Algebraic Topology IV

MATH4161

Lecturer : Dirk Schuetz

Term : Epiphany 2015

Lectures :
  • Monday 14:00 in CM 219
  • Friday 14:00 in CM 219

Problems classes:

  • Thursday 22.01, 05.02, 19.02 and 05.03 at 16:00 in CM 101

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 16.01. 26.01. pdf
Problem set 2 pdf 30.01. 09.02. pdf
Problem set 3 pdf 09.02. 23.02. pdf
Problem set 4 pdf 23.02. 09.03. pdf
Problem set 5 pdf 13.03. - pdf

Problem Class Date Solutions
pdf 22.01. pdf
pdf 05.02. pdf
pdf 19.02. pdf
pdf 05.03. pdf

Links

  • Allen Hatcher's Homepage

Lecture Outline

Date Outline
13.03. In this lecture we give more examples of maps from a space to itself which may or may not have fixed points.
09.03. In this lecture we will prove the Lefschetz Fixed Point Theorem, and give plenty of examples of it.
06.03. In this lecture we will prove the Simplicial Approximation Theorem, the Hopf Trace Formula and define the Lefschetz number.
02.03. In this lecture we will have a closer look at the Simplicial Approximation Theorem.
27.02. In this lecture we see more duality theorems and their applications.
23.02. In this lecture we see a few properties of the cap-product, and meet duality theorems.
20.02. In this lecture we see more of the fundamental class, and we define the cap-product.
16.02. In this lecture we describe the fundamental class of a compact manifold with triangulation.
13.02. In this lecture we discuss naturality of the cup product, and show how it can be used to calculate more cohomology rings.
09.02. In this lecture we calculate the cohomology ring of the torus, discuss commutativity of the cup product and give relative versions of it.
06.02. In this lecture we show some calculations of the cup product, especially for the torus and the projective plane.
02.02. In this lecture we define the cup-product, and show how it turns cohomology into a ring.
30.01. In this lecture we see how Ext makes its way into the Universal Coefficient Theorem, and what this means for singular cohomology.
26.01. In this lecture we see some more properties of Ext, and meet the Universal Coefficient Theorem.
23.01. In this lecture we define Ext, the right derived friend of Hom, and give some calculations.
19.01. In this lecture we obtain the long exact sequences in cohomology of a pair and of a union of two open sets.
16.01. In this lecture we define cochain complexes and the singular cohomology groups of a topological space. We will also consider certain properties and examples.
12.01. In this lecture we consider the set of homomorphisms between two abelian groups and study some of its properties.

Last modified: 05.05.2015.