
65401: 
Algebraic Topology
(Master)

Instructor: 
Christoph
Schweigert

Exercise Class: 
Severin Bunk

Office Hours: 
see
webpage

Content: 
1. Homology theory
 Chain complexes
 Singular homology; \(H_0\) and \(H_1\); homotopy invariance
 The long exact sequence in homology; the long exact sequence of a
pair of spaces
 Excision and the MayerVietoris sequence
 Reduced homology and suspension
 Mapping degree
 CW complexes and Cellular homology
 Homology with coefficients
 Tensor products, the universal coefficient theorem and the
topological Künneth formula
2. Singular cohomology
 Definition of singular cohomology
 Universal coefficient theorem for cohomology
 Axiomatic description of a cohomology theory
 Cap product and cup product
 Orientability of manifolds
 Cohomology with compact support and Poincaré duality
 AlexanderLefschetz duality and applications
 Duality and cup products
 The Milnor sequence
 Lens spaces

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
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.
For more information refer to:
http://www.math.unihamburg.de/home/schweigert/ss18/atopology.html

Prerequesites: 
This 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 a 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 my topology class in the winter
term. (The link to the lecture notes is below.)

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

When and where: 
Lectures: see STiNE
Exercice classes: see STiNE
Lecture starts on see STiNE. Exercise
classes start on see STiNE.

Problem sheets: 
will be linked here.

Exam: 
The final exam for this course is an oral exam at the end of term.
Periods in which the exam can be taken will be announced later.
The exam is by individual appointment.
In order to qualify for the exam, you have to present solutions to the weekly
exercises three times in the exercise classes. General
comments
on oral exams.

Lecture notes: 
Lecture Notes
of Topology (Bachelor), covering point sent topology and elementary homotopy
theory (fundamental group and coverings).
Lecture Notes for the current term.


