
Lecture: 
Algebraic Topology (Master)  Summer term 2020

Instructor: 
Sara Azzali

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
MayerVietoris sequence and the Künneththeorem allow many
concrete calculations. On the level of cohomology we have the
cupproduct. This multiplicative structure together with the
capproduct 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 12.1513.45 (group 1) on Big Blue Button
Thursday 14.1515.45 (group 2) on Big Blue Button

Office hours: 
Fridays 12.1513.45 on Big Blue Button:
https://lernen.min.unihamburg.de/mod/bigbluebuttonbn/view.php?id=9838
or by appointment (do not hesitate to contact me per email
sara.azzali (at) unihamburg.de)

Exam: 
The final exam for this course is an oral exam at the end of term. More details to come.


