# Smoothing closed gridded surfaces embedded in ${\mathbb R}^4$

## TL;DR

This paper proves that any closed, oriented cubical 2-manifold embedded in higher-dimensional Euclidean space can be smoothly approximated through a small ambient isotopy, extending smoothing techniques to cubical manifolds.

## Contribution

It establishes that all closed, oriented cubical 2-manifolds in -dimensional space are smoothable via a transverse 2-plane field, providing a new smoothing result for cubical manifolds.

## Key findings

- Any closed, oriented cubical 2-manifold admits a transverse 2-plane field.
- Such manifolds can be smoothed by a small ambient isotopy.
- The result extends smoothing techniques to cubical manifolds in -space.

## 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 $\mathcal{C}$ 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.