In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space X {\displaystyle X} in terms of the fundamental groups of two open, path-connected subspaces that cover X {\displaystyle X} . It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones.
Van Kampen's theorem for fundamental groups Let X be a topological space which is the union of two open and path connected subspaces U1, U2. Suppose U1 ∩ U2 is path connected and nonempty, and let x0 be a point in U1 ∩ U2 that will be used as the base of all fundamental groups. The inclusion maps of U1 and U2 into X induce group homomorphisms j 1 : π 1 ( U 1 , x 0 ) → π 1 ( X , x 0 ) {\displaystyle j_{1}:\pi _{1}(U_{1},x_{0})\to \pi _{1}(X,x_{0})} and j 2 : π 1 ( U 2 , x 0 ) → π 1 ( X , x 0 ) {\displaystyle j_{2}:\pi _{1}(U_{2},x_{0})\to \pi _{1}(X,x_{0})} . Then X is path connected and j 1 {\displaystyle j_{1}} and j 2 {\displaystyle j_{2}} form a commutative pushout diagram:
The natural morphism k is an isomorphism. That is, the fundamental group of X is the free product of the fundamental groups of U1 and U2 with amalgamation of π 1 ( U 1 ∩ U 2 , x 0 ) {\displaystyle \pi _{1}(U_{1}\cap U_{2},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.
Van Kampen's theorem for fundamental groupoids Unfortunately, the theorem as given above does not compute the fundamental group of the circle – which is the most important basic example in algebraic topology – because the circle cannot be realised as the union of two open sets with connected intersection. This problem can be resolved by working with the fundamental groupoid π 1 ( X , A ) {\displaystyle \pi _{1}(X,A)} on a set A of base points, chosen according to the geometry of the situation. Thus for the circle, one uses two base points. This groupoid consists of homotopy classes relative to the end points of paths in X joining points of A ∩ X. In particular, if X is a contractible space, and A consists of two distinct points of X, then π 1 ( X , A ) {\displaystyle \pi _{1}(X,A)} is easily seen to be isomorphic to the groupoid often written I {\displaystyle {\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. The groupoid I {\displaystyle {\mathcal {I}}} also allows for groupoids a notion of homotopy: it is a unit interval object in the category of groupoids.
The category of groupoids admits all colimits, and in particular all pushouts.
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