The <van Kampen theorem> says: if $X=U\cup V$ with $U,V$ open and path connected, $U\cap V$ path connected, and a base point in the intersection, then $\pi_1(X)$ is the <free product> $\pi_1(U)*\pi_1(V)$ modulo the <normal subgroup> generated by identifying the two images of every element of $\pi_1(U\cap V)$. Equivalently it is the <pushout> of the two inclusion-induced fundamental-group maps.

Solution

Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 1 / 21F / i / Solution

 Solution (source code)