%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: Van Kampen Theorem for groups and groupoids
%%% Primary Category Code: 00.
%%% Filename: VanKampenTheoremForGroupsAndGroupoids.tex
%%% Version: 19
%%% Owner: bci1
%%% Author(s): bci1
%%% PlanetPhysics is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}
\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}
\usepackage{html}
%\documentclass{amsart}
\usepackage{amsmath}
\usepackage[all,poly,knot,dvips]{xy}
%\usepackage{pstricks,pst-poly,pst-node,pstcol}
\usepackage{amssymb,latexsym}
\usepackage{amsthm,latexsym}
\usepackage{eucal,latexsym}
% THEOREM Environments --------------------------------------------------
\newtheorem{thm}{Theorem}
\newtheorem*{mainthm}{Main~Theorem}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{claim}[thm]{Claim}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}
\numberwithin{equation}{subsection}
%--------------------- Greek letters, etc -------------------------
\newcommand{\CA}{\mathcal{A}}
\newcommand{\CC}{\mathcal{C}}
\newcommand{\CM}{\mathcal{M}}
\newcommand{\CP}{\mathcal{P}}
\newcommand{\CS}{\mathcal{S}}
\newcommand{\BC}{\mathbb{C}}
\newcommand{\BN}{\mathbb{N}}
\newcommand{\BR}{\mathbb{R}}
\newcommand{\BZ}{\mathbb{Z}}
\newcommand{\FF}{\mathfrak{F}}
\newcommand{\FL}{\mathfrak{L}}
\newcommand{\FM}{\mathfrak{M}}
\newcommand{\Ga}{\alpha}
\newcommand{\Gb}{\beta}
\newcommand{\Gg}{\gamma}
\newcommand{\GG}{\Gamma}
\newcommand{\Gd}{\delta}
\newcommand{\GD}{\Delta}
\newcommand{\Ge}{\varepsilon}
\newcommand{\Gz}{\zeta}
\newcommand{\Gh}{\eta}
\newcommand{\Gq}{\theta}
\newcommand{\GQ}{\Theta}
\newcommand{\Gi}{\iota}
\newcommand{\Gk}{\kappa}
\newcommand{\Gl}{\lambda}
\newcommand{\GL}{\Lamda}
\newcommand{\Gm}{\mu}
\newcommand{\Gn}{\nu}
\newcommand{\Gx}{\xi}
\newcommand{\GX}{\Xi}
\newcommand{\Gp}{\pi}
\newcommand{\GP}{\Pi}
\newcommand{\Gr}{\rho}
\newcommand{\Gs}{\sigma}
\newcommand{\GS}{\Sigma}
\newcommand{\Gt}{\tau}
\newcommand{\Gu}{\upsilon}
\newcommand{\GU}{\Upsilon}
\newcommand{\Gf}{\varphi}
\newcommand{\GF}{\Phi}
\newcommand{\Gc}{\chi}
\newcommand{\Gy}{\psi}
\newcommand{\GY}{\Psi}
\newcommand{\Gw}{\omega}
\newcommand{\GW}{\Omega}
\newcommand{\Gee}{\epsilon}
\newcommand{\Gpp}{\varpi}
\newcommand{\Grr}{\varrho}
\newcommand{\Gff}{\phi}
\newcommand{\Gss}{\varsigma}
\def\co{\colon\thinspace}
\begin{document}
Van Kampen's \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html} for fundamental groups is stated as 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 inclusion \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} draw a commutative pushout \htmladdnormallink{diagram}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}: $$\begin{xy}
*!C\xybox{
\xymatrix{ {\pi_1({X_1 \cap X_2)}}\ar [r]^{i_1}\ar[d]^{i_2}
&\pi_1(X_1)\ar[d]^{j_1} \\
{\pi_1(X_2)}\ar [r]_{j_2}& {\pi_1(X)} } }\end{xy}$$
The natural morphism
$$\pi_1(X_1,*)\bigstar_{\pi_1(X_0,*)}\pi_1(X_2,*)\to \pi_1(X,*)\,,$$
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 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 \htmladdnormallink{injective}{http://planetphysics.us/encyclopedia/BCConjecture.html}, and the more precise version of the statement
is in terms of {pushouts} of \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}.
``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,
\cite{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 $\mathcal I$ 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.'' Dr. Ronald Brown also stated the extension of the van Kampen's theorem for \htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} and provided a proof of the new theorem:
\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
$$\begin{xy}
*!C\xybox{
\xymatrix{ {\pi_1(X_0,A)}\ar [r]^{\pi_1(i_1)}\ar[d]_{\pi_1(i_2)}
&\pi_1(X_1,A)\ar[d]^{\pi_1(j_1)} \\
{\pi_1(X_2,A)}\ar [r]_{\pi_1(j_2)}& {\pi_1(X,A)} } }\end{xy}$$
is a pushout diagram in the \htmladdnormallink{category of groupoids}{http://planetphysics.us/encyclopedia/GroupoidCategory4.html}.
\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}{8}
\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}
\end{document}
