Homotopy Theory in Digital Topology

Abstract

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.

Digital Topological Complexity of Digital Maps

“…If g is not a digital fibration, then the Schwarz genus of the digital map g is the digital Schwarz genus of the digital fibrational substitute of g. Given any digital images Y and Z, the function space of digital images [18] Z Y consists of a set of all digitally continuous functions from Y to Z and Z Y has an adjacency relation such that for all α, β ∈ Z Y and u, v ∈ Y , u κ v implies that α(u) λ β(v). Let g and h be two digital paths in Y [0,m] Z with ρ−adjacency.…”

Explicit motion planning in digital projective product spaces

“…Since [a, b] Z ⊂ Z, it has 2-adjacency. The notation in [27] I m represents the digital interval such that I m ⊂ Z includes integers from 0 to m in Z, and integers are consecutively adjacent. A digital path f in X from x to y is defined by a digital map f : I m → X is digital (2, κ)-continuous with f (0) = x and f (m) = y [5].…”

“…Since the topological complexity and its related invariants are homotopy invariants, the definition and properties of digital homotopy have gained importance and some features of digital homotopy have been generalized in [27]. For the Lusternik-Schnirelmann category, one of the most important related invariants of the topological complexity, the digital LS category is defined in [2] and the study is expanded by applying it to digital functions [28].…”

Certain topological methods for computing digital topological complexity

“…The adjacency relation varies in several digital images. For instance, an adjacency relation on the set of digital functions is discussed in [17]. For any images X and Y , a function space map in digital images is stated with the set of all maps X → Y with adjacency as follows: for any two maps f , g : X → Y , they are called adjacent in the set of digital function spaces if f (x) and g(x ) are adjacent points in Y whenever x and x are adjacent points in X.…”