Tricki

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.

Use topology to study your group

Example 1

Van Kampen's theorem (also called the Seifert-van Kampen theorem) describes how to compute the fundamental group of a topological space , written as the union of two open subsets and . If we work just with fundamental groups, then we should assume that , , , and 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 . The van Kampen theorem then states that there is a canonical isomorphism

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