Lecture:  Algebraic Topology (Master) - Summer term 2020
Instructor:
Exercise Class: Sara Azzali
Aim: 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.
Prerequesites: The lecture aims at students in the master programs of mathematics, mathematical physics and physics. It is accessible to advanced bachelor students as well.
Students who did not take an algebraic topology course during their Bachelor studies should still be able to follow this course (with some additional work in your own initiative). 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 the lecture notes of Julian Holstein in the last winter term, available here, or in Christoph Schweigert notes, available here
Literature: 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
Lecture notes: Lecture Notes by Birgit Richter
Lecture Notes by Christoph Schweigert.
When and where: Due to the current pandemic, the summer term 2020 starts in digital form on April 20th.
Videolectures are posted weekly on Lecture2Go and are accessible from Moodle.
Exercice classes:
Wednesday 12.15--13.45 (group 1) on Big Blue Button
Thursday 14.15--15.45 (group 2) on Big Blue Button
Office hours: Fridays 12.15-13.45 on Big Blue Button: https://lernen.min.uni-hamburg.de/mod/bigbluebuttonbn/view.php?id=9838
or by appointment (do not hesitate to contact me per email sara.azzali (at) uni-hamburg.de)
Exam: The final exam for this course is an oral exam at the end of term. More details to come.