Fundamental Groupoid and van Kampen’s Theorem

Abstract

In many geometrically relevant situations, a space can be decomposed into smaller spaces. This might either mean that one has a suitable cover by subspaces or the space arises as a gluing of subspaces right away. Our first goal is to find means that in both cases allow to compute the fundamental group of the space in terms of the fundamental groups of its constituents. We should keep in mind, however, that the fundamental group π₁(X, x₀) of a topological space X with base point x₀ ∈ X depends and informs on the path component of x₀ in X only. Consequently, any decomposition theorem on the fundamental group that we endeavor to come up with will have to include assumptions on path connectedness. The proof would moreover involve some cumbersome juggling with base points. To avoid these nuisances, we generalize the concept of fundamental group to the notion of fundamental groupoid for which we obtain a clean statement and proof of a decomposition result: the van Kampen theorem in the groupoid version due to R. Brown.