The Seifert-Van Kampen theorem via covering spaces

Abstract

[en] The Seifert-Van Kampen theorem describes a way of computing the fundamental group of a space X from the fundamental groups of two open subspaces that cover X, and the fundamental group of their intersection. The classical proof of this result is done by analyzing the loops in the space X and deforming them into loops in the subspaces. For all the details of such proof see [1, Chapter I]. The aim of this work is to provide an alternative proof of this theorem using covering spaces, sets with actions of groups and category theory. On this version of the theorem we are going to ask more conditions on the topological space than in the 'classical' proof. Nevertheless, the spaces which do not follow those requirements are a bit 'pathological'. First, we are going to introduce category theory, and all the concepts which will be needed to follow the proof. Then we are going to talk about group actions, focusing on its categorical implications. After we recall the basics of homotopy theory, we are going to see covering spaces and how do they relate with the fun- damental group. Finally, we are going to prove the theorem of Seifert-van Kampen using covering spaces.