Algebraic Topology II (V4D2), summer term 2026

Mondays, 14:15-16:00 in Großer Hörsaal and Wednesdays, 8:15-10:00 in Kleiner Hörsaal

Lecturer: Stefan Schwede
Email : schwede (at) math.uni-bonn.de
Assistant: Tobias Lenz
Email: lenz (at) math.uni-bonn.de

Topics

This class will be an introduction to stable homotopy theory, based on the category of orthogonal spectra as a 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, Adams spectral sequence

I will partially base the class on course notes (last updated April 20, 2026); these notes will occasionally be updated as we progress.

Prerequisites

This class is a continuation of my class "Algebraic Topology I" from the winter term 2025/26. Prerequisites are point set topology, fundamental group, covering space theory, CW-complexes, higher homotopy groups, singular homology and cohomology, cup product, Künneth and universal coefficient theorems, manifolds, Poincaré duality, fiber bundles, Serre fibrations, mapping spaces, loop spaces; Blakers-Massey theorem, Freudenthal suspension theorem, Hurewicz theorem; CW-approximation, Eilenberg-MacLane spaces, representability of singular cohomology; cohomology operations, Steenrod squares, Adem relations; geometric realization, spaces versus simplicial sets

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.

Exercises

We provide weekly exercise sheets on Fridays, available here for download. The completed exercises have to be handed in 10 days later before the Monday lecture. Exercise sheets may be handed in jointly by at most three students.

The registration for the exercise groups is administered online via eCampus during the first week of classes. To register for an exercise group, you must join the eCampus course "V4D2 Algebraic Topology II Problem Sessions".

Exercise 01, solutions due: 27.04.26

There are two exercise groups, both meeting in SR N0.003 (Annex) in the Mathematikzentrum (Endenicher Allee 60).

Time (c.t.)	Tutor
Thursdays, 12-14	Yifan Song
Thursdays, 14-16	Pedro Mayorga

Exam

There will be oral exams during the last week of classes (July 20-24, 2026), and a second round of oral exams during September 28-30, 2026.

The prerequisites for admission to the exam are:

to score at least 50% of the points for the exercises
to present at least two solutions to exercises during the course of the term.

S. Schwede, 20.04.2026