Lecture Course: Algebraic Topology (Master), Summer Term 2010
Birgit Richter, email: birgit.richter at 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. It would be good if you 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. Some of the basics will be recalled in the exercise classes.
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 have half the points in the weekly exercises. The exercise class on Friday is converted into a tutorial and that takes place from 12:00h to 13:00h in 432. The exercise class on Tuesday stays as it is (14:15-15:45h, 241). The lecture course is Tuesday, 12:00h to 13:30h, and Friday, 14:15h to 15:45h, in H5.
Notes for the course, use with caution, these notes are not proofread and probably full of typos and mistakes. Exercises No 1, No 2, No 3, No 4, No 5, No 6, No 7, No 8, No 9, No 10, No 11, No 12
Books: There a lot of good books on algebraic topology that cover the topics of this course, for instance: