Seifert-van Kampen theorem

Statement
Suppose $$X$$ is a fact about::path-connected space. Suppose $$U$$ and $$V$$ are nonempty open subsets of $$X$$ whose union is $$X$$ and whose intersection is also path-connected. Since $$X$$ is path-connected, the intersection $$U \cap V$$ must be a nonempty open subset of $$X$$, which we call $$W$$. Note that since all the four spaces $$U,V,W,X$$ are path-connected, the fundamental groups of each of these spaces is independent of the choice of respective basepoint for each space.

We have natural homomorphisms for the fact about::fundamental groups induced by inclusion maps of the spaces(with respect to any chosen basepoint in $$W$$, which we choose not to write for brevity):

$$\! \pi_1(W) \to \pi_1(U), \pi_1(W) \to \pi_1(V), \pi_1(U) \to \pi_1(X), \pi_1(V) \to \pi_1(X)$$

The Seifert-van Kampen theorem has the following equivalent formulations:


 * Category-theoretic version: The commutative diagram formed by the above four inclusions is a pushout.
 * Group-theoretic version: $$\pi_1(X)$$ is the amalgamated free product of $$\pi_1(U)$$ and $$\pi_1(V)$$ via identification of the images of $$\pi_1(W)$$ in each. In other words:

$$\pi_1(X) = \pi_1(U) *_{\pi_1(W)} \pi_1(V)$$

Note that since the natural maps from $$\pi_1(W)$$ to $$\pi_1(U)$$ and $$\pi_1(V)$$ need not be injective, this is not necessarily an amalgamated free product in the strict sense of a common subgroup being identified between $$\pi_1(U)$$ and $$\pi_1(V)$$.
 * Presentation version:

Applications

 * Union of two simply connected open subsets with path-connected intersection is simply connected
 * Suspension of path-connected space is simply connected (this is a special case of the preceding application)
 * n-sphere is simply connected for n greater than 1 (this is a special case of both the preceding applications)
 * Fundamental group of wedge of circles is free group of rank equal to the number of circles