Van Kampen’s theorem

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

Theorem 1.

Let $X$ be a topological space which is the union of the interiors of two path connected subspaces $X_{\mathrm{1}}\mathrm{,}{X}_{\mathrm{2}}$. Suppose $X_{\mathrm{0}}\mathrm{:=}{X}_{\mathrm{1}}\mathrm{\cap }{X}_{\mathrm{2}}$ is path connected. Let further $\mathrm{*}\mathrm{\in }{X}_{\mathrm{0}}$ and $i_k\mathrm{:}{\pi }_{\mathrm{1}}\mathit{}\mathrm{(}{X}_{\mathrm{0}}\mathrm{,}\mathrm{*}\mathrm{)}\mathrm{\to }{\pi }_{\mathrm{1}}\mathit{}\mathrm{(}{X}_k\mathrm{,}\mathrm{*}\mathrm{)}$, $j_k\mathrm{:}{\pi }_{\mathrm{1}}\mathit{}\mathrm{(}{X}_k\mathrm{,}\mathrm{*}\mathrm{)}\mathrm{\to }{\pi }_{\mathrm{1}}\mathit{}\mathrm{(}X\mathrm{,}\mathrm{*}\mathrm{)}$ be induced by the inclusions for $k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}$. Then $X$ is path connected and the natural morphism

$${\pi }_1(X_1,*){\mathrm{★}}_{{\pi }_1(X_0,*)}{\pi }_1(X_2,*)\to {\pi }_1(X,*),$$

is an isomorphism, that is, the fundamental group of $X$ is the free product of the fundamental groups of $X_{\mathrm{1}}$ and $X_{\mathrm{2}}$ with amalgamation of ${\pi }_{\mathrm{1}}\mathit{}\mathrm{(}{X}_{\mathrm{0}}\mathrm{,}\mathrm{*}\mathrm{)}$.

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 groupoids allows for a version of the theorem for the non path connected case, using the fundamental groupoid ${\pi }_1(X,A)$ on a set $A$ of base points, [rb1]. This groupoid consists of homotopy classes rel end points of paths in $X$ joining points of $A\cap X$. In particular, if $X$ is a contractible space, and $A$ consists of two distinct points of $X$, then ${\pi }_1(X,A)$ is easily seen to be isomorphic to the groupoid often written $ℐ$ 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 $X_{\mathrm{1}}\mathrm{,}{X}_{\mathrm{2}}$ and let $A$ be a set which meets each path component of $X_{\mathrm{1}}\mathrm{,}{X}_{\mathrm{2}}$ and $X_{\mathrm{0}}\mathrm{:=}{X}_{\mathrm{1}}\mathrm{\cap }{X}_{\mathrm{2}}$. Then $A$ meets each path component of $X$ and the following diagram of morphisms induced by inclusion