Homotopy Theory
Summer term 2015
lecture by Prof. Dr. E. Vogt, exercises joint with Filipp Levikov

Time and place: 15.0415.07. Wednesdays 1012 in SR031, Arminallee 6.
 Exercise Session: Thursdays 1214, 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 1sphere, 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 Ktheory 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
Assessment
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.
References
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).