Seifert-van Kampen theorem

Statement

Suppose $X$ is a 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 Fundamental group (?)s 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: Fill this in later

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