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Seifert-van Kampen theorem

In mathematics, the Seifert-van Kampen theorem of algebraic topology explains the structure of the fundamental group of a topological space X, in terms of those of two overlapping subspaces U and V, under certain hypothesis 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 π1(W,w) -> π1(U,w) and 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.

In the general case (with W still assumed at least connected, though) 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 from π1(W,w) (which might not be injective). In category theorists' terms, it is the pushout of the diagram.