Course info

Prerequisites

- 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.

- Familiarity with the definition of a differentiable manifold or at least a submanifold of R^n.

- Familiarity with the definition of a category and a functor is also helpful. For those who haven't seen this before, the "Intensive course on Categories and Modules" http://www.rolandvdv.nl/AT18/#linksat can be recommended. The most relevant bits are Lecture 1 and Lecture 3.3.

Aim of the course

The aim of the course is to introduce the basic tools of algebraic topology: homotopy and (co)homology groups. Homotopy groups are higher-dimensional generalizations of the fundamental group and classify maps from spheres to our chosen space up to homotopy. We will use differential topology to show that they can vanish in low enough degrees and to construct the mapping degree. As computations with homotopy groups are very difficult, we will introduce (co)homology groups of spaces, which are much more computable. On the other hand, they are more difficult to define and we will present a route via Eilenberg--Mac Lane spaces. Using (co)homology, we have tools to show that many spaces are not homeomorphic (and not even homotopy equivalent). We will also present other applications.