Algebraic Topology IV

MATH4161

Lecturer : Dirk Schuetz

Term : Epiphany 2013

Lectures :
  • Thursday 9:00am in CM 225a
  • Friday 10:00am in CM 225a

Literature

The material for the course follows mainly the book of Hatcher, which is available from the author's webpage (see link below) or through the library. The other books also contain some or all of the material and can offer a different viewpoint.
  • A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.
  • W. Fulton, Algebraic Topology: a first course, Springer Verlag, 1995.
  • W.S. Massey, A basic course in Algebraic Topology, Springer Verlag, 1991.
  • W.S. Massey, Singular homology theory, Springer Verlag 1980.
  • E. Spanier, Algebraic Topology. McGraw-Hill, 1966.

Assignments

Homework Date Hand in Solutions
Problem set 1 pdf 18.01. 31.01. pdf
Problem set 2 pdf 01.02. - pdf
Problem set 3 pdf 15.02. 01.03. pdf
Problem set 4 pdf 01.03. - pdf
Problem set 5 pdf 15.03. - pdf

Links

  • Allen Hatcher's Homepage

Lecture Outline

Date Outline
15.03. In this lecture we state the duality theorems of Poincaré, Lefschetz and Alexander, and use them prove some very interesting results.
14.03. In this lecture we define homology with coefficients, look at the impact for non-orientable manifolds and define the cap product.
08.03. In this lecture we will discuss Problem Sets 3 and 4.
07.03. In this lecture we look at unordered simplicial complexes, and determine the fundamental class of a compact manifold.
01.03. In this lecture we finish the calculation of the cohomology of the n-torus, and then look closer at topological manifolds and triangulations.
28.02. In this lecture we discuss the tensor product of two chain complexes, and use this to express the cohomology of a product of cell complexes in terms of the tensor product of cohomology groups of the factors. You can still email me if you read this before the lecture.
22.02. In this lecture we discuss the tensor product of two abelian groups, and show its relation to the product of two cell complexes. If you read this before the lecture, please inform me about it via email.
21.02. In this lecture we discuss the naturality of the cup product, and show how this can be used to calculate certain cup products.
15.02. In this lecture we finish the calculation of the cohomology ring of the torus, discuss commutativity of the cup product and give relative versions of it.
14.02. In this lecture we determine the cohomology ring of the projective plane with mod 2 coefficients, and of the torus with integer coefficients.
08.02. In this lecture we discuss problems, problems, problems.
07.02. In this lecture we introduce the cup product and show how it turns cohomology into a graded ring.
01.02. In this lecture we show how the Universal Coefficient Theorem can be used to calculate singular cohomology groups of various spaces.
31.01. In this lecture we give a few calculations of Ext, and meet the Universal Coefficient Theorem.
25.01. In this lecture we obtain the cohomology version of the Mayer-Vietoris sequence and define Ext, the right derived friend of Hom.
24.01. In this lecture we finish the calculation of the cohomology of the projective plane and obtain the long exact cohomology sequence of a pair.
18.01. In this lecture we define cochain complexes and the singular cohomology groups of a topological space. We will also consider certain properties and examples.
17.01. In this lecture we consider the set of homomorphisms between two abelian groups and study some of its properties.

Last modified: 03.05.2013.