Seifert–van Kampen theorem

Let \(X=U\cup V\), where \(U,V\subseteq X\) are open and path connected, \(U\cap V\) is path connected, and choose a basepoint \(x_0\in U\cap V\). Let \[ i_U:U\cap V\hookrightarrow U, \qquad i_V:U\cap V\hookrightarrow V \] be the inclusions. Then the inclusion maps \(U,V\hookrightarrow X\) induce a surjective homomorphism \[ \pi_1(U,x_0)*\pi_1(V,x_0)\twoheadrightarrow\pi_1(X,x_0), \] and its kernel is the normal closure of all elements \[ i_{U*}(\omega)i_{V*}(\omega)^{-1}, \qquad \omega\in\pi_1(U\cap V,x_0). \] Equivalently, \[ \pi_1(X,x_0) \cong \pi_1(U,x_0)*_{\pi_1(U\cap V,x_0)}\pi_1(V,x_0), \] the pushout (amalgamated free product) of the two induced maps on fundamental groups.