Van Kampen's Theorem and Fundamental Groupoids

Bachelor Thesis (2023)
Author(s)

Q. Donker (TU Delft - Electrical Engineering, Mathematics and Computer Science)

Contributor(s)

K.P. Hart – Graduation committee member (TU Delft - Analysis)

Robin de Jong – Mentor (Universiteit Leiden)

Faculty
Electrical Engineering, Mathematics and Computer Science
Copyright
© 2023 Quinten Donker
More Info
expand_more
Publication Year
2023
Language
English
Copyright
© 2023 Quinten Donker
Graduation Date
03-07-2023
Awarding Institution
Delft University of Technology
Programme
Applied Mathematics
Faculty
Electrical Engineering, Mathematics and Computer Science
Reuse Rights

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons.

Abstract

This thesis is about Van Kampen's theorem and fundamental groupoids. Van Kampen's Theorem is a classical result in algebraic topology, which proposes a way of calculating the fundamental group of a topological spaces using the fundamental groups of certain subspaces. In this thesis we will construct the fundamental group, which intuitively counts "holes" in a topological space and is mainly used to distinguish topological spaces. Van Kampen's theorem is then proven for a arbitrary large cover using covering spaces. In the proof we glue topological spaces together and an example of this gluing is given for the Klein bottle.

Van Kampen's theorem does not work for every topological space, so we take a look at a generalization of the fundamental group, the so-called fundamental groupoid. Van Kampen's theorem can be upgraded to calculate fundamental groupoid and this theorem is proven in this thesis as well.

Files

License info not available