Van Kampen’s theorem result

There is a more general version of the theorem of van Kampen which involvesthe fundamental groupoid ${\pi }_1(X,A)$ on a set $A$ of base points,defined as the full subgroupoid of ${\pi }_1(X)$ on the set $A\cap X$. Thisallows one to compute the fundamental group of the circle $S^1$ and many morecases.

Theorem If $X$ is the union of open sets $U,V$ with intersection $W$,and $A$ meets each path component of $U,V,W$ then the following induced diagram

is a pushout in the category of groupoids.

This may be found in R. Brown’s book “Topology: a geometric account of general topologyand the fundamental groupoid”, Ellis Horwood 1988 (first edition McGraw Hill, 1968).It has many useful applications, and was the guide for higher dimensional theoremsinvolving higher homotopy groupoids.