Variants on Digital Covering Maps
Laurence Boxer

TL;DR
This paper explores different variants of digital covering maps, establishing their equivalences and addressing limitations in previous work to advance understanding in digital topology.
Contribution
It clarifies the relationships among various digital covering map variants and corrects inaccuracies in Han's earlier analysis.
Findings
Established equivalences among digital covering map variants
Identified and discussed shortcomings in Han's original paper
Provided clearer framework for digital covering maps
Abstract
SE Han's paper [11] discusses several variants of digital covering maps. We show several equivalences among these variants and discuss shortcomings in Han's paper.
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Taxonomy
TopicsDigital Image Processing Techniques · Advanced Numerical Analysis Techniques · Computational Geometry and Mesh Generation
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Variants on Digital Covering Maps
Laurence Boxer
Department of Computer and Information Sciences, Niagara University, Niagara University, NY 14109, USA; and Department of Computer Science and Engineering, State University of New York at Buffalo. email: [email protected]
Abstract
S-E Han’s paper [11] discusses several variants of digital covering maps. We show several equivalences among these variants and discuss shortcomings in Han’s paper.
Key words and phrases: digital topology, digital image, covering map
MSC: 54B20, 54C35
1 Introduction
The notion of a covering map has been adapted from classical algebraic topology to digital topology, where it is an important tool for computing digital versions of fundamental groups for binary digital images. With varying success, attempts have been made to modify the notion of a digital covering map to obtain related results under less restrictive conditions. Among these attempts are Han’s paper [11], which contains a proof that is murky (see section 4 for clarification) and citations that are inappropriate. We also discuss a strangely presented example in Han’s related paper [9] (see Remark 6.6). Further, it turns out that some of Han’s variants on covering maps don’t really vary from covering maps (see Theorem 6.5). Also, some of material of [11] is superseded by other papers including [3, 12, 13]. We justify these claims in the current paper.
2 Preliminaries
We use for the set of natural numbers, for the set of integers, and for the number of distinct members of .
We typically denote a (binary) digital image as , where for some and represents an adjacency relation of pairs of points in . Thus, is a graph, in which members of may be thought of as black points, and members of as white points, of a picture of some “real world” object or scene.
2.1 Adjacencies
Let , . Han’s papers use “-adjacency” sometimes to mean an arbitrary adjacency, sometimes as an abbreviation for what he calls “-adjacency,” where the digital image satisfies and are -adjacent if and only if
- •
, and
- •
for at most indices , , and
- •
for all indices such that , we have .
Other authors refer to this adjacency as -adjacency. We will prefer the latter notation in the current paper. The adjacencies are the adjacencies most used in digital topology, especially and .
In low dimensions, it is also common to denote a adjacency by the number of points that can have this adjacency with a given point in . E.g.,
- •
For subsets of , -adjacency is 2-adjacency.
- •
For subsets of , -adjacency is 4-adjacency and -adjacency is 8-adjacency.
- •
For subsets of , -adjacency is 6-adjacency, -adjacency is 18-adjacency, and -adjacency is 26-adjacency.
We use the notations , or, when the adjacency can be assumed, , to mean and are -adjacent. The notations , or, when can be assumed, , mean either or . For , let
[TABLE]
When the image under discussion is clear, we will use the notations or as follows.
[TABLE]
A sequence in a digital image is a -path from to if , , and for .
is -connected [14], or connected when is understood, if for every pair of points there exists a -path in from to .
A (digital) -closed curve is a path such that , and implies . If, also, implies
[TABLE]
then is a (digital) -simple closed curve.
2.2 Digitally continuous functions
Digital continuity is defined to preserve connectedness, as at Definition 2.1 below. By using adjacency as our standard of “closeness,” we get Theorem 2.2 below.
Definition 2.1**.**
[2] (generalizing a definition of [14]) Let and be digital images. A function is -continuous if for every -connected we have that is a -connected subset of .
If either of or is a subset of the other, we use the abbreviation -continuous for -continuous.
When the adjacency relations are understood, we will simply say that is continuous. Continuity can be expressed in terms of adjacency of points:
Theorem 2.2**.**
A function is continuous if and only if in implies .*
Han’s papers generally use the equivalent formulation that is continuous if and only for every , .
See also [5, 6], where similar notions are referred to as immersions, gradually varied operators, and gradually varied mappings.
A digital isomorphism (called homeomorphism in [1]) is a -continuous surjection such that is -continuous.
The literature uses path polymorphically: a -continuous function is a -path if is a -path as described above from to .
3 Han’s variants on local isomorphisms
The definition [8] of a digital covering map was simplified to the following.
Definition 3.1**.**
[4]
Let be a continuous surjection of digital images. The map is a * covering map* if and only if
- •
for every , there is an index set such that
[TABLE]
- •
if , , then ; and
- •
is a -isomorphism.
We find the following definition in Han’s paper [10] (not in [7] despite the claims to the contrary in [10, 11]).
Definition 3.2**.**
A digitally continuous map is a pseudo-local (PL) isomorphism if for every , is -isomorphic to .
In his paper [7], Han gives the following.
Definition 3.3**.**
A digitally continuous map is a local homeomorphism [in more recent terminology, a local isomorphism] if for all , is a -homeomorphism [-isomorphism] onto .
We have the following.
Proposition 3.4**.**
Let be a digitally continuous map. If is a local isomorphism then is a PL isomorphism.
Proof.
Elementary and left to the reader. ∎
Theorem 3.5**.**
([13], correcting an error of [7])*
Let be a continuous surjection. Then is a digital covering map if and only if is a local isomorphism.*
We will also discuss the following notion.
