Abstract. In this paper a notion of lighting function is introduced as an axiomatized formalization of the "face membership ruleg' 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 (a, fl)-connectedness (~, fl E {6, 18, 26}) defined on 2[ 3 within the graph-based approach to digital topology. Furthermore, the Kong-Roscoe (c~,fl)-surfaces, with (a,~) 5~ (6,6), (18,6), are also found as particular cases of a more general notion of digital surface.
Abstract. A finitely presented group G is said to be properly 3-realizable if there exists a compact 2-polyhedron K with π 1 (K) ∼ = G and whose universal coverK has the proper homotopy type of a (p.l.) 3-manifold with boundary. In this paper we show that, after taking wedge with a 2-sphere, this property does not depend on the choice of the compact 2-polyhedron K with π 1 (K) ∼ = G. We also show that (i) all 0-ended and 2-ended groups are properly 3-realizable, and (ii) the class of properly 3-realizable groups is closed under amalgamated free products (HNN-extensions) over a finite cyclic group (as a step towards proving that ∞-ended groups are properly 3-realizable, assuming 1-ended groups are).
The goal of this work is to study the structure of the pure Morse complex of a graph, that is, the simplicial complex given by the set of all possible classes of discrete Morse functions (in Forman's sense) defined on it. First, we characterize the pure Morse complex of a tree and prove that it is collapsible. In order to study the general case, we consider all the spanning trees included in a given graph G and we express the pure Morse complex of G as the union of all pure Morse complexes corresponding to such trees.
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 G is said to be properly 3-realizable if there exists a compact 2-polyhedron K with π 1 (K) ∼ = G whose universal coverK has the proper homotopy type of a PL 3-manifold (with boundary). In this paper, we study the asymptotic behavior of finitely generated one-relator groups and show that those having finitely many ends are properly 3-realizable, by describing what the fundamental pro-group looks like, showing a property of one-relator groups which is stronger than the QSF property of Brick (from the proper homotopy viewpoint) and giving an alternative proof of the fact that one-relator groups are semistable at infinity.
Following the techniques of ordinary homotopy theory, a theoretical treatment of proper homotopy theory, including the known proper homotopy groups, is provided within Baues's theory of cofibration categories.
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-Rosenfeld's one for (26, 6)-surfaces in 7/3 [9]. The proof of this generalization is based on a new approach to digital topology sketched in [5] and developed in [2].
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