Van Kampen’s theorem

Van Kampen’s theorem for fundamental groupsMathworldPlanetmathPlanetmath may be stated as follows:

Theorem 1.

Let X be a topological spaceMathworldPlanetmath which is the union of the interiors of two path connected subspacesMathworldPlanetmath X1,X2. Suppose X0:=X1X2 is path connected. Let further *X0 and ik:π1(X0,*)π1(Xk,*), jk:π1(Xk,*)π1(X,*) be induced by the inclusions for k=1,2. Then X is path connected and the natural morphismMathworldPlanetmath


is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath, that is, the fundamental group of X is the free productMathworldPlanetmath of the fundamental groups of X1 and X2 with amalgamation of π1(X0,*).

Usually the morphisms induced by inclusion in this theorem are not themselves injective, and the more precise version of the statement is in terms of pushouts of groups.

The notion of pushout in the category of groupoidsPlanetmathPlanetmath allows for a version of the theorem for the non path connected case, using the fundamental groupoidMathworldPlanetmathPlanetmathPlanetmath π1(X,A) on a set A of base points, [rb1]. This groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath consists of homotopy classes rel end points of paths in X joining points of AX. In particular, if X is a contractible space, and A consists of two distinct points of X, then π1(X,A) is easily seen to be isomorphic to the groupoid often written with two vertices and exactly one morphism between any two vertices. This groupoid plays a role in the theory of groupoids analogous to that of the group of integers in the theory of groups.

Theorem 2.

Let the topological space X be covered by the interiors of two subspaces X1,X2 and let A be a set which meets each path component of X1,X2 and X0:=X1X2. Then A meets each path component of X and the following diagram of morphisms induced by inclusion