Homotopy Theory in Digital Topology

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.Comment: 45 pages, 2 figure

On digital simply connected spaces and manifolds: a digital simply connected 3-manifold is the digital 3-sphere

In the framework of digital topology, we study structural and topological properties of digital n-dimensional manifolds. We introduce the notion of simple connectedness of a digital space and prove that if M and N are homotopy equivalent digital spaces and M is simply connected, then so is N. We show that a simply connected digital 2-manifold is the digital 2-sphere and a simply connected digital 3-manifold is the digital 3-sphere. This property can be considered as a digital form of the Poincar\'e conjecture for continuous three-manifolds.Comment: Key words: Graph; Dimension; Digital manifold; Simply connected space; Spher

Classification of graphs based on homotopy equivalence. Homotopy equivalent graphs. Basic graphs and complexity of homotopy equivalence classes of graphs

Graph classification plays an important role is data mining, and various methods have been developed recently for classifying graphs. In this paper, we propose a novel method for graph classification that is based on homotopy equivalence of graphs. Graphs are called homotopy equivalent if one of them can be converted to the other one by contractible transformations. A basic graph and the complexity of a homotopy equivalence class are defined and investigated. It is shown all graphs belonging to a given homotopy equivalence class have similar topological properties and are represented by a basic graph with the minimal number of points and edges. Diagrams are given of basic graphs with the complexity N<7. The advantage of this classification is that it relies on computer experiments demonstrating a close connection between homotopy equivalent topological spaces and homotopy equivalent graphs.Comment: In press. Asian Journal of Mathematics and Computer Research. http://www.ikpress.org/articles-press/4

Proximal Planar Cech Nerves. An Approach to Approximating the Shapes of Irregular, Finite, Bounded Planar Regions

This article introduces proximal Cech nerves and Cech complexes, restricted to finite, bounded regions $K$ of the Euclidean plane. A Cech nerve is a collection of intersecting balls. A Cech complex is a collection of nerves that cover $K$. Cech nerves are proximal, provided the nerves are close to each other, either spatially or descriptively. A Cech nerve has an advantage over the usual Alexandroff nerve, since we need only identify the center and fixed radius of each ball in a Cech nerve instead of identifying the three vertices of intersecting filled triangles (2-simplexes) in an Alexandroff nerve. As a result, Cech nerves more easily cover $K$ and facilitate approximation of the shapes of irregular finite, bounded planar regions. A main result of this article is an extension of the Edelsbrunner-Harer Nerve Theorem for descriptive and non-descriptive Cech nerves and Cech complexes, covering $K$.Comment: 11 pages, 2 figures, keywords: Ball, Cech Complex, Cech Nerve, Cover, Homotopic Equivalence, Proximit

Remarks on pointed digital homotopy

We present and explore in detail a pair of digital images with $c_u$-adjacencies that are homotopic but not pointed homotopic. For two digital loops $f,g: [0,m]_Z \rightarrow X$ with the same basepoint, we introduce the notion of {\em tight at the basepoint (TAB)} pointed homotopy, which is more restrictive than ordinary pointed homotopy and yields some different results. We present a variant form of the digital fundamental group. Based on what we call {\em eventually constant} loops, this version of the fundamental group is equivalent to that of Boxer (1999), but offers the advantage that eventually constant maps are often easier to work with than the trivial extensions that are key to the development of the fundamental group in Boxer (1999) and many subsequent papers. We show that homotopy equivalent digital images have isomorphic fundamental groups, even when the homotopy equivalence does not preserve the basepoint. This assertion appeared in Boxer (2005), but there was an error in the proof; here, we correct the error.Comment: major new section, some errors correcte

An intrinsic homotopy theory for simplicial complexes, with applications to image analysis

