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Lecture Course in Algebraic Topology (Master)
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Birgit Richter, email: birgit.richter at
uni-hamburg.de
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| 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 \(f\colon X \rightarrow Y\) there are homomorphisms
\(H_n(f) \colon H_n(X) \rightarrow H_n(Y)\) 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 algrebraic means in order to get geometric
statements. We will discuss several examples and some geometric
applications such as Poincare duality.
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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
Laures-Szymik, 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
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| Exam: |
The final exam for this course is an oral exam after the end of term. In
order to qualify for the exam, you have to present solutions to the
weekly exercises four times in the exercise classes.
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| When? Where? |
We 10-12h H4, Fr 14-16h H3,
Exercise classes: We, 14-16h, 241
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