Algebraic Topology is a continuation of the idea of the fundamental group, i.e. of assigning a group (or other simple algebraic object) to a space in order to measure its properties. We will look at the homology and cohomology groups of a space, which are invariants defined by using the machinery of homological algebra. These groups are easier to compute than the fundamental group but nevertheless powerful invariants. Here are some striking results that are easy to prove with the basic tools of algebraic topology:

There are always two opposite points on the earth with the exact same temperature and humidity.
There is always a place on the earth with no wind.
However messily made, any sandwich with bread, cheese, and tomato can be cut by a straight cut into two halves with the exact same amount of bread, cheese, and tomato in each half.

Algebraic topology is a major branch of pure mathematics with many interconnections to other areas such as geometry, algebra, physics, and even data science.

Course contents

singular homology and cohomology of topological spaces
exact sequences, chain complexes and homology
homotopy invariance of singular homology
the Mayer-Vietoris sequence and excision
cell complexes and cellular homology
the cohomology ring
homology and cohomology of spheres and projective spaces
applications such as the Brouwer Fixed Point theorem, the Borsuk-Ulam theorem and theorems about vector fields on spheres.

Prerequisites: abstract algebra (groups and rings), topology.

For information about lecture planning and what is expected of you, consult the Syllabus tab on the left.

The first part of the course (period 3) is taught by Greg Arone (SU) and the second part (period 4) by Tilman Bauer (KTH). The teaching assistant for the course is Tomas Zeman.

Here is the schedule for the first half of the course.

Literature

Hatcher, Alan: Algebraic Topology, Cambridge University Press 2001. Freely available online and cheaply in print.