Seifert–van Kampen theorem

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In mathematics, the Seifert–van Kampen theorem of algebraic topology, sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space X, in terms of the fundamental groups of two open, path-connected subspaces U and V that cover X. It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones.

The underlying idea is that paths in X can be partitioned: into journeys through the intersection W of U and V, through U but outside V, and through V outside U. In order to move segments of paths around, by homotopy to form loops returning to a base point w in W, we should assume U, V and W are path-connected; and that W isn't empty. We assume also that U and V are open subspaces with union X.

Under these conditions, π1(U,w), π1(V,w), and π1(W,w), together with the inclusion homomorphisms (induced by the inclusion map):

I : π1(W,w) → π1(U,w)

and

J : π1(W,w) → π1(V,w),

are sufficient data to determine π1(X,w). The maps I and J extend to an epimorphism

Φ : π1(U,w) * π1(V,w) → π1(X,w),

where π1(U,w) * π1(V,w) is the free product of π1(U,w) and π1(V,w). The kernel of the map Φ are the loops in W that, when viewed in X, are homotopic to the trivial one at w. The group π1(X,w) is therefore isomorphic to π1(U,w) * π1(V,w) modulo such elements.

In particular, when W is simply connected (so that its fundamental group is the trivial group), the theorem says that π1(X,w) is isomorphic to the free product π1(U,w) * π1(V,w).

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[edit] Equivalent formulations

In the language of combinatorial group theory, π1(X,w) is the free product with amalgamation of those of U and V, with respect to the homomorphisms I and J (which might not be injective): given group presentations

π1(U,w) = <u1,...,uk | α1,...,αl>
π1(V,w) = <v1,...,vm | β1,...,βn>
π1(W,w) = <w1,...,wp | γ1,...,γq>

the amalgamation can be written in terms of generators and relations as π1(X,w) = <u, v | α, β, γ, I(wr)·J(wr)-1> where each letter u, v, w, α, β, γ stands for the respective set of generators or relators, and the final relator means that the images of each generator wr under the inclusions I, J are equivalent in the fundamental group of X.

In category theory, the fundamental group of X is a colimit of the diagram of those of U, V and W. More precisely, π1(X,w) is the pushout of the diagram.

[edit] Examples

One can use Van Kampen's theorem to calculate fundamental groups for topological spaces that can be decomposed into simpler spaces. For example, consider the sphere S2. Pick open sets A = S2 − n and B = S2 − s where n and s denote the north and south poles respectively. Then we have the property that A, B and A ⋂ B are open path connected sets. Thus we can see that there is a commutative diagram including A ⋂ B into A and B and then another inclusion from A and B into S2 and that there is a corresponding diagram of homomorphisms between the fundamental groups of each subspace. Applying Van Kampen's theorem gives the result π1(S2) = π1(A) * π1(B) / ker(Φ). However A and B are both homeomorphic to \mathbf{R^2} which is simply connected, so both A and B have trivial fundamental groups. It is clear from this that the fundamental group of S2 is trivial.


A more complicated example is the calculation of the fundamental group of a genus n orientable surface S, 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 compliment 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 \pi_1(A \cap B)=\pi_1(S^1), which is the integers, and π1(A) = π1(D2) = 1. Thus the inclusion of \pi_1(A \cap B) into π1(A) sends any generator to the trivial element. However, the inclusion of \pi_1(A \cap B) into π1(B) is not trivial. In order to understand this, first one must calculate π1(B). This is easily done as one can deformation retract B (which is S with one point deleted) onto the edges labeled by A1B1A1-1B1-1A2B2A2-1B2-1... AnBnAn-1Bn-1. This space is known to be the wedge product of 2n 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: \{A_1,B_1,\ldots,A_n,B_n\}. We now have enough information to apply Van Kampen's theorem. The generators are the loops \{A_1,B_1,\ldots,A_n,B_n\} (A is simply connected, so it contributes no generators) and there is exactly one relation: A1B1A1-1B1-1A2B2A2-1B2-1... AnBnAn-1Bn-1 = 1. Using generators and relations, this group is denoted

\langle A_1,B_1,\ldots,A_n,B_n|A_1B_1A_1^{-1}B_1^{-1}\ldots A_nB_nA_n^{-1}B_n^{-1}\rangle.

[edit] Generalizations

This theorem has been extended to the non-connected case by using the fundamental groupoid π1(X,A) on a set A of base points, which consists of homotopy classes of paths in X joining points of X which lie in A. The connectivity conditions for the theorem then become that A meets each path-component of U,V,W. The pushout is now in the category of groupoids. This extended theorem allows the determination of the fundamental group of the circle, and many other useful cases. Applications of the fundamental groupoid to the Jordan Curve theorem, covering spaces, and orbit spaces are given in Ronald Brown's book cited below.

There is also a version that allows more than two overlapping sets; for more information on this, see Allen Hatcher's book below, theorem 1.20.

In fact, we can extend van Kampen's theorem significantly farther by considering the fundamental groupoid Π(X), an element of the category of small categories whose objects are points of X and whose arrows are paths between points modulo homotopy equivalence. In this case, to determine the fundamental groupoid of a space, we need only know the fundamental groupoids of the open sets covering X as follows: create a new category in which the objects are fundamental groupoids of the open sets, with an arrow between groupoids if the domain space is a subspace of the codomain. Then van Kampen's theorem is the assertion that the fundamental groupoid of X is the colimit of the diagram. For details, see Peter May's book, chapter 2.

[edit] References