= Degree-one maps between closed oriented surfaces
{title2=$\Sigma_g\to\Sigma_h\text{ of degree }1\iff g\geq h$}
A <degree of a map between oriented manifolds> equal to one makes the pullback on degree-one <cohomology> injective by the <Poincare duality pairing>. Thus a degree-one map of closed oriented surfaces requires $2h\leq2g$. Conversely, collapse the extra handles in $\Sigma_h\#\Sigma_{g-h}$ onto a point. The retained oriented disc shows that this map has degree one.
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