A simplicial complex is a set equipped with a down-closed family of distinguished finite subsets. This structure, usually viewed as codifying a triangulated space, is used here directly, to describe "spaces" whose geometric realisation can be misleading. An intrinsic homotopy theory, not based on such realisation but agreeing with it, is introduced. The applications developed here are aimed at image analysis in metric spaces and have connections with digital topology and mathematical morphology. A metric space X has a structure of simplicial complex at each (positive) resolution e; the resulting n-homotopy group detects those singularities which can be captured by an n-dimensional grid, with edges bound by e; this works equally well for continuous or discrete regions of euclidean spaces. Its computation is based on direct, intrinsic methods.Comment: 46 page

Geodesics of Triangulated Image Object Shapes. Approximating Image Shapes via Rectilinear and Curvilinear Triangulations

This paper introduces the geodesics of triangulated image object shapes. Both rectilinear and curvilinear triangulations of shapes are considered. The triangulation of image object shapes leads to collections of what are known as nerve complexes that provide a workable basis for the study of shape geometry.A nerve complex is a collection of filled triangles with a common vertex. Each nerve complex triangle has an extension called a spoke, which provides an effective means of covering shape interiors. This leads to a geodesic-based metric for shape approximation which offers a straightforward means of assessing, comparing and classifying the shapes of image objects with high acuity.Comment: 32 pages, 11 figure

The Jordan-Brouwer theorem for graphs

We prove a discrete Jordan-Brouwer-Schoenflies separation theorem telling that a (d-1)-sphere H embedded in a d-sphere G defines two different connected graphs A,B in G such a way that the intersection of A and B is H and the union is G and such that the complementary graphs A,B are both d-balls. The graph theoretic definitions are due to Evako: the unit sphere of a vertex x of a graph G=(V,E) is the graph generated by {y | (x,y) in E} Inductively, a finite simple graph is called contractible if there is a vertex x such that both its unit sphere S(x) as well as the graph generated by V-{x} are contractible. Inductively, still following Evako, a d-sphere is a finite simple graph for which every unit sphere is a (d-1)-sphere and such that removing a single vertex renders the graph contractible. A d-ball B is a contractible graph for which each unit sphere S(x) is either a (d-1)-sphere in which case x is called an interior point, or S(x) is a (d-1)-ball in which case x is called a boundary point and such that the set of boundary point vertices generates a (d-1)-sphere. These inductive definitions are based on the assumption that the empty graph is the unique (-1)-sphere and that the one-point graph K_1 is the unique 0-ball and that K_1 is contractible. The theorem needs the following notion of embedding: a sphere H is embedded in a graph G if it is a sub-graph of G and if any intersection with any finite set of mutually adjacent unit spheres is a sphere. A knot of co-dimension k in G is a (d-k)-sphere H embedded in a d-sphere G.Comment: 26 pages, 1 figur

Proximal Planar Shapes. Correspondence between Shapes and Nerve Complexes

This article considers proximal planar shapes in terms of the proximity of shape nerves and shape nerve complexes. A shape nerve is collection of 2-simplexes with nonempty intersection on a triangulated shape space. A planar shape is a shape nerve complex, which is a collection of shape nerves that have nonempty intersection. A main result in this paper is the homotopy equivalence of a planar shape nerve complex and the union of its nerve sub-complexes.Comment: 13 pages, 5 figures. arXiv admin note: text overlap with arXiv:1704.0590

Proximal Planar Shape Signatures. Homology Nerves and Descriptive Proximity

This article introduces planar shape signatures derived from homology nerves, which are intersecting 1-cycles in a collection of homology groups endowed with a proximal relator (set of nearness relations) that includes a descriptive proximity. A 1-cycle is a closed, connected path with a zero boundary in a simplicial complex covering a finite, bounded planar shape. The signature of a shape sh A (denoted by sig(sh A)) is a feature vector that describes sh A. A signature sig(sh A) is derived from the geometry, homology nerves, Betti number, and descriptive CW topology on the shape sh A. Several main results are given, namely, (a) every finite, bounded planar shape has a signature derived from the homology group on the shape, (b) a homology group equipped with a proximal relator defines a descriptive Leader uniform topology and (c) a description of a homology nerve and union of the descriptions of the 1-cycles in the nerve have same homotopy type.Comment: 15 pages; 4 figure