= Standard polygon words for closed surfaces
Read a boundary word counterclockwise, giving every paired edge a label. A positive exponent means the edge arrow agrees with the boundary traversal, and an inverse means it disagrees. Identify equally labelled edges in their arrow directions. The <sphere> has a square word $abb^{-1}a^{-1}$; the vertex count is three, giving <Euler characteristic> $3-2+1=2$. A <closed orientable surface> of <genus> $g\geq1$ has word
$$
\prod_{j=1}^g a_jb_ja_j^{-1}b_j^{-1}.
$$
A <nonorientable surface> which is the <connected sum of surfaces> of $k$ <real projective planes> has word $c_1c_1\cdots c_kc_k$. For $k=1$, this is a digon; the square word $abab$, identifying antipodal boundary points, is another model of the <real projective plane>. The <Klein bottle> has square word $aba^{-1}b$, representing the nonorientable case $k=2$. The <classification theorem for closed combinatorial surfaces> says that these orientable and nonorientable families exhaust the connected closed cases.
\Image[/past-exam-of-the-topology-course-of-the-university-of-oxford/2017-surface-polygon-words.png]
{title=Square polygon models of four closed surfaces}
{description=Read the boundary word counterclockwise and identify paired edges in their arrow directions. The sphere model has three vertex classes; the projective-plane square has two.}
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