A. Quintero

Digital Lighting Functions

In this paper a notion of lighting function is introduced as an axiomatized formalization of the “face membership rules” suggested by Kovalevsky. These functions are defined in the context of the framework for digital topology previously developed by the authors. This enlarged framework provides the (α, β)-connectedness (α, β e {6,18, 26}) defined on ℤ3 within the graph-based approach to digital topology. Furthermore, the Kong-Roscoe (α, β)-surfaces, with (α, β) ≠ (6, 6), (18, 6), are also found as particular cases of a more general notion of digital surface.

Determining the components of the complement of a digital (n-1)-manifold in Zn

The goal of this paper is to determine the components of the complement of digital manifolds in the standard cubical decomposition of Euclidean spaces for arbitrary dimensions. Our main result generalizes the Morgenthaler-Rosenfelds one for (26, 6)-surfaces in ℤ3 [9]. The proof of this generalization is based on a new approach to digital topology sketched in [5] and developed in [2].

A digital index theorem

This paper is devoted to state and prove a Digital Index Theorem for digital (n - 1)-manifolds in a digital space (Rn, f), where f belongs to a large family of lighting functions on the standard cubical decomposition Rn of the n-dimensional Euclidean space. As an immediate consequence we obtain the corresponding theorems for all (α, β)-surfaces of Kong–Roscoe, with α, β ∈ {6, 18, 26} and (α, β) ≠ (6, 6), (18, 26), (26, 26), as well as for the strong 26-surfaces of Bertrand–Malgouyres.

A maximum set of (26,6)-connected digital surfaces

In the class

One-relator groups and proper

\mathcal{H}

3

of (26,6)–connected homogeneous digital spaces on R3 we find a digital space EU with the largest set of digital surfaces in that class. That is, if a digital objet S is a digital surface in any space

-realizability

E \epsilon \mathcal{H}

Amalgamated products and properly 3-realizable groups

then S is a digital surface in EU too.

An Elementary Approach to the Projective Dimension in Proper Homotopy Theory

How different is the universal cover of a given finite 2-complex from a 3-manifold (from the proper homotopy viewpoint)? Regarding this question, we recall that a finitely presented group

Local characterization of a maximum set of digital (26,6)-surfaces

G