VanKampenTheoremForGroupsAndGroupoids 2010-12-13 13:54:25 2010-12-13 13:54:25 Theorem fundamental groups pushout frundamental groups pushouts Van Kampen's theorem for groups topological spaces algebraic topology %\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} \textbf{Van Kampen's theorem for fundamental groups} may be stated as follows: \begin{theorem} \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 $\pi_1(X_0,*)$.}\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} 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}