What compact surface is represented by the regular $10$-gon with edges identified in pairs, as indicated by the symbol: $$abcdec^{-1}da^{-1}b^{-1}e^{-1}?$$
It's an exercise (Exercise 8.7 of chapter 1) from Massey's book. I solved this problem using Euler characteristics. From the representation, it is clear that the surface is nonorientable because of the pair $(d, d$). Also, I found that there exactly is one vertex class in this polygon. Therefore, the Euler characteristics is: $$1-5+1 =-3. $$ The only nonorientable compact surface with the Euler characteristics $-3$ is the connected sum of $3$ projective planes. I am not that comfortable with this kind of problem. Could anyone please check whether my computation is correct or not? Also, is there any quicker way to solve this problem? Thanks so much.
Edit: $2- n = -3$ gives $n= 5$. Sorry, $3$ was a typo. Thank you, William.