- Background in point-set topology: topological spaces, continuous maps, compactness, quotients and products, and ideally a first encounter with the fundamental group
- Knowledge about basic constructions with vector spaces and abelian groups.
- Some familiarity with categories and functors is also helpful. For those who haven't seen this before, the "Intensive course on Categories and Modules" at the start of the term is recommended.
Aim of the course
This course is an introduction to Algebraic Topology. Its main topic is the study of homology groups of topological spaces and their applications. These homology groups provide algebraic invariants of topological spaces which can be computed in many examples of interest. They are a bit like the fundamental group, only that they are abelian and also better adapted to study higher-dimensional phenomena.
In the first part of the course we will construct the singular homology groups of topological spaces and establish their basic properties, such as homotopy invariance and long exact sequences. In the second part of the course we will introduce CW-complexes. These provide a useful class of topological spaces with favorable properties, and we will explain how the homology of CW-complexes can be computed using cellular homology. We will also discuss some basic concepts from homotopy theory, in particular the generalization of the fundamental group to arbitrary homotopy groups.