Abstract: We prove that for every polyhedral complex, K, and every dimension, p, there is a partition of the p-cells into a maximal p-tree, a maximal p-cotree, and the remaining p-cells defining the p-th homology of K. As an application, we consider the manipulation of the hole structure in geometric shapes, using the tri-partition to facilitate the opening and closing of holes. In a concrete application, we let K be the Delaunay mosaic of a finite set, and we extract a partial order on the filtration induced by the radius function, whose cuts define the subcomplexes that can be constructed with this method.

Joint work with Katharina Oelsboeck.