PlanetPhysics/Van Kampen Theorems

Van Kampen Theorems

Van Kampen Theorems for Groups and Groupoids
The following two theorems are cited here as originally stated by Ronald Brown in 1983; the full citation follows: \begin{thm} Let $$X$$ be a topological space which is the union of the interiors of two path connected subspaces $$X_1, X_2$$. Suppose $$X_0:=X_1\cap X_2$$ is path connected. Let further $$*\in X_0$$ and $$i_k\co \pi_1(X_0,*)\to\pi_1(X_k,*)$$, $$j_k\co\pi_1(X_k,*)\to\pi_1(X,*)$$ be induced by the inclusions for $$k=1,2$$. Then $$X$$ is path connected and the natural morphism $$\pi_1(X_1,*)\bigstar_{\pi_1(X_0,*)}\pi_1(X_2,*)\to \pi_1(X,*)\,,$$ is an isomorphism, that is, the fundamental group of $$X$$ is the free product of the fundamental groups of $$X_1$$ and $$X_2$$ with amalgamation of $$\pi_1(X_0,*)$$. \end{thm}

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]{http://planetphysics.us/encyclopedia/Pushout.html} 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, . 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 $$\mathcal I$$ 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.

\begin{thm} Let the topological space $$X$$ be covered by the interiors of two subspaces $$X_1, X_2$$ and let $$A$$ be a set which meets each path component of $$X_1, X_2$$ and $$X_0:=X_1 \cap X_2$$. Then $$A$$ meets each path component of $$X$$ and the following diagram of morphisms induced by inclusion $$\begin{xy} \xymatrix{ {\pi_1(X_0,A)}\ar [r]^{\pi_1(i_1)}\ar[d]_{\pi_1(i_2)} &\pi_1(X_1,A)\ar[d]^{\pi_1(j_1)} \\ {\pi_1(X_2,A)}\ar [r]_{\pi_1(j_2)}& {\pi_1(X,A)} } }\end{xy}$$ is a pushout diagram in the category of groupoids. \end{thm}
 * !C\xybox{

The interpretation of this theorem as a calculational tool for fundamental groups needs some development of `combinatorial groupoid theory',. This theorem implies the calculation of the fundamental group of the circle as the group of integers, since the group of integers is obtained from the groupoid $$\mathcal I$$ by identifying, in the category of groupoids, its two vertices.

There is a version of the last theorem when $$X$$ is covered by the union of the interiors of a family \{U_\lambda : \lambda \in \Lambda\}$$ of subsets,. The conclusion is that if $$A meets each path component of all 1,2,3-fold intersections of the sets $$U_\lambda$$, then A meets all path components of $$X$$ and the diagram $$ \bigsqcup_{(\lambda,\mu) \in \Lambda^2} \pi_1(U_\lambda \cap U_\mu, A) \rightrightarrows \bigsqcup_{\lambda \in \Lambda} \pi_1(U_\lambda, A)\rightarrow \pi_1(X,A) $$ of morphisms induced by inclusions is a coequaliser in the category of groupoids.