Let \(A\), \(B\), and \(C\) be groups, and let \(i\colon C\to A\) and \(j\colon C\to B\) be homomorphisms. The free product with amalgamation of \(A\) and \(B\) over \(C\) is \[ A \ast_C B \coloneqq \qty{A \ast B} / N, \] where \(A\ast B\) is the free product and \(N\) is the normal subgroup of \(A\ast B\) generated by \(\ts{ i(c) \inverseof{j(c)} \st c\in C }\).
Together with the homomorphisms \(A\to A\ast_C B\) and \(B\to A\ast_C B\) induced by the inclusions of \(A\) and \(B\) into \(A\ast B\), the group \(A\ast_C B\) is the pushout of \(A\xleftarrow{i} C\xrightarrow{j} B\) in \(\Grp\).
By van Kampen's theorem, if \(X = U_1\union U_2\) with \(U_1\), \(U_2\) and \(U_1\intersect U_2\) open and path-connected, then \(\pi_1(X)\) is the free product of \(\pi_1(U_1)\) and \(\pi_1(U_2)\) amalgamated over \(\pi_1(U_1\intersect U_2)\) [@Hat02].
[@Hat02].