I am trying to prove the following excersise (1.2.8) from Hatcher's Algebraic Topology: Given 2 tori $S^1\times S^1$ and identifying $S^1\times \{x_0\}$} compute the fundamental group.
My approach is the following: Consider the first torus as an identified square via the scheme $aba^{-1}b^{-1}$ and the second torus via $cdc^{-1}d^{-1}$.
To idintify the two circles is to "glue" $a$ with $c$. Here I must make 2 remarks. The first is that we could have picked $b,d$ with no substantial change to our proof. The second is that since Hatcher doesnt describe how to glue them I assume he means in the trivial way.
Therefore , having identified $a$ with $c$ with the arrow pointing the same way we get a new CW complex with scheme $bdad^{-1}b^{-1}a^{-1}$ and therefore, from Van Campen's theorem, we the fundamental group has representation $<a,b,d|bdad^{-1}b^{-1}a^{-1}>$.
I apologise for the quality of the figure:
hint: can you use your figure to see that $X= (S^1 \vee S^1) \times S^1$?