= Closed surfaces obtained from a square
There are exactly four types: the <two-sphere>, <real projective plane>, <torus>, and <Klein bottle>. Four sides admit three pairings and four direction choices for each pairing. Up to cyclic relabelling, labels are adjacent, $aa^sbb^t$, or alternating, $aba^sb^t$. For adjacent labels, the vertex counts for $(s,t)=(-1,-1)$, a mixed-sign pair, and $(1,1)$ are $3,2,1$. For alternating labels they are $1,1,2$. There are two edge classes and one face, so <Euler characteristic> is $V-1$. Orientability occurs exactly when $s=t=-1$. The <classification theorem for surfaces> identifies every case, leaving no additional closed surfaces from a square. Examples are $aa^{-1}bb^{-1}$ for the sphere, $aabb^{-1}$ for the projective plane, $aba^{-1}b^{-1}$ for the torus and $aba^{-1}b$ for the Klein bottle.
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