We're supposed to use cutting and gluing to find out, whether a given surface is a connected sum of tori or a connected sum of $\mathbb{R}P^2$. Now, there's one surface I'm stuck with. It is a 10-gon with edges (a,b,c,d,e) and presentation: $abcdea^{-1}b^{-1}c^{-1}d^{-1}e^{-1}$.
I connect the crossed opposing pairs and end up with $fgf^{-1}g^{-1}e^{-1}hjh^{-1}j^{-1}e$. But according to our book, this should not happen if all points are identified.
Does anybody see what's going wrong here?
That's because not all the points are identified. In fact, they are divided into two groups of five, each of which is identified as a separate point. See the diagram below and try to chase them.