Vorlesung Topologie (Master)
Birgit Richter, email: richter at math.uni-hamburg.de
Plan: This topology course deals with singular homology and cohomology of topological spaces. Homology groups H_n(X), for n = 0,1,2... are abelian groups and they are assigned to a space in a functorial way, i.e. for any continuous map f: X --> Y there are homomorphisms f_*: H_n(X) --> H_n(Y) for n=0,1,2.... Homology groups are in general easier to calculate than homotopy groups, because they have several structural properties (homotopy invariance, long exact sequences for pairs of spaces, additivity, excision etc). Cellular homology, the Mayer-Vietoris sequence and the Künneth-theorem allow many concrete calculations. On the level of cohomology we have the cup-product. This multiplicative structure together with the cap-product that combines cohomology and homology, is a further feature that allows us to use algrebraic means in order to get geometric statements. We will discuss several examples and some geometric applications such as Poincare duality.
Students who did not take an algebraic topology course during their Bachelor studies should still be able to follow this course, but of course this involves some work. You should read something about the basics of algebraic topology (topological spaces, fundamental group, covering spaces). These topics are covered for instance in Bredon, Topology and Geometry, (Chapter I (1,2,3,8,13,14), Chapter III) or Laures-Szymik, Grundkurs Topologie, Kapitel 1,2,6,7,8.
Books:
• A. Hatcher, Algebraic Topology, Cambridge University Press, 2002, available online here
• G. Bredon, Topology and Geometry, Springer, 2010
• R. Stöcker, H. Zieschang, Algebraische Topologie, Teubner 1994
• G. Laures, M. Szymik, Grundkurs Topologie, Spektrum, 2009
Exam: The final exam for this course is an oral exam at the end of term. In order to qualify for the exam, you have to present solutions to the weekly exercises five times in the exercise classes.
When and where: Tu, Fr 14:15--15:45, H5.

Exercise sheets: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

Course notes. These notes are probably full of typos and mistakes. Use at your own risk! I thank Nina Friedrich for spotting numerous typos.