|| Algebraic Topology
1. Homology theory
2. Singular cohomology
- 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 Mayer-Vietoris 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
- 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
- Alexander-Lefschetz duality and applications
- Duality and cup products
- The Milnor sequence
- Lens spaces
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
\(f_*\colon H_n(X) \rightarrow H_n(Y)\) 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.
For more information refer to:
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.)
A. Hatcher, Algebraic Topology, Cambridge University Press, 2002,
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.
will be linked here.
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
on oral exams.
of Topology (Bachelor), covering point sent topology and elementary homotopy
theory (fundamental group and coverings).
Lecture Notes for the current term.