Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 19I / a

The pushout of groups isIts maps from $G_1,G_2$ agree on $H$, and every other pair of group homomorphisms agreeing on $H$ factors through one unique homomorphism from $G$. This is the universal property. When the $i_j$ are injective, it is the amalgamated free product, and the two factors and the identified copy of $H$ embed in it. Some conventions use this name for the quotient even without injectivity; the embedding assertion requires injectivity.

The Seifert-van Kampen theorem states: if $X=U\cup V$, where $U,V$ are open and path connected, $U\cap V$ is path connected and contains a basepoint $x$, then ${\pi }_1(X,x)$ is the pushout of groups of ${\pi }_1(U,x)$ and ${\pi }_1(V,x)$ over ${\pi }_1(U\cap V,x)$. Thus the fundamental group is obtained by amalgamating the overlap relations. Openness can be replaced by standard neighborhood hypotheses, as in the wedge calculation.