Algebraic Topology I (V4D1), winter term 2025/26

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

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

Topics

Blakers-Massey theorem, Freudenthal suspension theorem, Hurewicz theorem, CW-approximation, Eilenberg-MacLane spaces, representability of singular cohomology, cohomology operations, Steenrod squares, geometric realization, spaces versus simplicial sets

Literature:
- G. Bredon, Topology and Geometry (Springer)
- A. Hatcher, Algebraic Topology (Cambridge University Press)
- W. Lück, Algebraic Topology (script)
- T. tom Dieck, Algebraic topology (EMS Textbooks in Mathematics)
- S. Schwede, Kohomologie (script, in german)
- S. Schwede, Cohomology operations and the Steenrod algebra (script, last updated Dec. 1, 2025)
- S. Schwede, Spaces versus simplicial sets (script)

Prerequisites

This class is a continuation of my class "Topology II" from the summer term 2025. Prerequisites are point set topology, fundamental group, covering space theory, CW-complexes, higher homotopy groups, singular homology and cohomology, cup product, Künneth and unviersal coefficient theorems, manifolds, Poincaré duality, fiber bundles, Serre fibrations, mapping spaces, loop spaces.

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.

Exercise 01, solutions due: 27.10.25
Exercise 02, solutions due: 03.11.25
Exercise 03, solutions due: 10.11.25
Exercise 04, solutions due: 17.11.25
Exercise 05, solutions due: 24.11.25
Exercise 06, solutions due: 01.12.25
Exercise 07, solutions due: 08.12.25
Exercise 08, solutions due: 15.12.25

There are three exercise groups, all meeting in SR 0.006 in the Mathematikzentrum (Endenicher Allee 60).

Time	Tutor
Mondays, 16-18	Yehor Avdieiev
Tuesdays, 8-10	Pedro Mayorga
Wednesdays, 14-16	Ben Steffan

Exam

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.

The written exam will take place February 4, 2026. The second exam will take place on March 24, 2026.

S. Schwede, 03.12.2025