Avsnittsöversikt
-
This course is owned jointly by SU and KTH.
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.
The first part of the course (period 3) is taught by Alexander Berglund (SU) and the second part (period 4) by Tilman Bauer (KTH). The teaching assistant for the course is Sylvain Douteau.
Schedule
Classes are on Mondays:
8:30-9:15 exercise session with Sylvain
9:30-11:15 lecture with Alexander/Tilman.
The first lecture is on January 24 (no exercise session this day).
The second half of the course will start on March 14 and be given by Tilman Bauer at KTH, room 3721. Classes will be in-person and no longer be streamed.
Examination
The examination is based on weekly short homework sets (given out on Mondays, due on the following Thursday), along with individual oral presentations of one or two of the homework problems at the end of the course. We will tell you in advance which problem we want you to present in this oral exam, which will be graded on a pass/fail scale.
Literature
Hatcher, Alan: Algebraic Topology, Cambridge University Press 2001. Freely available online and cheaply in print.