Classification theorem for surfaces

For a closed connected nonorientable surface, the crosscap number is the number of real projective plane summands in its connected sum classification. It is at least one, and its Euler characteristic is $2-h$. The projective plane has one crosscap and the Klein bottle has two.

For a closed connected orientable surface, its orientable genus is the number of torus summands in its connected sum classification, with the two-sphere assigned genus zero. The Euler characteristic is $2-2g$, so it determines this number within the orientable family.

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, $a{a}^{s}b{b}^t$, or alternating, $ab{a}^s{b}^t$. 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 $a{a}^{-1}b{b}^{-1}$ for the sphere, $aab{b}^{-1}$ for the projective plane, $ab{a}^{-1}{b}^{-1}$ for the torus and $ab{a}^{-1}b$ for the Klein bottle.