise course in algebraic topology. University of Chicago Press, Chicago, IL, 1999.

Model Categories

At some point you might want to learn about model categories, one of many abstraction that allows you to do homotopy theory in contexts where the objects no longer need to be topological spaces, but where you can still make sense of fibrations, cofibrations, and equivalences. Here are some references that are suitable introductions.

[Dwyer-Spalinski] W.G. Dwyer, J. Spaliński. Homotopy theories and model categories. Handbook of algebraic topology, 73–126. North-Holland, Amsterdam, 1995.

[Hirschhorn] P.S. Hirschhorn. The Quillen model category of topological spaces. Expo. Math. 37 (2019) 2–24.

2020-12-18, Drew Kenneth Heard