Abstract
In this paper, we consider the digital cohomology modules of a digital image consisting of a bounded and finite subset of Zn and an adjacency relation. We construct a contravariant functor from the category of digital images and digital continuous functions to the category of unitary R-modules and R-module homomorphisms via the category of cochain complexes of R-modules and cochain maps, where R is a commutative ring with identity 1R. We also examine the digital primitive cohomology classes based on digital images and find the relationship between R-module homomorphisms of digital cohomology modules induced by the digital convolutions and digital continuous functions.
Highlights
We explore the digital homology and cohomology modules of digital images and construct a contravariant functor from the category of digital images and digital continuous functions to the category of unitary R-modules and R-module homomorphisms, where R is a commutative ring with identity 1R
We define a digital primitive cohomology class and find out the relationship between R-module homomorphisms of digital cohomology R-modules induced by the digital convolutions and digital continuous functions based on the digital Hopf spaces with digital homotopy multiplications as the immediate application of a Hopf space in algebraic topology; see [8,41]
We investigated some fundamental properties of the digital cohomology modules and the primitive cohomology classes of digital images
Summary
A cohomology group is a general mathematical term for a (positive integer graded or representation ring graded) sequence of Z-modules associated with a topological space (or a spectrum) usually defined from a usual cochain complex by taking a Homfunctor on a chain complex, where Z is the ring of integers. The singular (or simplicial) cochains sn , n ≥ 0 are group homomorphisms (or, equivalently, Z-module homomorphisms) from the free Abelian group Sn ( X ) to an Abelian group G in classical homology and cohomology theories; that is, sn : Sn ( X ) → G is a group homomorphism, where Sn ( X ) is the n-th singular chain group of a topological space X. The digital versions of classical homology and cohomology groups (or Z-modules) may be important algebraic tools to classify (pointed) digital images from the digital homotopy theoretic point of view. The (pointed) Hopf spaces have been a direct and natural generalization of Lie groups in classical homotopy theory as nicely presented in [5,6,7]. Hopf space on the homology level as one of the advantages in classical homology theory
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