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 .[1]
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 on a set A of base points, chosen according to the geometry of the situation. Thus for the circle, one uses two base points.[2]
is a pushout diagram in the category of groupoids.[4]
This theorem gives the transition from
topology to
algebra, in determining completely the fundamental groupoid ; 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'.[5][6] 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 X is covered by the union of the interiors of a family of subsets.[7][8]
The conclusion is that if A meets each path component of all 1,2,3-fold intersections of the sets , then A meets all path components of X and the diagram
of morphisms induced by inclusions is a
coequaliser in the category of groupoids.
[...] people still obstinately persist, when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under the symmetries of the situation, which thus get lost on the way. In certain situations (such as descent theorems for fundamental groups
à la Van Kampen) it is much more elegant, even indispensable for understanding something, to work with fundamental groupoids with respect to a suitable packet of base points [...]
In the language of
combinatorial group theory, if is a topological space; and are open, path connected subspaces of ; is nonempty and path-connected; and ; then is the
free product with amalgamation of and , with respect to the (not necessarily injective) homomorphisms and . Given
group presentations:
However, A and B are both
homeomorphic to R2 which is
simply connected, so both A and B have
trivial fundamental groups. It is clear from this that the fundamental group of is trivial.
If admits a contractible open
neighborhood and admits a contractible open neighborhood (which is the case if, for instance, and are
CW complexes), then we can apply the Van Kampen theorem to by taking and as the two open sets and we conclude that the fundamental group of the wedge is the
free product
of the fundamental groups of the two spaces we started with:
.
Orientable genus-g surfaces
A more complicated example is the calculation of the fundamental group of a
genus-n orientable surfaceS, otherwise known as the genus-n surface group. One can construct S using its
standard fundamental polygon. For the first open set A, pick a disk within the center of the polygon. Pick B to be the complement in S of the center point of A. Then the intersection of A and B is an
annulus, which is known to be
homotopy equivalent to (and so has the same fundamental group as) a circle. Then , which is the integers, and . Thus the inclusion of into sends any generator to the trivial element. However, the inclusion of into is not trivial. In order to understand this, first one must calculate . This is easily done as one can
deformation retractB (which is S with one point deleted) onto the edges labeled by
This space is known to be the
wedge sum of 2n circles (also called a
bouquet of circles), which further is known to have fundamental group isomorphic to the
free group with 2n generators, which in this case can be represented by the edges themselves: . We now have enough information to apply Van Kampen's theorem. The generators are the
loops (A is simply connected, so it contributes no generators) and there is exactly one relation:
Using generators and relations, this group is denoted
As explained above, this theorem was extended by
Ronald Brown to the non-connected case by using the fundamental groupoid on a set A of base points. The theorem for arbitrary covers, with the restriction that A meets all threefold intersections of the sets of the cover, is given in the paper by Brown and Abdul Razak Salleh.[11] The theorem and
proof for the fundamental group, but using some groupoid methods, are also given in
J. Peter May's book.[12] The version that allows more than two overlapping sets but with A a
singleton is also given in
Allen Hatcher's book below, theorem 1.20.
Applications of the fundamental groupoid on a set of base points to the
Jordan curve theorem,
covering spaces, and
orbit spaces are given in Ronald Brown's book.[13] In the case of orbit spaces, it is convenient to take A to include all the fixed points of the action. An example here is the conjugation action on the circle.
References to higher-dimensional versions of the theorem which yield some information on homotopy types are given in an article on higher-dimensional group theories and groupoids.[14] Thus a 2-dimensional Van Kampen theorem which computes nonabelian second relative
homotopy groups was given by Ronald Brown and Philip J. Higgins.[15] A full account and extensions to all dimensions are given by Brown, Higgins, and Rafael Sivera,[16] while an extension to n-cubes of spaces is given by Ronald Brown and
Jean-Louis Loday.[17]
^Lee, John M. (2011). Introduction to topological manifolds (2nd ed.). New York: Springer.
ISBN978-1-4419-7939-1.
OCLC697506452. pg. 252, Theorem 10.1.
^R. Brown, Groupoids and Van Kampen's theorem, Proc. London Math. Soc. (3) 17 (1967) 385–401.
^P.J. Higgins, Categories and Groupoids, Van Nostrand, 1971, Reprints of Theory and Applications of Categories, No. 7 (2005), pp 1–195.
^R. Brown, Topology and Groupoids., Booksurge PLC (2006).
^Ronald Brown, Philip J. Higgins and Rafael Sivera. Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids, European Mathematical Society Tracts vol 15, August, 2011.
^Brown, Ronald; Salleh, Abdul Razak (1984). "A Van Kampen theorem for unions of nonconnected spaces". Archiv der Mathematik. 42 (1). Basel: 85–88.
doi:
10.1007/BF01198133.
^May, J. Peter (1999). A Concise Introduction to Algebraic Topology. chapter 2.
^Brown, Ronald, "Topology and Groupoids", Booksurge, (2006)
R. Brown and A. Razak, A Van Kampen theorem for unions of non-connected spaces, Archiv. Math. 42 (1984) 85–88. (This paper gives probably the optimal version of the theorem, namely the groupoid version of the theorem for an arbitrary open cover and a set of base points which meets every path component of every 1-.2-3-fold intersections of the sets of the cover.)
Ronald Brown, Higher-dimensional group theory (2007) (Gives a broad view of higher-dimensional Van Kampen theorems involving multiple groupoids).
Greenberg, Marvin J.; Harper, John R. (1981), Algebraic topology. A first course, Mathematics Lecture Note Series, vol. 58, Benjamin/Cummings,
ISBN0805335579
E. R. van Kampen. On the connection between the fundamental groups of some related spaces. American Journal of Mathematics, vol. 55 (1933), pp. 261–267.
Brown, R., Higgins, P. J, On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc. (3) 36 (1978) 193–212.
Brown, R., Higgins, P. J. and Sivera, R.. 2011, EMS Tracts in Mathematics Vol.15 (2011)
Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids; (The first of three Parts discusses the applications of the 1- and 2-dimensional versions of the Seifert–van Kampen Theorem. The latter allows calculations of nonabelian second relative homotopy groups, and in fact of homotopy 2-types. The second part applies a Higher Homotopy van Kampen Theorem for crossed complexes, proved in Part III.)
R. Brown, H. Kamps, T. Porter : A homotopy double groupoid of a Hausdorff space II: a Van Kampen theorem', Theory and Applications of Categories, 14 (2005) 200–220.
Dylan G.L. Allegretti,
Simplicial Sets and Van Kampen's Theorem(Discusses generalized versions of Van Kampen's theorem applied to topological spaces and simplicial sets).
R. Brown and J.-L. Loday, "Van Kampen theorems for diagrams of spaces", Topology 26 (1987) 311–334.