  A small triangulation of the sphere product can be found in lower dimensions
by computer search and is known in few other cases: Klee and Novik constructed
a centrally symmetric triangulation of $\mathbb{S}^i\times \mathbb{S}^{d-i-1}$
with $2d+2$ vertices for all $d\geq 3$ and $1\leq i\leq d-2$; they also
proposed a balanced triangulation of $\mathbb{S}^1\times \mathbb{S}^{d-2}$ with
$3d$ or $3d+2$ vertices. In this paper, we provide another centrally symmetric
$(2d+2)$-vertex triangulation of $\mathbb{S}^2\times \mathbb{S}^{d-3}$. We also
construct the first balanced triangulation of $\mathbb{S}^2\times
\mathbb{S}^{d-3}$ with $4d$ vertices, using a sphere decomposition inspired by
handle theory.
