Let $T=S^{1} \times S^{1}$ be the 2-dimensional torus. Let $\alpha: S^{1} \rightarrow T$ be the inclusion of the coordinate circle $S^{1} \times\{1\}$, and let $X$ be the result of attaching a 2-cell along $\alpha$.

(a) Write down a presentation for the fundamental group of $X$ (with respect to some basepoint), and identify it with a well-known group.

(b) Compute the simplicial homology of any triangulation of $X$.

(c) Show that $X$ is not homotopy equivalent to any compact surface.