PlanetPhysics/Van Kampen Theorems

Van Kampen Theorems

Van Kampen Theorems for Groups and Groupoids

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The following two theorems are cited here as originally stated by Ronald Brown in 1983; the full citation follows: \begin{thm} Let   be a topological space which is the union of the interiors of two path connected subspaces  . Suppose   is path connected. Let further   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  . Then   is path connected and the natural morphism   is an isomorphism, that is, the fundamental group of   is the free product of the fundamental groups of   and   with amalgamation of  . \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   on a set   of base points, [1]. This groupoid consists of homotopy classes rel end points of paths in   joining points of  . In particular, if   is a contractible space, and   consists of two distinct points of  , then   is easily seen to be isomorphic to the groupoid often written   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   be covered by the interiors of two subspaces   and let   be a set which meets each path component of   and  . Then   meets each path component of   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', [2]. 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   by identifying, in the category of groupoids, its two vertices.

There is a version of the last theorem when   is covered by the union of the interiors of a family </math>\{U_\lambda : \lambda \in \Lambda\} A , then A meets all path components of   and the diagram   of morphisms induced by inclusions is a coequaliser in the category of groupoids.

All Sources

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[1] [3] [4] [5]

References

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  1. 1.0 1.1 R. Brown, "Groupoids and Van Kampen's theorem", Proc. London Math. Soc. , (3), 17 ,(1967) 385--401.
  2. Cite error: Invalid <ref> tag; no text was provided for refs named rb,higgins
  3. R. Brown, Topology and Groupoids , Booksurge PLC (2006).
  4. R. Brown and A. Razak, "A van Kampen theorem for unions of non--connected spaces", Archiv. Math. 42, (1984), 85--88.
  5. P.J. Higgins, Categories and Groupoids , van Nostrand, 1971, Reprints of Theory and Applications of Categories, No. 7 (2005), pp 1--195.

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