Dual handle presentations of a genus-g surface

Start with a disk. In each of $g$ repetitions, attach an orientable band along two arcs of the current single boundary circle so that the boundary splits into two circles; attach a second orientable band between these two circles. The boundary is again one circle and the surface has acquired one punctured-torus piece. This is the usual genus-$g$ orientable surface with one disk removed. Capping its final circle gives ${\mathrm{\Sigma }}_g$, using exactly one disk, $2g$ bands and one cap.

Read the disk, the bands and the cap as handles of indices $0,1,2$. By [F1] the resulting finite handle presentation is realized by an adapted excellent $h:{\mathrm{\Sigma }}_g\to [0,1]$ with one minimum, $2g$ saddles of distinct values and one maximum. No arbitrary embedded height function is being assumed excellent or already in the adapted range. For $g=0$ the construction is two disks glued along their circle.

Patch a background metric from [F8] to Euclidean metrics in smaller disjoint Morse charts of the realizing function of step 2.1, using the finite chart cutoffs. Its negative gradient is strictly descending off the critical points and equals $(2u,-2v)$ in these charts. It is complete by [F8] because the closed surface is compact. Thus it is an adapted field, and [F4] applies to this pair. The negated function $1-h$ is adapted excellent with the same $2g+2$ critical points, and the indices are exchanged by $k\mapsto 2-k$: the maximum of $h$ has index $0$ for $1-h$, the $2g$ saddles keep index $1$, and the minimum of $h$ has index $2$ for $1-h$ (this is the general fact that negating a function changes the index of a nondegenerate critical point from $k$ to $n-k$, here $n=2$). By [F1] applied to $1-h$, the dual presentation has one $0$-handle, $2g$ $1$-handles and one $2$-handle.

Identify the handles of the two presentations through [F3]: the dual $0$-handle is the original $2$-handle, the $2g$ one-handles are self-dual since $2-1=1$, and the dual $2$-handle is the original $0$-handle; the order of attachment is reversed and attaching and belt spheres are interchanged. For $g=0$ this says that the dual presentation of the sphere's two-handle presentation is the same pair of handles read in reverse order, which agrees with the explicit picture of two disks glued along their boundary circle.

The index exchange is verified in every surface degree: $0\leftrightarrow 2$ and $1\leftrightarrow 1$, so no handle of the dual presentation has an index outside $\{0,1,2\}$ and the numbers of handles of each index are $1,2g,1$ in both presentations. This is exactly the surface case of the duality theorem.