Digital homotopy relations and digital homology theories

  title={Digital homotopy relations and digital homology theories},
  author={P. Christopher Staecker},
In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images.We introduce a new type of homotopy relation for digitally continuous functions which we call ``strong homotopy.'' Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane.We also consider four… 

Digital topological groups

. In this article, we develop the basic theory of digital topological groups. The basic definitions directly lead to two separate cat-egories, based on the details of the continuity required of the

Computability of digital cubical singular homology of $c_1$-digital images

Digital cubical singular homology dH q ( X ) for digital images X was developed by the first and third authors, and digital analogues to various results in classical algebraic topology were proved.

Lefschetz numbers and fixed point theory in digital topology

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 $\bar L(f)$. We show that

Hyperspaces and Function Graphs in Digital Topology

We adapt the study of hyperspaces and function spaces from classical topology to digital topology. We define digital hyperspaces and digital function graphs, and study some of their relationships and

Digital Lefschetz numbers and related fixed point theorems



Digital Hurewicz theorem and digital homology theory

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 Properties of Digital Simplicial Homology Groups

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

A Classical Construction for the Digital Fundamental Group

  • L. Boxer
  • Mathematics
    Journal of Mathematical Imaging and Vision
  • 2004
The digital fundamental group is constructed based on the notions of digitally continuous functions presented in [10] and digital homotopy and yields isomorphic fundamental groups for the digital images considered in the latter papers (for certain connectedness types).

A basic course in algebraic topology

1: Two-Dimensional Manifolds. 2: The Fundamental Group. 3: Free Groups and Free Products of Groups. 4: Seifert and Van Kampen Theorem on the Fundamental Group of the Union of Two Spaces.

A fundamental group for digital images

A functor from digital images to groups, which closely resembles the ordinary fundamental group from algebraic topology, is constructed, which shows that the fundamental group is preserved by subdivision.

Computing Homology

The aim of this paper is to provide a short introduction to computational homology based on cubical complexes. The discussed topics include cubical complexes, a reduction algorithm for computing

'Continuous' functions on digital pictures

Conference on Algebraic Topological Methods in Computer Science

  • V. Pratt
  • Computer Science, Mathematics
  • 2001
Algebraic topology has been demonstrated to be a useful tool in solving combinatorial problems of relevance to computing and an extremely useful framework for analyzing problems in distributed computing, such as concurrency.

Remarks on fixed point assertions in digital topology

Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces and homotopy invariant fixed

Non-product property of the digital fundamental group