| 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
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.
| 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
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| 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
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