Lecturer: Thomas Nikolaus
Office: B 10 (MPI)
eMail: thoni at mpim-bonn.mpg.de
Assistent: Fabian Hebestreit
Office: 3.023
eMail: f.hebestreit at math.uni-bonn.de
Excercise Sessions:
Group 1: Michael Stahlhauer, tuesdays, 10 -12 am
eMail: mstahlhauer at uni-bonn.de
Group 2: Malte Lackmann, thursdays, 10 -12 am
eMail: maltelackmann googlemail.com

Algebraic Topology II


Sheet no 10 has arrived! It contained a typo in exercise 2 (thanks for making us aware). I screwed up the direction of the extension of A by B and correspondingly the domain of the boundary map was false. This is now corrected.

The script is now finally available. Sorry for the delay. Nevertheless it is still very, very, very rough and all comments are welcome. Here it is.


The lectures will take place on mondays 2-4 pm and wednesdays 8-10 am in the 'Zeichensaal' of the Wegelerstraße. The exercise sessions are scheduled for tuesdays 10 am - 12 pm and thursday 10 am - 12 pm in room N 0.007.

There will be oral exams at the end of the semester.

In the lecture we will cover the main modern computational tool in algebraic topology, namely spectral sequences. We will particularly aim for the cohomology of fibre bundles and computations of homotopy groups of spheres. Along the way we will also develop the necessary theoretical background in homotopy theory and (time permitting) present the basics of stable homotopy theory. More specifically we plan to cover the following topics (roughly in order):

  • Spectral sequences in general
  • Serre spectral sequence
  • Some more foundational homotopy theory
  • Computation of rational and low-dimensional homotopy groups of spheres
  • Steenrod operations
  • Complex K-Theory
  • Atiyah-Hirzebruch spectral sequence
  • Adams spectral sequence
  • Excercise sheets

    Sheet no 1
    Sheet no 2 In exercise no 1, it should say rational cohomology!
    Sheet no 3
    Sheet no 4
    Sheet no 5
    Sheet no 6
    Sheet no 7
    Sheet no 8
    Sheet no 9
    Sheet no 10

    Recommended Literature

    Adams: Stable Homotopy and generalised homology
    Ausoni: Algebraische Topologie II Kurzskript (notes for a lecture course, available here, spectral sequences start around december 16th, in german)
    Hatcher: Spectral sequences in algebraic topology (unfinished project, available here)
    McLeary: A user's guide to spectral sequences
    Switzer: Algebraich Topology - Homotopy and Homology
    Weibel: Homological Algebra