= Fixed-point-free self-maps homotopic to the identity on products of surfaces
{title2=$(g_1-1)(g_2-1)=0$}
Such a self-map of $\Sigma_{g_1}\times\Sigma_{g_2}$ exists exactly when one factor is a torus. Necessity follows because its <Lefschetz number> equals the <Euler characteristic> $4(1-g_1)(1-g_2)$ and must vanish. Sufficiency is translation by a nonzero point in the torus factor, times the identity on the other factor. The condition does not determine the product $g_1g_2$, since the other genus can be arbitrary.
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