Smoothing closed gridded surfaces embedded in ℝ4

Abstract

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 C of R n+k (k ≥ 1). In this paper, we prove that any closed, oriented cubical 2manifold has a transverse field of 2-planes in the sense of Whitehead and therefore it is smoothable by a small ambient isotopy. * 2010 Mathematics Subject Classification. Primary: 57R10. Secondary: 57R55, 57R25.

Cubulated moves for 2-knots

“…(i) K 2 ∩ Q 4 j is the union of the two 2-faces F 2 7 and F 2 4 (see Figure 5.4). Then there exist two 3-faces M 3 4 and…”

“…Then we have only one possibility for K 2 ∩ Q 4 j (see Figure 6.2). There exist three 3faces M 3 1 , M 3 −2 and M 3 −1 which contain two 2-faces of (…”

“…M 3 m which are enumerated in such a way that M 3 n+1 is a neighbor of M 3 n for all n, i.e., M 3 n+1 ∩ M 3 n consists of a 2-face F 2 n . We will use induction on m. Consider M 3 1 . We apply (M2)-moves on M 3 1 in such a way that the faces (of any dimension) belonging to K 2 are replaced by those belonging to K 2 (by construction M 3 1 ∩ K 2 = ∅ and M 3 1 ∩ K 2 = ∅).…”

“…We will use induction on m. Consider M 3 1 . We apply (M2)-moves on M 3 1 in such a way that the faces (of any dimension) belonging to K 2 are replaced by those belonging to K 2 (by construction M 3 1 ∩ K 2 = ∅ and M 3 1 ∩ K 2 = ∅). Next, we consider M 3 2 .…”

“…The first cube Q 3 1 contains a 2-face of type T − , and given the cube Q 3 n , the cube Q 3 n+1 shares a 2-face F 2 n with Q 3 n and whenever it is possible, we choose Q 3 n+1 in such a way that F 2 n is parallel to F 2 n−1 ; otherwise we choose Q 3 n+2 such that F 2 n+1 is parallel to F 2 n−1 , at the end we continue on the level k 1 + 1 and so on. We will use induction on m. Consider the cube Q 3 1 . We apply the (M 2)-move to Q 3 1 replacing the 2-faces of type T by 2-faces of type T − .…”