65-401:  Algebraic Topology (Master)
Instructor: Christoph Schweigert
Exercise Class: Severin Bunk
Office Hours: see webpage
Content: 1. Homology theory
  • 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
2. Singular cohomology
  • 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
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.
For more information refer to: http://www.math.uni-hamburg.de/home/schweigert/ss18/atopology.html
Prerequesites: 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.)
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
When and where: Lectures: see STiNE
Exercice classes: see STiNE
Lecture starts on Wednesday, April 4. Exercise classes start on Friday, April 13. Note the change of time from Wednesday to Friday
Problem sheets: can be found here
Exam: The final exam for this course is an oral exam at the end of term. Periods in which the exam can be taken are: Monday, July 16, Wednesday - Friday August 22-24, Monday - Wednesday September 10-12. The exam is by individual appointment, to be taken 2 weeks before the exam. In order to qualify for the exam, you have to present solutions to the weekly exercises three times in the exercise classes. General comments on oral exams.
Lecture notes: Lecture Notes of Topology (Bachelor), covering point sent topology and elementary homotopy theory (fundamental group and coverings).
Lecture Notes for the current term.