Let $X$ the quotient of an octagon by the edgeword $aba^{-1}bcdc^{-1}d^{-1}$. Justify that $X$ is homeomorphic to a connected sum of 4 copies of a certain compact surface $Y$.
I believe that from the Classification Theorem, this space $Y$ must be either a Torus or the Projective Plane (my intuition goes for the projective plane) but how do I proceed to "cut" this word into 4 projective planes ?