= Minimal Morse critical count on a circle times an orientable surface
{title2=$\min\#\operatorname{Crit}(f|_{S^1\times\Sigma_g})=4g+4$}
For a closed connected <orientable surface> of <genus> $g$, the <Künneth theorem> gives the <Betti numbers> of $S^1\times\Sigma_g$ as $(1,2g+1,2g+1,1)$. The weak <Morse inequalities> give the displayed lower bound. A surface <Morse function> with one minimum, $2g$ saddles and one maximum exists from its standard <handle decomposition>. Adding a positive multiple of the cosine <Morse function> on the <circle> makes a <product of Morse functions>, whose critical counts are precisely these four Betti numbers. The lower bound is attained.
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