Computing Cup Products in $$\mathbb {Z}_2$$ Z 2 -Cohomology of 3D Polyhedral Complexes

Abstract

Let I = (Z 3 , 26, 6, B) be a 3D digital image, let Q(I) be the associated cubical complex and let ∂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 Z 3 B in the background -the ambient space. We show how to simplify the combinatorial structure of ∂Q(I) and obtain a 3D polyhedral complex P (I) homeomorphic to ∂Q(I) but with fewer cells. We introduce an algorithm that computes cup products on H * (P (I); Z 2 ) directly from the combinatorics. The computational method introduced here can be effectively applied to any polyhedral complex embedded in R 3 .

An A ∞-coalgebra structure on a closed compact surface

“…A more general exposition of Kravatz's diagonal appears in [1]. For k > 2, define the k-ary A ∞ -coalgebra operation ∆ k : C * (P ) → C * (P ) ⊗k by…”

Distributed computation of low-dimensional cup products

“…To implement (2) one can use the Alexander-Whitney diagonal approximation formula in the case of simplicial spaces, and Serre's analogue of this for cubical spaces. Details of the cubical analogue can be found in [26], and details on practical implementations of these two formulae can be found in [16,18,14,15,28,17,24]. In this section we assume that X is an arbitrary connected regular finite CW-space and observe that for k = 2 the homomorphism (2), and hence the cup product (1), can be read directly from a group presentation P = x | r for the fundamental group π 1 X.…”

“…In Section 3 we illustrate the method on the integral cohomology ring of a 3-dimensional digital image. Previous papers [16,18,14,15] have described different approaches to computing the cohomology ring, over Z/2Z, of cubical and simplicial spaces arising from 3-dimensional digital images; these papers are based on techniques in [28,17,24]. The fundamental group algorithm in [1] involves the construction of an admissible discrete vector field on X, and this construction can consume significant memory and time for large CW-spaces X.…”

Making Sullivan Algebras Minimal Through Chain Contractions

“…The (co)homology of (M, d M ) is isomorphic to the one of (KH, d H = 0) since (f, g, φ) is a chain contraction. Besides, Algorithm 1 can be used to compute more sophisticated topology information such as cup products on cohomology [9] or persistent homology [8].…”