  We say that a topological $n$-manifold $N$ is a cubical $n$-manifold if it is
contained in the $n$-skeleton of the canonical cubulation $\mathcal{C}$ of
${\mathbb{R}}^{n+k}$ ($k\geq1$). In this paper, we prove that any closed,
oriented cubical $2$-manifold has a transverse field of 2-planes in the sense
of Whitehead and therefore it is smoothable by a small ambient isotopy.
