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Homotopy Theory

Summer term 2015

lecture by Prof. Dr. E. Vogt, exercises joint with Filipp Levikov

  • Time and place: 15.04-15.07. Wednesdays 10-12 in SR031, Arminallee 6. 

  • Exercise Session: Thursdays 12-14, SR005, Arnimallee 3.

Course Overview

Roughly, we are interested in studying the set [(X,x),(Y,y)] of homotopy classes of continuous maps between the topological spaces X and Y preserving the base points x and y. For example, if X is the 1-sphere, we obtain the fundamental group. For higher dimensional spheres we even get abelian groups. These are the homotopy groups of (Y,y). Slogan: easy to define but hard to compute. The notions of fibrations and cofibrations and other related concepts help us to get some systematic approach in understanding what is going on.
As in homology theory we will encounter long exact sequences relating these groups in a fairly general setting. We also plan to give some glimpses into stable homotopy theory. We will see that its objects, the so called spectra, give rise to (generalized) homology theories. We already know singular homology theory from Topology 2, but there are many more and very interesting ones, like K-theory and Bordism theory.

Exercise sheets

Every week (Wednesday/Thursday) we will put an exercise sheet online. The solutions have to be submitted before the exercise session the week after.

Lecture Notes

Here you can find scanned lecture notes. It is not impossible that an attentive reader spots a mistake. Those who are less attentive should use these notes at their own risk.

  • Lecture 01
  • Lecture 02
  • Lecture 03 (+ more)
  • Lecture 04
  • Lecture 05
  • Lecture 06
  • Lecture 07 (flaw in the proofs of 4.1 and 4.2 corrected)
  • Lecture 08
  • Lecture 09
  • Lecture 10
  • Lecture 11
  • Lecture 12 + Extra (on the Hurewicz homomorphism)
  • Lecture 13
  • Lecture 14


To receive credits fo the course you need to

  • actively participate in the exercise session and successfully solve the weekly exercises (obtain at least 50% of total points)
  • pass the final exam

To register for the final exam it suffices to register for the course via CMS.


Allen Hatcher, Algebraic Topology. Also available online from the author's website.

Wolfgang Lück, Algebraische Topologie, Vieweg Verlag, (in german). 

Tammo tom Dieck, Algebraic Topology, EMS Textbooks in Mathematics, (in german).