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\begin{document}
Van Kampen Theorems
\section{Van Kampen Theorems for Groups and Groupoids}
The following two \htmladdnormallink{theorems}{http://planetphysics.us/encyclopedia/Formula.html} are cited here as originally stated by Ronald Brown in 1983; the
full citation follows:
\begin{thm}
Let $X$ be a \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} space which is the \htmladdnormallink{union}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} 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 \htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $$\pi_1(X_1,*)\bigstar_{\pi_1(X_0,*)}\pi_1(X_2,*)\to \pi_1(X,*)\,,$$
is an \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html}, that is, the \htmladdnormallink{fundamental group}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} of $X$ is the
free product of the
\htmladdnormallink{fundamental groups}{http://planetphysics.us/encyclopedia/HomotopyCategory.html} 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 \htmladdnormallink{injective}{http://planetphysics.us/encyclopedia/BCConjecture.html}, and the more precise version of the statement
is in terms of \htmladdnormallink{{pushouts}}{http://planetphysics.us/encyclopedia/Pushout.html} of \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}.
The notion of pushout in the \htmladdnormallink{category of groupoids}{http://planetphysics.us/encyclopedia/GroupoidCategory4.html} allows for a
version of the theorem for the non path connected case, using the
\htmladdnormallink{fundamental groupoid}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} $\pi_1(X,A)$ on a set $A$ of base points,
\cite{rb1}. This \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} consists of \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} 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 \htmladdnormallink{diagram}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of morphisms induced
by inclusion
$$\begin{xy}
*!C\xybox{
\xymatrix{ {\pi_1(X_0,A)}\ar [r]^{\pi_1(i_1)}\ar[d]_{\pi_1(i_2)}
&\pi_1(X_1,A)\ar[d]^{\pi_1(j_1)} \\
{\pi_1(X_2,A)}\ar [r]_{\pi_1(j_2)}& {\pi_1(X,A)} } }\end{xy}$$
is a pushout diagram in the category of groupoids.
\end{thm}
The interpretation of this theorem as a calculational tool for
fundamental groups needs some development of `combinatorial groupoid
theory', \cite{rb,higgins}. 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, \cite{brs}. 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.
\begin{thebibliography}{9}
\bibitem{rb1} R. Brown, ``Groupoids and Van Kampen's theorem'', {\em Proc. London Math. Soc.}, (3), 17 ,(1967)
385--401.
\bibitem{rb} R. Brown, {\em Topology and Groupoids}, Booksurge PLC (2006).
\bibitem{brs} R. Brown and A. Razak, ``A van Kampen theorem for unions of non--connected spaces'', {\em Archiv. Math.} 42, (1984), 85--88.
\bibitem{higgins} P.J. Higgins, {\em Categories and Groupoids}, van Nostrand, 1971, Reprints of Theory and Applications of Categories, No. 7 (2005), pp 1--195.
\end{thebibliography}
"Van Kampen's theorem" is owned by Ronald Brown.
\end{document}
