I am studying some algebraic topology and have come across a question that asks to find the fundamental group of a space given below. Naturally, my inclination is to use the Seifert-Van Kampen Theorem, but I am not quite sure what to choose as my sets. Can anyone point me in the right direction or explain what would be reasonable choices of sets?
A topological space X consists of four triangles with a common side $PQ$. Their sides $AP$, $PB$, $CP$, and $PD$ are glued together (note the directions!) and their sides $CP$, $PQ$ and $QC$ are also glued together. Compute the fundamental group of $X$ in terms of generators.
First I wanted to write this as a comment .... but it was too long :)
I would do it as follows: Before starting the gluing construction remove the triangle $PQC$ "from the picture". Of course the side $PQ$ is still in the picture of the remaining three triangles $PQA$, $PQB$ and $PQD$.
The "first" gluing construction (the "simple arrows") is only inside your picture with the three traingles.
The "second" gluing construction (the "double arrows") is only inside your pictucre with the single triangle $PQC$.
The last gluing (in the box it is actually the first sentence) is putting your two pictures together along the edge $PQ$.
I hope you see what I am trying to say ;)