How to compute the fundamental group of a space

Quick description

This article describes various methods for computing the fundamental group of a topological space.

Prerequisites

Example 1

Van Kampen's theorem (also called the Seifert-van Kampen theorem) describes how to compute the fundamental group of a topological space $X$ , written as the union of two open subsets $U$ and $V$ . If we work just with fundamental groups, then we should assume that $X$ , $U$ , $V$ , and $U\cap V$ are all path-connected. If one is willing to work with fundamental groupoids, then these assumptions are not necessary.

Assume first that all the spaces involved are path-connected and non-empty, and fix a base-point $x \in U\bigcap V$ . The van Kampen theorem then states that there is a canonical isomorphism

$\pi_1(X,x) \cong \pi_1(U,x) {*_{\pi_1(U\cap V,x)}} \pi_1(V,x).$

(The construction on the right hand side is an amalgamated product.)

General discussion

This article is incomplete. This article is evidently incomplete, in need of examples, techniques, etc.

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