VanKampenTheoremForGroupsAndGroupoids 2010-12-13 13:54:25 2010-12-13 14:29:36 Theorem fundamental groups pushout frundamental groups pushouts Van Kampen's theorem for groups topological spaces algebraic topology \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\grpL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\rO}{{\rm O}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\SL}{{\rm Sl}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\Symb}{{\rm Symb}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} %\newcommand{\grp}{\mathcal G} \renewcommand{\H}{\mathcal H} \renewcommand{\cL}{\mathcal L} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathsf{G}}} \newcommand{\dgrp}{{\mathsf{D}}} \newcommand{\desp}{{\mathsf{D}^{\rm{es}}}} \newcommand{\grpeod}{{\rm Geod}} %\newcommand{\grpeod}{{\rm geod}} \newcommand{\hgr}{{\mathsf{H}}} \newcommand{\mgr}{{\mathsf{M}}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathsf{G)}}} \newcommand{\obgp}{{\rm Ob(\mathsf{G}')}} \newcommand{\obh}{{\rm Ob(\mathsf{H})}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\grphomotop}{{\rho_2^{\square}}} \newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\grplob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\grpa}{\grpamma} %\newcommand{\grpa}{\grpamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} %\newcommand{\>}{{\rangle}} %\usepackage{geometry, amsmath,amssymb,latexsym,enumerate} %\usepackage{xypic} \def\baselinestretch{1.1} \hyphenation{prod-ucts} %\grpeometry{textwidth= 16 cm, textheight=21 cm} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} %\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$ %{\mbox{}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} \textbf{Van Kampen's theorem for fundamental groups} may be stated as follows: \emph{Let $X$ be a topological space which is the union of the interiors of two path connected subspaces $X_1, X_2$. Suppose that $X_0:=X_1\cap X_2$ 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 morphisms draw a commutative pushout diagram}: $$\xymatrix{{\pi_1({U_1 \cap U_2})}\ar [r]^{i_1}\ar[d]^{i_2} &\pi_1(U_1)\ar[d]^{j_1} \\ {\pi_1(U_2)}\ar [r]_{j_2}&{\pi_1(X)}}$$ 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 $\pi_1(X_0,*)$. 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, \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. \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 $$\xymatrix{{\pi_1(X_0,A)}\ar [r]^{\pi_1(i_1)}\ar[d]_{\pi_1(i_2)} &\pi_1(X_1,A)\ar[d]^{\pi_1(j_1)} \\ {\pi_1(X_2,A)}\ar [r]_{\pi_1(j_2)}& {\pi_1(X,A)}} $$ 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}{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}