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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. enter image description here

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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 ;)

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  • $\begingroup$ So if I'm reading your suggestion correctly, we should take a fattened neighborhood, i.e., a small open neighborhood of $PQ$ in the first gluing construction, and have that be one of the open sets. Call it $U$. Likewise, let the other open, $V$, be a fattened neighborhood of the second gluing construction. Is that correct? $\endgroup$ – swygerts Dec 30 '16 at 23:34
  • $\begingroup$ yes. I personally like to "search", when applying SvK, for the intersection of my two open sets. Here, in my opinion, it is natural to look at (some neighbourhood) of the side PQ as "the intersection area". $\endgroup$ – M.U. Dec 30 '16 at 23:45
  • $\begingroup$ So does this require multiple applications of SvK? I don't immediately see what the fundamental group for your first gluing construction should be, but maybe could figure it out by applying SvK to similar subsets and onward. $\endgroup$ – swygerts Dec 30 '16 at 23:48

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