

Summer Term 2015: Lecture Course: Algebraic Topology (Master)


Birgit Richter, email: birgit.richter at
unihamburg.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
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.


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
LauresSzymik, 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 12:1513:45h H5, Th 12:1513:45h H6, Geomatikum. Exercise class:
We 12:1513:45h 415 (Julia Schulz)

