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, Adem relations, 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 Jan. 8, 2026)
- S. Schwede, Spaces versus simplicial sets (script, last updated Jan. 18, 2026)
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 universal 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.
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Exercise 01,
solutions due: 27.10.25
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Exercise 02,
solutions due: 03.11.25
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Exercise 03,
solutions due: 10.11.25
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Exercise 04,
solutions due: 17.11.25
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Exercise 05,
solutions due: 24.11.25
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Exercise 06,
solutions due: 01.12.25
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Exercise 07,
solutions due: 08.12.25
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Exercise 08,
solutions due: 15.12.25
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Exercise 09,
solutions due: 12.01.26
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Exercise 10,
solutions due: 19.01.26
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, between 8:30-10:00 in "Großer Hörsaal" (Wegelerstr. 10). The review of the marked exams takes place February 5, 2026, between 13:00-13:30 in SR 0.011.
The second exam will take place on March 24, 2026.
Instructions for the written exam:
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During the exam, please have an official picture ID
(national ID card, EU drivers licence, or passport) openly visible at your
place for inspection. We do not need to see your student ID.
- We will provide paper, including scrap paper.
- Coats, jackets, bags and backpacks must be left at the front of the room.
We recommend leaving valuables at home.
- Only writing utensils are allowed at your place during the exam.
In particular, the lecture script, personal notes, or text books are not allowed.
- During the exam, you must not have any wireless devices near or on you.
This includes in particular smart phones, tablets, laptops, and smart watches.
- If we detect any items that are not allowed on you or at your place during the exam, we will view this
as an attempt of fraud, and you fail the exam; it is irrelevant whether or not the forbidden
item or device was actually used.
S. Schwede, 18.01.2026