http://www.staff.science.uu.nl/~caval101/
After a quick review of the notion of manifold, the first goal of this course is the classification of compact surfaces. On our way towards proving that theorem, the operation of connected sum and the notions of orientability and of Euler characteristic will be introduced. The course will then focus on a more general class of topological spaces known as cell complexes, or CW-complexes.
We will introduce the notions of homotopy and homotopy equivalence between topological spaces.
The latter is an equivalence relation, and we will only be interested in properties that are invariant under that equivalence relation. We will then introduce an important homotopy invariant called the fundamental group. The fundamental group of a topological space and its relationship to coverings of that space will occupy a major part of the course. The path lifting property for covers will be used to compute the fundamental group of S^1, and we will use van Kampen's theorem to get presentations of the fundamental groups of any CW-complex. On our way, we will also prove a correspondence between covers and subgroups of the fundamental group.
Finally, time permitting, we will end up with a brief introduction to homology.
Prerequisites:
A good understanding of the notion of topological space and of continuous maps.
Some familiarity with the basic operations on topological spaces:
products, quotients, disjoint unions.
Exams forms:
There are two exams for this class both of which count for 45% of the final grade.
Furthermore, there is homework which counts for 10% of the final grade.
Course material:
We will start by following these notes on the classification
book).