Getting rid of topological pinches in nD cubical complexes to make homological computations easier

José María Tornero Sánchez
Seminario II (IMUS), Edificio Celestino Mutis
Nicolas Boutry
Event type

Some years ago, a method able to topologically repair 3D cubical complexes into homotopically-equivalent polyhedral complexes was developed in Digital Topology community. It was proved that this method provides topological manifolds, which means that we can compute homology of the initial complex based on the homology of the boundary of the repaired complex, which fasten computations and then lead to faster feature recognition based on the homology of a complex (Euler component and so on). This method has been extended to nD but topological manifoldness of the repaired (simplicial) complex has not been provided yet due to the difficulty to prove such a property in nD. We will develop this point in this presentation. In particular, we will recall the different flavors of well-composednesses that should be proved for this repaired complex by order of difficulty, and we will recall which formulas make able to compute the homology of the initial complex based on the homology of the surface of the repaired complex in nD. A relation between this nD repairing method and the nD Marching Cubes will also be discussed.