Digital homotopy fixed point theory

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 & 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.

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.

Arabian Journal of Mathematics

The purpose of this paper is to introduce the class of multi-valued operators by the technique of interpolation of operators. Our results extend and generalize several results from the existing literature. Moreover, we also study the data dependence problem of the fixed point set and Ulam–Hyers stability of the fixed point problem for the operators introduced herein. Moreover, as an application, we obtain a homotopy result.

arXiv (Cornell University), 2020

In this paper, we present two types of Lefschetz numbers in the topology of digital images. Namely, the simplicial Lefschetz number L(f ) and the cubical Lefschetz number L(f ). We show that L(f ) is a strong homotopy invariant and has an approximate fixed point theorem. On the other hand, we establish that L(f ) is a homotopy invariant and has an n-approximate fixed point result. In essence, this means that the fixed point result for L(f ) is better than that for L(f ) while the homotopy invariance of L(f ) is better than that of L(f ). Unlike in classical topology, these Lefschetz numbers give lower bounds for the number of approximate fixed points. Finally, we construct some illustrative examples to demonstrate our results.

Commentarii Mathematici Helvetici, 1988

2007

We use the ideas of Lusternik–Schnirelmann theory to describe the set of fixed points of certain homotopy equivalences of a general space. In fact, we extend Lusternik–Schnirelmann theory to pairs (φ, f), where φ is a homotopy equivalence of a topological space X and where f : X → R is a continuous function satisfying f(φ(x)) < f(x) unless φ(x) = x; in addition, the pair (φ, f) is supposed to satisfy a discrete analogue of the Palais–Smale condition. In order to estimate the number of fixed points of φ in a subset of X, we consider different relative categories. Moreover, the theory is carried out in an equivariant setting.

2015

A point of a digital space is called simple if it can be deleted from the space without altering topology. This paper introduces the notion simple set of points of a digital space. The definition is based on contractible spaces and contractible transformations. A set of points in a digital space is called simple if it can be contracted to a point without changing topology of the space. It is shown that contracting a simple set of points does not change the homotopy type of a digital space, and the number of points in a digital space without simple points can be reduces by contracting simple sets. Using the process of contracting, we can substantially compress a digital space while preserving the topology. The paper proposes a method for thinning a digital space which shows that this approach can contribute to computer science such as medical imaging, computer graphics and pattern analysis.

Discrete Applied Mathematics, 2004

K-Theory, 2000

Let G be a finite group, let X and Y be finite G-complexes, and suppose that for each K ___ G, yK is dim(X x)-connected and simple. G acts on the function complex F(X, Y) by conjugation of maps. We give a complete analysis of the homotopy fixed point set of the space 92~E~ Y). As a corollary, we are able to analyze at a prime p, the homotopy fixed point set of the circle action on f~E~AX, where AX denotes the free loop space of X, and X is a simply connected finite complex.