

Lecture Course: Algebraic Topology (Master), Summer Term 2010


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

Plan: 
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: X > Y there are homomorphisms
f_*: H_n(X) > 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 algrebraic means in order to get geometric
statements. We will discuss several examples and some geometric
applications such as Poincare duality.


Students who did not take an algebraic topology course
during their Bachelor studies should still be able to
follow this course. It would be good if you 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. Some of the basics will be recalled in the
exercise classes. 

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 have half
the points in the weekly exercises.
The exercise class on Friday is converted into a tutorial and that
takes place from 12:00h to 13:00h in 432. The exercise class on
Tuesday stays as it is (14:1515:45h, 241).
The lecture course is Tuesday, 12:00h to 13:30h, and Friday, 14:15h to
15:45h, in H5.


Notes for the course, use with caution,
these notes are not proofread and probably full of typos and mistakes.
Exercises No 1, No 2, No 3, No 4, No 5, No 6, No 7, No 8,
No 9, No 10,
No 11, No 12

Books: 
There a lot of good
books on algebraic topology that cover the topics of this
course, for instance:
 A. Hatcher, Algebraic Topology, Cambridge University Press
2002 (free online version
here)

G. Bredon, Topology and Geometry, Springer, 3rd Edition 1997
 R. Stöcker, H. Zieschang, Algebraische Topologie, Teubner
1994

E. Ossa, Topologie, Vieweg+Teubner, 2.Auflage 2009

