Inhaltsbeschreibung / course description and syllabus

Moodle
(für Einschreibeschlüssel siehe Inhaltsbeschreibung oben / see course description above for the enrolment key)

Vorlesungsskript / lecture notes
This is a complete set of lecture notes for the courses Topology I-III.

Ankündigungen / Announcements

• 13.04.2026: Die Übung findet in der ersten Semesterwoche nicht statt. / There will be no problem session (Übung) in the first week of the semester.
• 20.04.2026: Starting from 21.04.2026 (the third lecture), the lectures for this course will take place in room 1.013 instead of room 1.115.
• 20.04.2026: The first (and only the first!) meeting of the problem session / Übung, on 22.04.2026, will take place in room 1.013.
• 20.04.2026: Starting from 29.04.2026 (the third week), the problem session / Übung will be divided into two sections meeting at the same time (Wednesdays 15:15-16:45) in separate rooms:
  • Group 1 (J. Campos): room 1.011
  • Group 2 (S. Kumar): room 3.006
  A tool is available on the moodle for students to select which section they attend.
• 11.05.2026: As of Problem Set 5, it is possible to submit solutions and receive grades for select problems on the problem sets. See the syllabus linked above for more on the incentive for doing this.

Exam info (NEW)

The results of the exam on 6.10.2026 are now available on the moodle. Klausureinsicht will take place on Wednesday, 14.10.2026 between 14:30 and 15:30 in 1.301.

Here is a practice exam. It has the same instructions and format that the real exam will have, though it's possible that the real exam may be slightly shorter. It is inevitable on any timed exam that there will not be enough room for some important topics, e.g. this one does not include any problem related to the Galois correspondence, though a problem of that kind could very plausibly appear on the real exam (a nice example is Problem 2 on the Take-Home Midterm). There could also be a problem involving the axiomatic properties of singular homology, such as the long exact sequence, which does not appear on the practice exam.

Update (17.07.26): There is now also a set of handwritten solutions to the practice exam. I have tried to write this as a model of what a perfect exam should look like, with all details of arguments adequately justified, and relavant results from the lectures or the exercises cited. In a few places, one could write less than I wrote here and still get full credit (the solution set contains some added comments about this, written in red).

As indicated in the instructions on the practice exam, the actual exam will be open book, so you will be allowed to use any non-electronic resources you bring with you (i.e. books and notes, printouts of the lecture notes, but not smartphones or laptops). Any material that was covered in the lectures in this course should be considered examinable; all of it is in the lecture notes, though the notes also contain some non-examinable material. To be precise, you may consider each of the following specific topics in the lecture notes to be off limits for the exam:

• Everything in Lecture 1 other than the basic definitions that appear there (the rest is just for motivational purposes)
• The notion of "total boundedness", briefly mentioned at the end of Lecture 5 as an explanation for why compactness and sequential compactness are equivalent in arbitrary metric spaces (that is material for an analysis course, not this one)
• The (sketched) proof of Lemma 14.7, on why the connected sum of two surfaces does not depend on choices
• Everything that involves smooth structures on manifolds, mainly in Lectures 18 and 19
• Everything involving higher homotopy groups and bordism theory (Lecture 21) that we didn't prove, which is most things; bordism theory is nice to know about for motivation, but we can't yet do anything serious with it in this course
• Section 23.4 on fundamental classes and how to visualize relative cycles; this is yet another area in which knowing something about it may be useful for your intuition, but you are not officially required at this point to know anything about fundamental classes of manifolds.
• Section 24.3 on the chain-level version of the excision theorem
• Everything from Section 25.4 onward (including equivalence of simplicial and singular homology, Euler characteristic, axiomatic homology theories, Cech homology and cell complexes)

Übungsblätter / Problem sets

Weekly problem sets (consisting of exercises contained in the lecture notes) will be posted here by Wednesday of each week and discussed in the Übung on the following Wednesday. The exercises marked with an asterisk(*) should be considered essential, but it is highly recommended that you at least think through all exercises as a form of mental preparation for the following week's lectures.

New (11.05.2026): From Problem Set 5 onward, it is possible to submit solutions to the problems labelled with the symbol (†) and receive grades for them. Solutions can be uploaded to the moodle until exactly 15:15 on Wednesday, i.e. the beginning of the Übung.

Important: Press the reload button on the lecture notes to make sure you have the most recent version before looking for the week's exercises; their numbers will change, so you are unlikely to find them if you do not do this!

• Problem Set 1 (to be discussed 22.04.2026): Exercises 2.3, 2.4 (especially parts (a)(*) and (d)(*)), 2.5 (especially part (c)(*)), 2.6(*) and 2.7
• Problem Set 2 (to be discussed 29.04.2026): Exercises 3.3, 3.4 (especially part (c)(*)), 3.5, 3.6, 4.1(*), 4.2 (especially part (b)(*)), 4.3 and 4.4(*)
• Problem Set 3 (to be discussed 6.05.2026): Exercises 5.2 (especially part (b)(*)), 5.3 (especially part (a)(*)), 5.4, 6.2 (especially part (d)(*)), 6.3, 6.4 (especially part (c)(*)) and 6.5 (especially part (b)(*))
• Problem Set 4 (to be discussed 13.05.2026): Exercises 7.3, 7.4, 7.5, 7.6(*), 7.7 (especially part (a)(*)), 7.8 (especially part (b)(*)), 7.9 (especially parts (d)(*) and (e)(*))
• Problem Set 5 (due 20.05.2026): Exercises 8.2, 9.1 (especially part (b)(†)), 10.3 (especially part (c)(†)), 10.5 (especially parts (b)(†), (d)(†) and (e)(†)), 10.6
• Problem Set 6 (due 27.05.2026): Exercises 12.1(†), 12.2 (especially parts (d)(†) and (e)(†)), 12.3, 12.4(†)
• Problem Set 7 (due 3.06.2026): Exercises 14.1, 14.2 (all (†) except part (e)), 14.3 (especially part (a)(†))
• Problem Set 8 (due 10.06.2026): Exercises 15.1 (especially part (e)(†)), 15.2 (especially parts (a)(†) and (c)(†)), 16.2 (especially part (d)(†)), 16.3, 17.1 (especially part (a)(†))
• Take-Home Midterm (due 24.06.2026)
• Problem Set 9 (due 1.07.2026): Exercises 20.3(†) and 21.1
• Problem Set 10 (due 8.07.2026): Exercises 21.2 (especially parts (b)(†) and (c)(†)), 22.1(†), 22.2 (especially parts (c)(†) and (f)(†))
• Problem Set 11 (due 15.07.2026): Exercises 23.1, 23.2, 23.3, 23.4 (especially part (a)(†)), 25.1(†)