Let $X$ be the quotient space obtained from an 8-sided polygonal region $P$ by pasting its edges together according to the labelling scheme $acadbcd^{-1}d$. a) Check that all vertices of $P$ are mapped to the same point of the quotient space $X$ by the pasting map. b) Calculate $H_{1}(X)$ c) Assuming $X$ is homeomorphic to one of the surfaces among $S^{2}, T_{1}, T_{2} \ldots, P_{1}, P_{2}, \ldots$, where $T_{n}$ and $P_{m}$ denote the $n$-fold connected sum of tori and the m-fold connected sum of projective planes, respectively, which surface is it?
I think I proved that $H_{1}(X)$ is $\mathbb{Z}_{2} * \mathbb{Z}^{3}$. However, I would like to have some feedback.
Thanks.