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:\pi_1(U_1,x_0)\to \pi_1(X,x_0) and j_2:\pi_1(U_2,x_0)\to \pi_1(X,x_0). Then
X is path connected and j_1 and 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 \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 \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 \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. The groupoid \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. :
Theorem. Let the topological space
X be covered by the
interiors of two subspaces
X1,
X2 and let
A be a set which meets each
path component of
X1,
X2 and
X0 =
X1 ∩
X2. Then
A meets each path component of
X and the diagram
P of morphisms induced by inclusion :::: :is a pushout diagram in the category of groupoids. This theorem gives the transition from
topology to
algebra, in determining completely the fundamental groupoid \pi_1(X,A); one then has to use algebra and
combinatorics to determine a fundamental group at some basepoint. One interpretation of the theorem is that it computes homotopy 1-types. To see its utility, one can easily find cases where
X is connected but is the union of the interiors of two subspaces, each with say 402 path components and whose intersection has say 1004 path components. 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 \{U_\lambda : \lambda \in \Lambda\} of subsets. 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. == Equivalent formulations ==