= Classification of compact surfaces
A <connected> <compact> <surface> is determined by its <orientability>, number $b$ of <boundary> circles and <genus>. An <orientable> <surface> of <genus> $g$ has <Euler characteristic> $2-2g-b$; a <surface> which is not <orientable> with $r$ crosscaps has <Euler characteristic> $2-r-b$. In particular, a closed <connected> <surface> with vanishing first mod-two <homology> is a <sphere>, and a <connected> closed <orientable> <surface> of <Euler characteristic> zero is a <torus>.
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