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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.

See also

Use topology to study your group

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

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