Algebraic Topology II (V4D2), summer term 2022

Stefan Schwede
Mondays, 14:15-16:00 and Wednesdays, 8:15-10:00
Kleiner Hörsaal (Wegelerstr. 10)


This class will be an introduction to stable homotopy theory, based on the category of orthogonal spectra as our model. I plan to cover: orthogonal spectra, stable homotopy groups, smash product, triangulated categories, the stable homotopy category, Eilenberg-MacLane spectra, K-theory spectra, Thom spectra

I am preparing course notes for this class; these notes will occasionally be updated as we progress.


Prerequisites are point set and algebraic topology, in particular covering space theory, CW-complexes, homotopy groups, singular homology and cohomology, Hurewicz theorem, fiber bundles, Serre fibrations, loop spaces, Eilenberg-MacLane spaces.

We will work in the category of compactly generated spaces, which includes the weak Hausdorff condition. References with background information about this category include
- Section 2 of: MC McCord, Classifying spaces and infinite symmetric products. Trans. Amer. Math. Soc. 146 (1969), 273-298
- Section 7.9 of: T tom Dieck, Algebraic Topology, EMS Textbooks in Mathematics, 2008. xii+567 pp.
- Appendix A of: S Schwede, Global homotopy theory, New Mathematical Monographs 34. Cambridge University Press, Cambridge, 2018. xviii+828 pp.
- N Strickland: The category of CGWH spaces. Preprint.


The weekly exercise sheets will be made available on this webpage for download on Fridays, beginning April 15; the completed exercises are due 10 days later before the Monday lecture. To be admitted to the final exam, you have to obtain 50% of the points in the exercises over the course of the term and present at least two solutions on the blackboard. You can submit individual solution sheets, or team up with at most one other person.

There are 3 exercise groups, all meeting in SR 0.007 in the Mathematikzentrum (Endenicher Allee 60).

Time/Place Tutor
Thursdays, 14-16 D. Kirstein
Thursdays, 16-18 F. Zillinger
Fridays, 10-12 U. Flock


The first exams took place July 11-14, 2022. The second exam takes place on September 29, 2022.

S. Schwede, 14.07.2022