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# Asian Journal of Mathematics

## Volume 20 (2016)

### Number 4

### Normal smoothings for smooth cube manifolds

Pages: 709 – 724

DOI: https://dx.doi.org/10.4310/AJM.2016.v20.n4.a6

#### Author

#### Abstract

A smooth cube manifold $M^n$ is a smooth $n$-manifold $M$ together with a smooth cubulation on $M$. (A smooth cubulation is similar to a smooth triangulation, but with cubes instead of simplices). The cube structure provides, for each open $k$-subcube $\dot{\sigma}^k$, rays that are normal to $\dot{\sigma}$. Using these rays we can construct *normal charts* of the form $\mathbb{D}^{n-k} \times \dot{\sigma} \to M$, where we are identifying $\mathbb{D}^{n-k}$ with the cone over the link of $\dot{\sigma}$ (these identifications are arbitrary and the identification between $\partial \mathbb{D}^{n-k} +$ and the link of $\dot{\sigma}$ is called a *link smoothing*). These normal charts respect the product structure of $\mathbb{D}^{n-k} \times \dot{\sigma}$ and the radial structure of $\mathbb{D}^{n-k}$. A complete set of normal charts gives a (topological) *normal atlas* on $M$. If this atlas is smooth it is called a *normal smooth atlas* on $M$ and induces a *normal smooth structure* on $M$ (normal with respect to the cube structure). In this paper we prove that every smooth cube manifold has a normal smooth structure, which is diffeomorphic to the original one. This result also holds for smooth all-right-spherical manifolds.

#### Keywords

smoothings, cubifications

#### 2010 Mathematics Subject Classification

57R10, 57R55

Published 1 November 2016