Van Kampen's theorem for fundamental groups is stated as follows:
The natural morphism
is an isomorphism, that is, the fundamental group of
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
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.
is a pushout diagram in the category of groupoids.
The interpretation of this theorem as a calculational tool for
fundamental groups needs some development of `combinatorial groupoid
theory', [2,4]. 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
of subsets, [3]. The conclusion is that if
meets each path component of all 1,2,3-fold intersections of the
sets
, then A meets all path components of
and the
diagram
of morphisms induced by inclusions is a coequaliser in the category of groupoids.
As of this snapshot date, this entry was owned by bci1.