= Solution
Work over $\mathbb Q$. A closed connected <orientable surface> of <genus> $g$ has <Betti numbers> $(1,2g,1)$; the <circle> has Betti numbers $(1,1)$. The <Künneth theorem> therefore gives
$$
(b_0,b_1,b_2,b_3)(S^1\times\Sigma_g)=(1,2g+1,2g+1,1).
$$
We use the following precise consequence of the <Morse handle-attachment theorem>: a closed <manifold> with a <Morse function> has the <homotopy type> of a finite <CW complex> with one index-$j$ cell for each index-$j$ <critical point>. Its degree-$j$ rational <cellular chain group> has dimension $c_j$, so its <homology (mathematics)> has dimension at most $c_j$. Consequently $c_j\geq b_j$, the weak <Morse inequalities>. Summing yields
$$
\boxed{\#\operatorname{Crit}(f)\geq1+(2g+1)+(2g+1)+1=4g+4.}
$$
This counts every critical point, regardless of whether some critical values coincide.
To attain the bound, construct a surface <Morse function> $h:\Sigma_g\to\mathbb R$ from the standard orientable <handle decomposition>: start with a disk, attach $2g$ orientable bands in the usual paired pattern forming the genus-$g$ surface with one boundary component, and cap that boundary with a disk. Give the first disk a minimum, each band one saddle with local form $-x^2+y^2$, and the final cap a maximum. Regular collar coordinates join successive levels without additional critical points. This constructs $h$ with counts $(1,2g,1)$.
On the product take
$$
F(\theta,x)=h(x)+\delta\cos\theta,\qquad\delta>0.
$$
Its <critical points> are exactly the pairs of a critical point of $h$ with $\theta=0,\pi$. The <Hessian matrix> is block diagonal, with surface block $\operatorname{Hess}h$ and circle entry $-\delta\cos\theta$. It is nonsingular and its <Morse index> is the sum of the two factor indices. Thus this <product of Morse functions> has counts
$$
(c_0,c_1,c_2,c_3)=(1,2g+1,2g+1,1),
$$
and exactly \b[$4g+4$ critical points]. If distinct critical values are desired, choose $\delta$ outside the finite set producing collisions after first choosing distinct critical values for $h$. The <minimal Morse critical count on a circle times an orientable surface> is therefore attained for every $g\geq0$.
Back to article page
