Summer Term 2015: Lecture Course: Algebraic Topology (Master)
Birgit Richter, email: birgit.richter at uni-hamburg.de
Outline: 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 $$f\colon X \rightarrow Y$$ there are homomorphisms $$f_*\colon H_n(X) \rightarrow H_n(Y)$$ 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 algebraic means in order to get geometric statements. We will discuss several examples and some geometric applications such as Poincaré 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: Mo 12:15-13:45h H5, Th 12:15-13:45h H6, Geomatikum. Exercise class: We 12:15-13:45h 415 (Julia Schulz)

Exercise sheets: no 1, no 2, no 3, no 4, no 5, no 6, no 7, no 8, no 9, no 10, no 11, no 12, no 13

Lecture notes: Use at own risk!