Prerequisites
Background in point-set topology: topological spaces, continuous maps, compactness, quotients and products, and maybe 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: http://www.rolandvdv.nl/AT18/#links - The most relevant parts are Lecture 1 and Lecture 3.3.
Aim of the course
This course is an introduction to Algebraic Topology. Its main topic is the study of homology groups of topological spaces. These homology groups provide algebraic invariants of topological spaces which can be computed in many examples of interest.
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.
Rules about Homework/Exam
There will be four hand-in homework sets (to be done in pairs or groups of three) throughout the course and there will be a written exam at the end.
The lowest homework score will be dropped. The score from the remaining homework assignments will count as a bonus for the final examination: if the grade from the written exam is at least 5.0 and the average grade from the homework assignments is higher, then the written exam counts 75% and the homework assignments count 25%. The homework bonus will still be valid for the retake. In case you take the retake, whether the better grade or automatically the retake grade counts depends on your university policy.
Lecture notes/Literature
We will follow these lecture notes: https://www.math.ru.nl/~sagave/teaching/at-lecturenotes-2023.pdf
As for a textbook, Hatcher’s “Algebraic Topology” (freely available online) will be the most relevant book.
- Docent: Renee Hoekzema
- Docent: Inbar Klang