Homotopy Theory in Digital Topology

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.

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.

Discrete Applied Mathematics, 2004

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.

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

American Mathematical Monthly, 1991

Google, Inc. (search). SIGN IN SIGN UP. A topological approach to digital topology. Authors: T. Yung Kong, Ralph Kopperman, Paul R. Meyer, Published in: · Journal. American Mathematical Monthly archive. Volume 98 Issue 12, Dec. 1991 Mathematical Association of America Washington, DC, USA table of contents doi>10.2307/2324147. 1991 Article. Bibliometrics. · Downloads (6 Weeks): n/a · Downloads (12 Months): n/a · Citation Count: 24. Tools and Resources. Save to Binder; Export Formats: BibTeX; EndNote; ACM Ref. Share: ...

1972

Spaces of continuous m~s Preliminaries to Part I 15-21 15 §1 Basic definitions " 2 Evaluation fibrations defined by a space of mappf.ng e with a suspension as domain. §3 A :rundanental t.re or-em of O.W.Whitehead and some facts about \'lhitehead products in spheres .. • 19 17 Chapter 2: Evaluation fibrat10ns §1 Sect ions in evalua tion fibrations••• .... 22-34 .. 22 §2 The structure of a neutral evaluation fibration 24 §3 Fibre homotopy equivalence of evaluation fibrations. . 28 §4 Strong fibre homotopy equivalence of evaluation fibrations 31 Chapter 3: Homotopy equivalence of components in spaces of maps between spheres,. Tte order of ?t n _ l (Go. (s", Sn» for n even and a. E 7t n (Sn) §2 Characterization of tbe homotopy type of the. 35-42 §1 .. 35 neutral component. 37 §3 The division int 0 homotopy types of the components in the mapping spaces G~S.r",Sn) and G(sn+l,Sn) ....

Contemporary Mathematics, 2010

We survey research on the homotopy theory of the space map(X, Y ) consisting of all continuous functions between two topological spaces. We summarize progress on various classification problems for the homotopy types represented by the path-components of map(X, Y ). We also discuss work on the homotopy theory of the monoid of self-equivalences aut (X) and of the free loop space LX. We consider these topics in both ordinary homotopy theory as well as after localization. In the latter case, we discuss algebraic models for the localization of function spaces and their applications.

Applied General Topology

Y. Rudyak develops the concept of the topological complexity TC(X) defined by M. Farber. We study this notion in digital images by using the fundamental properties of the digital homotopy. These properties can also be useful for the future works in some applications of algebraic topology besides topological robotics. Moreover, we show that the cohomological lower bounds for the digital topological complexity TC(X,κ) do not hold.