= Solution
The <classification theorem for surfaces> states that every connected <closed combinatorial surface> is <homeomorphic> to exactly one of:
* the <two-sphere>;
* an <orientable surface> of <orientable genus> $g\geq1$, the <connected sum> of $g$ <tori>;
* a <nonorientable surface> of <crosscap number> $h\geq1$, the <connected sum> of $h$ <real projective planes>.
The orientable family has <Euler characteristic> $2-2g$; the nonorientable family has Euler characteristic $2-h$. Orientability together with Euler characteristic distinguishes the entries.
Boundary words giving the required <polygonal schemas> are
$$
\boxed{\begin{aligned}
S^2:&\quad aa^{-1},\\
\Sigma_g:&\quad a_1b_1a_1^{-1}b_1^{-1}\cdots a_gb_ga_g^{-1}b_g^{-1},\\
N_h:&\quad a_1a_1a_2a_2\cdots a_ha_h.
\end{aligned}}
$$
A two-sided polygon here means a disk with its boundary split into two arcs; subdivision produces an ordinary polygon with more sides without changing the quotient. Alternatively the square word $aa^{-1}bb^{-1}$ represents the sphere. Each four-letter block in the orientable word adds a torus handle; each repeated-letter block in the nonorientable word adds a projective-plane summand.
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