Definition 3.6**.**
[9]
A function is a weakly local (WL) isomorphism if for all , is an isomorphism onto .
4 Theorem 3.15(3) of [11]
Part (3) of Theorem 3.15 of [11] states that
Neither of a PL--isomorphism and a WL--isomorphism implies the other.
The assertion is correct, but Han’s argument for the existence of , , and a WL--isomorphism that is not a PL--isomorphism, is not as clear as it could be. In the following, we clarify Han’s argument.
In his example, Han makes use of an unstated assumption, namely that is connected. He also assumes , that
[TABLE]
and that is the inclusion map, trivially a WL--isomorphism.
Note that since is connected, (1) implies there is a -path such that , . Therefore,
[TABLE]
Hence is not a PL--isomorphism.
5 Theorem 3.20 of [11]
Let and be digital simple closed curves of and points, respectively. Theorem 3.20 of [11] states that embeds into if and only if . Since a connected nonempty subset of is either itself or is isomorphic to a digital interval - which is not even of the same digital homotopy type as - Han’s assertion is an easy consequence of the much older Theorem 5.1 of [3], which states that and have the same digital homotopy type if and only if .
6 Han’s pseudo-covering maps in [11]
Han defines a digital pseudo-covering as follows.
Definition 6.1**.**
[9] Let be a surjection such that for every ,
there is an index set such that , where ; 2. 2.
if and , then ; and 3. 3.
is a WL-isomorphism for all .
Then is a pseudo-covering map.
However, A. Pakdaman shows in [12] that Han’s definition does not effectively give us a new object of study. In particular, Pakdaman shows the following.
Theorem 6.2**.**
A digital pseudo-covering map as defined in Definition 6.1 is in fact a digital covering map.
Definition 6.3**.**
[8] Let be -continuous. Let be -continuous. A -continuous function such that is a (digital) path lifting of . If for every , every , and every path such that ,
[TABLE]
then has the unique path lifting property.
Theorem 6.4**.**
[8]* A digital covering map has the unique path lifting property.*
Next, we show that several of the variants of covering maps that we have discussed are equivalent.
Theorem 6.5**.**
Let be a continuous surjection. Then the following are equivalent.
* is a digital covering map.* 2. 2.
* is a local isomorphism.* 3. 3.
* is a pseudo-covering in the sense of Definition 6.1.* 4. 4.
* is a WL-isomorphism with the unique path lifting property.*
Proof.
That 1) and 2) are equivalent is stated in Theorem 3.5.
That 1) implies 3) follows from Definitions 3.1 and 6.1.
That 3) implies 1) is stated in Theorem 6.2.
It follows from Theorem 6.4 that 1) implies 4).
To show 4) implies 2): suppose is a WL-isomorphism with the unique path lifting property. Let , , . Then is a path in , hence lifts to a unique path in with , . Thus . Since continuity implies , we have . Since is a WL-isomorphism, we have that is a local isomorphism. ∎
Remark 6.6**.**
Han’s Example 4.3(4) of [9] considers (please note here “” is the label of a point, so we will avoid using this notation for 4-adjacency) , , where
[TABLE]
See Figure 1. Han’s claim, that is not a pseudo-4-covering (a pseudo-covering when and both use 4-adjacency), is correct, but this example should not have been considered since is not 4-continuous:
[TABLE]
Since is (4,8)-continuous, perhaps Han intended to show that is not a (4,8)-pseudocovering as defined at Definition 6.1. This can be done by observing that
[TABLE]
Therefore, is not a (4,8)-local isomorphism, so by Theorem 6.5 is not a (4,8)-pseudocovering as defined at Definition 6.1.
In the first paragraph of page 5104 of [11], Han attributes the definition of a digital pointed continuous function to his paper [8]. The definition should be attributed to the earlier paper [2].
Pakdaman modifies Han’s Definition 6.1 as follows.
Definition 6.7**.**
[12] Let be a surjection of digital images. Suppose for all we have the following.
for some index set , where ; 2. 2.
if and then ; and 3. 3.
for all , is a -isomorphism.
Then is a -pseudocovering map.
Pakdaman proceeds to compare unique path lifting results for pseudocovering maps based on Definition 6.7 with those asserted by Han in [11] based on Definition 6.1. He showed that Definition 6.7 gives something not equivalent to a covering map, since such a pseudocovering need not have the unique path lifting property.
7 Further remarks
We have discussed various flaws in Han’s paper [11]. We have shown that several variants of digital covering maps that were presented in [11] are in fact equivalent.
Corrections and suggestions of an anonymous referee are acknowledged with gratitude.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] L. Boxer, Digitally continuous functions, Pattern Recognition Letters 15 (1994), 833-839.
- 2[2] L. Boxer, A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), 51-62
- 3[3] L. Boxer, Properties of digital homotopy, Journal of Mathematical Imaging and Vision 22 (2005), 19-26.
- 4[4] L. Boxer, Digital products, wedges, and covering spaces, Journal of Mathematical Imaging and Vision 25 (2006), 159-171
- 5[5] L. Chen, Gradually varied surfaces and its optimal uniform approximation, SPIE Proceedings 2182 (1994), 300-307.
- 6[6] L. Chen, Discrete Surfaces and Manifolds , Scientific Practical Computing, Rockville, MD, 2004
- 7[7] S.-E. Han, Digital ( k 0 , k 1 ) subscript 𝑘 0 subscript 𝑘 1 (k_{0},k_{1}) -covering map and its properties, Honam Math. J. 26 (2004) 107-117
- 8[8] S-E Han, Non-product property of the digital fundamental group, Information Sciences 171 (2005), 73-91
