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 Failed to parse (unknown function "\co"): {\displaystyle i_k\co \pi_1(X_0,*)\to\pi_1(X_k,*)} , Failed to parse (unknown function "\co"): {\displaystyle 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 Failed to parse (unknown function "\begin{xy}"): {\displaystyle \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', . 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 </math>\{U_\lambda : \lambda \in \Lambda\}$ofsubsets,<refname='brs'/>.Theconclusionisthatif$A$meetseachpathcomponentofall1,2,3-foldintersectionsofthesets<math>U_{\lambda }$, 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.

References

"Van Kampen's theorem" is owned by Ronald Brown.