Van Kampen’s theorem

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

Theorem 1.

Let $X$ be a topological space which is the union of the interiors of two path connectedsubspaces $X_{\mathrm{1}}\mathrm{,}{X}_{\mathrm{2}}$. Suppose $X_{\mathrm{0}}\mathrm{:=}{X}_{\mathrm{1}}\mathrm{\cap }{X}_{\mathrm{2}}$ is path connected. Letfurther $\mathrm{*}\mathrm{\in }{X}_{\mathrm{0}}$ 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 thefree product of thefundamental 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 notthemselves injective, and the more precise version of the statementis in terms of pushouts of groups.

The notion of pushout in the category of groupoids allows for aversion of the theorem for the non path connected case, using thefundamental groupoid ${\pi }_1(X,A)$ on a set $A$ of base points,[rb1]. This groupoid consists of homotopy classes rel endpoints of paths in $X$ joining points of $A\cap X$. In particular,if $X$ is a contractible space, and $A$ consists of two distinctpoints of $X$, then ${\pi }_1(X,A)$ is easily seen to be isomorphic tothe groupoid often written $ℐ$ with two vertices andexactly one morphism between any two vertices. This groupoid plays arole in the theory of groupoids analogous to that of the group ofintegers in the theory of groups.

Theorem 2.

Let the topological space $X$ be covered by the interiors of twosubspaces $X_{\mathrm{1}}\mathrm{,}{X}_{\mathrm{2}}$ and let $A$ be a set which meets each pathcomponent of $X_{\mathrm{1}}\mathrm{,}{X}_{\mathrm{2}}$ and $X_{\mathrm{0}}\mathrm{:=}{X}_{\mathrm{1}}\mathrm{\cap }{X}_{\mathrm{2}}$. Then $A$ meets eachpath component of $X$ and the following diagram of morphisms inducedby inclusion