Computing Higher Dimensional Digital Homotopy Groups

Discrete Applied Mathematics, 2003

The main contribution of this paper is a new "extrinsic" digital fundamental group that can be readily generalized to define higher homotopy groups for arbitrary digital spaces. We show that the digital fundamental group of a digital object is naturally isomorphic to the fundamental group of its continuous analogue. In addition, we state a digital version of the Seifert-Van Kampen theorem.

Discrete Applied Mathematics, 2004

Acta Applicandae Mathematicae, 2009

Recent papers have partially discussed the multiplicative or the non-multiplicative property of the digital fundamental group. Thus, the paper studies a condition of which the multiplicative property of the digital fundamental group holds. Precisely, for two digital spaces with k i -adjacencies of Z n i , denoted by (X i , k i ), i ∈ {1, 2}, using the L HSor L HC -property of the digital product (or Cartesian product of digital spaces) with k-adjacency (X 1 × X 2 , k), a k-homotopic thinning of the digital product, and various properties from digital covering and digital homotopy theories, we provide a method of calculating the k-fundamental group of the digital product. Furthermore, the notion of HT-(k 0 , k 1 )isomorphism is established and used in calculating the k-fundamental group of a digital product. Finally, we find a condition of which the multiplicative property of the digital fundamental group holds. This property can be used in classifying digital spaces from the view points of digital homotopy theory, mathematical morphology, and digital geometry.

Discrete & Computational Geometry, 2021

Digital topology is part of the ongoing endeavour to understand and analyze digitized images. With a view to supporting this endeavour, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the most basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik-Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics.

Journal of Applied and Computational Topology, 2021

We define a fundamental group for digital images. Namely, we construct a functor from digital images to groups, which closely resembles the ordinary fundamental group from algebraic topology. Our construction differs in several basic ways from previously established versions of a fundamental group in the digital setting. Our development gives a prominent role to subdivision of digital images. We show that our fundamental group is preserved by subdivision.

Acta Applicandae Mathematicae, 2011

The development of digital imaging (and its subsequent applications) has led to consider and investigate topological notions, well-defined in continuous spaces, but not necessarily in discrete/digital ones. In this article, we focus on the classical notion of path. We establish in particular that the standard definition of path in algebraic topology is coherent w.r.t. the ones (often empirically) used in digital imaging. From this statement, we retrieve, and actually extend, an important result related to homotopy-type preservation, namely the equivalence between the fundamental group of a digital space and the group induced by digital paths. Based on this sound definition of paths, we also (re)explore various (and sometimes equivalent) ways to reduce a digital image in a homotopy-type preserving fashion.

In this paper, we study digital homotopy of digital paths due to Laurence Boxer. We give some theorems, propositions and definitions on digital paths, digital path connectedness and introduce digital convex set and digital contractible spaces.

TURKISH JOURNAL OF MATHEMATICS, 2020

In this paper, we develop homology groups for digital images based on cubical singular homology theory for topological spaces. Using this homology, we present digital Hurewicz theorem for the fundamental group of digital images. We also show that the homology functors developed in this paper satisfy properties that resemble the Eilenberg-Steenrod axioms of homology theory, in particular, the homotopy and the excision axioms. We finally define axioms of digital homology theory. Keywords digital topology • digital homology theory • digital Hurewicz theorem • cubical singular homology for digital images • digital excision Mathematics Subject Classification (2010) 55N35 • 68U05 • 68R10 • 68U10

In this article we give characteristic properties of the simplicial homology groups of digital images which are based on the simplicial homology groups of topological spaces in algebraic topology. We then investigate Eilenberg-Steenrod axioms for the simplicial homology groups of digital images. We state universal coefficient theorem for digital images. We conclude that the Künneth formula for simplicial homology doesn't be hold in digital images. Moreover we show that the Hurewicz Theorem need not be hold in digital images. Copyright © AJCTA, all rights reserved.

Comptes Rendus Mathematique, 2015

In this paper, we construct a framework which is called the digital homotopy fixed point theory. We get new results associating digital homotopy and fixed point theory. We also give an application on this theory.