Algebraic Topology IV

MATH4161

Lecturer : Dirk Schuetz

Term : Epiphany 2014

Lectures :
  • Monday 9:00am in CM 225a
  • Wednesday 9:00am 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.
Here are some notes on what has been covered so far. Notice that examples are rather short and proofs are omitted, so you should still get your notes from the lectures.

Assignments

Homework Date Hand in Solutions
Problem set 1 pdf 22.01. 12.02. pdf
Problem set 2 pdf 05.02. 26.02. pdf
Problem set 3 pdf 19.02. 12.03. pdf
Problem set 4 pdf 05.03. 30.04. pdf
Problem set 5 pdf 19.03. - pdf

Problem Class Date Solutions
pdf 10.02. pdf
pdf 10.03. pdf

Links

  • Allen Hatcher's Homepage

Lecture Outline

Date Outline
19.03. In this lecture we look we prove Alexander duality for spheres embedded in spheres, and use this to prove the generalized Jordan Curve Theorem.
17.03. In this lecture we look at the direct limit without calling it the direct limit, and use this to prove special cases of Alexander duality.
12.03. In this lecture we see the Alexander and Lefschetz Duality Theorems, and some applications of them.
10.03. In this lecture we see have a problems, problems, Problems Class.
05.03. In this lecture we see some properties of the cap product, and state the Poincaré Duality Theorem.
03.03. In this lecture we describe the fundamental class of a compact manifold with triangulation and the cap-product.
26.02. In this lecture we define homology with coefficients and look at the homology of manifolds.
24.02. In this lecture we discuss naturality of the cup product, and show how it can be used to calculate more cohomology rings.
19.02. In this lecture we calculate the cohomology ring of the torus, discuss commutativity of the cup product and give relative versions of it.
17.02. In this lecture we show some calculations of the cup product, especially for the torus and the projective plane.
12.02. In this lecture we define the cup-product, and show how it turns cohomology into a ring.
05.02. In this lecture we see applications of the Universal Coefficient Theorem, and use it to calculate cohomology groups of certain spaces.
03.02. In this lecture we see some more properties of Ext, and meet the Universal Coefficient Theorem.
29.01. In this lecture we define Ext, the right derived friend of Hom, and give some calculations.
27.01. In this lecture we obtain the long exact sequences in cohomology of a pair and of a union of two open sets.
22.01. In this lecture we define cochain complexes and the singular cohomology groups of a topological space. We will also consider certain properties and examples.
20.01. In this lecture we consider the set of homomorphisms between two abelian groups and study some of its properties.

Last modified: 09.05.2014.