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, explains the structure of the fundamental group of a topological space X, in terms of two overlapping subspaces U and V, under certain hypotheses about connectedness. It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones. It expresses the idea that paths in X can be separated out: 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.

The conditions are then enough to ensure that π₁(U,w), π₁(V,w), and π₁(W,w), together with the inclusion homomorphisms

I : π₁(W,w) → π₁(U,w)

and

J : π₁(W,w) → π₁(V,w),

are sufficient data to determine π₁(X,w). It is easier to state the result in case W is simply connected, so that its fundamental group is {e}. In that case the theorem says simply that the fundamental group of X is the free product of those of U and V.

1 Equivalent Statements
2 Generalizations
3 Reference
4 External links

[edit] Equivalent Statements

In the general case (that is, dropping the assumption that W is simply connected) the fundamental group of X is a colimit of the diagram of those of U, V and W. In group theorists' terms, it 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,...,u_k | α₁,...,α_l>

π₁(V,w) = <v₁,...,v_m | β₁,...,β_n>

π₁(W,w) = <w₁,...,w_p | γ₁,...,γ_q>

the amalgamation can be written in terms of generators and relations as π₁(X,w) = <u, v, w | α, β, γ, I(w_r)·J(w_r)^-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 w_r under the inclusions I, J are equivalent in the fundamental group of X. Finally, in category theorists' terms, π₁(X,w) is the pushout of the diagram mentioned above.

[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.

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.