Normal smoothings for smooth cube manifolds

Ontaneda Pedro
2016 Asian Journal of Mathematics  
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-subcubeσ k , rays that are normal toσ. Using these rays we can construct normal charts of the form D n−k ×σ → M , where we are identifying D n−k with the cone over the link ofσ (these identifications are arbitrary and the identification between ∂D n−k and the link
more » ... ofσ is called a link smoothing). These normal charts respect the product structure of D n−k ×σ and the radial structure of 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.
doi:10.4310/ajm.2016.v20.n4.a6 fatcat:34cuyakc4jbijomtvv7xaarcpu