Cups Products in Z2-Cohomology of 3D Polyhedral Complexes

TL;DR

This paper presents a method to simplify the combinatorial structure of 3D digital image boundaries and introduces an algorithm to compute cup products in cohomology directly from these simplified polyhedral complexes.

Contribution

It introduces a novel simplification technique for 3D digital boundary complexes and an algorithm for direct cup product computation in cohomology.

Findings

Simplified boundary complexes with fewer cells while preserving topology.

Algorithm effectively computes cup products from combinatorial data.

Applicable to any polyhedral complex in 3D space.

Abstract

Let $I=(\mathbb{Z}^3,26,6,B)$ be a 3D digital image, let $Q(I)$ be the associated cubical complex and let $\partial Q(I)$ be the subcomplex of $Q(I)$ whose maximal cells are the quadrangles of $Q(I)$ shared by a voxel of $B$ in the foreground -- the object under study -- and by a voxel of $\mathbb{Z}^3\smallsetminus B$ in the background -- the ambient space. We show how to simplify the combinatorial structure of $\partial Q(I)$ and obtain a 3D polyhedral complex $P(I)$ homeomorphic to $\partial Q(I)$ but with fewer cells. We introduce an algorithm that computes cup products on $H^*(P(I);\mathbb{Z}_2)$ directly from the combinatorics. The computational method introduced here can be effectively applied to any polyhedral complex embedded in $\mathbb{R}^3$.