Smoothing closed gridded surfaces embedded in ℝ4

We say that a topological [Formula: see text]-manifold [Formula: see text] is a cubical [Formula: see text]-manifold if it is contained in the [Formula: see text]-skeleton of the canonical cubulation [Formula: see text] of [Formula: see text] ([Formula: see text]). In this paper, we prove that any closed, oriented cubical [Formula: see text]-manifold has a transverse field of 2-planes in the sense of Whitehead and therefore it is smoothable by a small ambient isotopy.