Algebraic Topology II

Summer term 2020
Lecturer: Christoph Winges

Office: 4.014

email: winges at math dot uni-bonn dot de

(mentored by Wolfgang Lück)

 Monday, 2.15 pm
 Wednesday, 8.30 am

 Location: your home

  Wednesday, 2-4 pm
  Location: your home
 Tutor: Thorben Kastenholz

 Thursday, 8-10 am
  Location: your home
 Tutor: Florian Kranhold


All password-protected content on this page can be accessed using the password provided in the first lecture.

Everyone who is entitled to take the second exam has been notified mail. If you think you are eligible for the second exam but have not received a mail, please write to me.

The literature section has been updated, and I have added some details how various references relate to different parts of the lecture course.


We will discuss the general concept of a spectral sequence and one concrete incarnation of this concept, the Serre spectral sequence, together with its applications in computing the homotopy groups of spheres. If time permits, we will study certain operations on the cohomology of spaces called Steenrod squares in the second part of the lecture.

Prerequisites: basics of algebraic topology like singular (co)homology, homotopy groups and some elementary homological algebra

Lecture notes: Below you find the notes of all lectures. To access them, use the password provided in the lecture.
Starting from lecture 3, there are also video recordings available for those who cannot attend the live lectures. To save bandwidth, please download those only if you really intend to watch them.

Lecture 1 from 2020-04-20
Lecture 2 from 2020-04-22
Cheat sheet for lecture 3ff
Lecture 3 from 2020-04-27, video recording (ca 240 MB)
Lecture 4 from 2020-04-29, video recording (ca 175 MB)
Lecture 5 from 2020-05-04, video recording (ca 170 MB)
Lecture 6 from 2020-05-06, video recording (ca 200 MB)
Lecture 7 from 2020-05-11, video recording (ca 200 MB); 2.3.8 was misstated in the lecture, see notes for correction
Lecture 8 from 2020-05-13, video recording (ca 160 MB)
Lecture 9 from 2020-05-18, video recording (ca 180 MB)
Lecture 10 from 2020-05-20, video recording (ca 170 MB); the discussion for Stiefel manifolds was incorrect; please consult the lecture on 2020-05-25
Lecture 11 from 2020-05-25, video recording (ca 205 MB)
Lecture 12 from 2020-06-03, video recording (ca 135 MB)
Lecture 13 from 2020-06-08, video recording (ca 185 MB)
Lecture 14 from 2020-06-10, video recording (ca 220 MB)
Lecture 15 from 2020-06-15, video recording (ca 180 MB)
Lecture 16 from 2020-06-17, video recording (ca 190 MB)
Lecture 17 from 2020-06-22, video recording (ca 150 MB)
Lecture 18 from 2020-06-24, video recording (ca 230 MB)
Lecture 19 from 2020-06-29, video recording (ca 235 MB)
Lecture 20 from 2020-07-01, video recording (ca 210 MB)
Lecture 21 from 2020-07-06, video recording (ca 250 MB)
Lecture 22 from 2020-07-08, video recording (ca 180 MB)

Exercise sheets

Sheet 0 Updated exercise sheet (2020-04-21), solutions
Sheet 1 Due date: 2020-04-27, 2 pm, template, solutions
Sheet 2 Due date: 2020-05-04, 2 pm, template, amended exercise 2 to account for the fact that one might be unable to solve exercise 1, (2020-05-01), solutions
Sheet 3 Due date: 2020-05-11, 2 pm, template, solutions
Sheet 4 Due date: 2020-05-18, 2 pm, template (includes some comments on typesetting spectral sequences), solutions
Sheet 5 Due date: 2020-05-25, 2 pm, template, solutions
Sheet 6 Due date: 2020-06-02, 2 pm, template, corrected wrong dimension in exercise 3 (2020-05-29), solutions
Sheet 7 Due date: 2020-06-08, 2 pm, template, new exercise 1! (2020-05-29), added missing assumption in exercise 2.1 (2020-06-04), solutions
Sheet 8 Due date: 2020-06-15, 2 pm, template, solutions
Sheet 9 Due date: 2020-06-22, 2 pm, template, fixed assertion of 2.4 (2020-06-17), solutions
Sheet 10 Due date: 2020-06-29, 2 pm, template
Sheet 11 Due date: 2020-07-06, 2 pm, template, solutions

There will be some modifications to the usual routine: Exercise sessions will also be conducted using Zoom. You are required to submit TeX'ed solutions to the exercises. To compensate for the increased workload, there will be a reduced number of exercises per sheet and you will be allowed to hand in solutions in small groups of up to three (3) people. Templates for solutions are provided above, but feel free to use your own.


The final examination will take the form of an oral exam conducted via Zoom.

The second batch of exams will be scheduled for the second half of September.


General references about spectral sequences include:
  • Hatcher. Spectral sequences. Unfinished book chapter, available online.
  • McCleary. A User's Guide to Spectral Sequences, Second Edition. Cambridge studies in advanced mathematics 58, Cambridge University Press.
  • Weibel. An introduction to homological algebra. Volume 38 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1994.
Dold's rational Chern character is treated in Section 5 of
  • Hilton. General Cohomology Theory and K-Theory (London Mathematical Society Lecture Note Series). Cambridge University Press, 1971.
If you want to read about surgery and exotic spheres, a good starting point is
  • Crowley, Macko, Lück. Surgery Theory Foundations. Book project, available online
The construction of spectral sequences via filtered complexes and double complexes is treated in Weibel's book. Dress' construction of the Serre spectral sequence is the subject of
  • Dress. Zur Spectralsequenz von Faserungen. Invent. Math. 3:172-178, 1967.
The original reference for the Serre spectral sequence is
  • Serre. Homologie singulière des espaces fibrés. I. La suite spectrale. Ann. Math. (2) 54:425-505, 1951.
and also includes the cohomological Serre spectral sequence including the multiplicative structure, but uses the description of singular homology using cubes instead of simplices. Other references for the multiplicative spectral sequence are
  • Brown. The Serre spectral sequence theorem for continuous and ordinary cohomology. Topol. Appl. 56:235-248, 1994.
and Chapter 5.3 in McCleary's book which is based on Brown's article. A discussion of the multiplicative structure in terms of Dress' construction is contained in lecture notes by Hebestreit, Krause and Nikolaus.
Serre classes, the Hurewicz and Whitehead theorems modulo a Serre class as well as the applications to homotopy groups of spheres are due to
  • Serre. Groupes d'homotopie et classes de groupes abéliens. Ann. of Math. (2), 58:258-294, 1953.
Other general references that I have used in preparing the course are the lecture notes by Hebestreit, Krause and Nikolaus (see above) as well as notes from a lecture course given by Christian Ausoni.

For a comparison of different constructions of the Serre spectral sequence, see
  • Barnes. Spectral sequence constructors in algebra and topology. Mem. Amer. Math. Soc. 317, 1985.
For a comparison of the different approaches to the Hochschild-Lyndon-Serrre spectral sequence, see
  • Beyl. The spectral sequence of a group extension. Bull. Sci. Math. (2) 105:417-434, 1981.

Last modified 2020-07-27