DeepAI AI Chat
Log In Sign Up

Digital homotopy relations and digital homology theories

by   P. Christopher Staecker, et al.

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 different digital homology theories: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by D. W. Lee, a cubical homology theory by Jamil and Ali, and a new kind of cubical homology for digital images with c_1-adjacency which is easily computed, and generalizes a construction by Karaca & Ege. We show that the two simplicial homology theories are isomorphic to each other, but distinct from the two cubical theories. We also show that homotopic maps have the same induced homomorphisms in the cubical homology theory, and strong homotopic maps additionally have the same induced homomorphisms in the simplicial theory.


page 1

page 2

page 3

page 4


Strong homotopy of digitally continuous functions

We introduce a new type of homotopy relation for digitally continuous fu...

Connectivity Preserving Multivalued Functions in Digital Topology

We study connectivity preserving multivalued functions between digital i...

Integrated Information in Process Theories

We demonstrate how the key notions of Tononi et al.'s Integrated Informa...

Digital Manifolds and the Theorem of Jordan-Brouwer

We give an answer to the question given by T.Y.Kong in his article "Can ...

Good Pairs of Adjacency Relations in Arbitrary Dimensions

In this text we show, that the notion of a "good pair" that was introduc...

Theories of real addition with and without a predicate for integers

We show that it is decidable whether or not a relation on the reals defi...