Normal smoothings for smooth cube manifolds

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

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Published 1 November 2016