# Computability of digital cubical singular homology of $c_{1}$-digital images

Samira Sahar Jamil, P. Christopher Staecker, Danish Ali

###### Abstract

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. Another homology denoted
$H_{q}^{c_{1}}(X)$ was developed by the second author for $c_{1}$-digital
images, which is computationally much simpler than $dH_{q}(X)$. This paper
shows an isomorphism between $dH_{q}(X)$ and $H_{q}^{c_{1}}(X)$ when $X$ is a
$c_{1}$-digital image.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Digital cubical singular homology, $dH_{q}(X)$
    2. 2.2 $c_{1}$-cubical homology, $H^{c_{1}}_{q}(X)$
  3. 3 Properties of digital $q$-cubes
    1. 3.1 Properties of $c_{1}$-neighborhoods
    2. 3.2 Properties of the elementary cube $I^{q}$ and digital $q$-cube
    3. 3.3 Operators on digital $q$-cubes
    4. 3.4 Properties of injective digital $q$-cubes
  4. 4 Classification of degree-$(q-1)$ digital $q$-cubes
  5. 5 A chain map from $dC_{q}(X)$ to $C^{c_{1}}_{q}(X)$
  6. 6 $dH_{q}(X)$ is isomorphic to $H^{c_{1}}_{q}(X)$

##  1 Introduction

In digital topology, digital images are modeled usually as a set of points in
$\mathbb{Z}^{n}$ with an adjacency relation.

To study the topological properties of digital images, notions like
continuity, connectedness, and homotopy were developed using adjacency
relations, which led to the development of homotopy and homology groups.
Digital cubical singular homology $dH_{q}(X)$ for digital images (see Section
2.1 for a precise definition) was developed in [9] based on cubical singular
homology of topological spaces defined in classical texts such as [10]. Some
important results analogous to the results in algebraic topology were obtained
in [9], which include functoriality and homotopy invariance of $dH_{q}(X)$,
and a digital Hurewicz theorem. Although $dH_{q}(X)$ is theoretically
interesting, it has the disadvantage of being computationally expensive,
because the number of generators of the chain groups grow superexponentially
with $q$.

In [11], the second author develops another cubical homology
$H_{q}^{c_{1}}(X)$ (defined in Section 2.2) for certain digital images, called
$c_{1}$-digital images. It was conjectured in [11, Conjecture 6.3] that the
two homologies $dH_{q}(X)$ and $H_{q}^{c_{1}}(X)$ are isomorphic for all $q$.
This paper will prove this conjecture, thus combining the theoretical merits
of $dH_{q}(X)$ with the computability of $H_{q}^{c_{1}}(X)$.

A partial proof of functoriality of $H_{q}^{c_{1}}(X)$ based on computer
enumerations was presented in [11]. We will give a complete analytical proof
of in Theorem 5.6.

Cubical homology of digital images is related to a “discrete homology theory”
for metric spaces introduced in [1]. A discrete cubical singular homology
$\mathcal{H}_{q}(G)$ for simple finite undirected graphs $G$ was studied in
[2] as a special case of discrete homology for metric spaces. Our cubical
homology theory of a digital image $X$ is the same as the discrete cubical
homology $\mathcal{H}_{q}(X)$, if we view $X$ as a graph with edges defined by
the adjacency relation. Like $dH_{q}(X)$, the homology $\mathcal{H}_{q}(G)$ is
computationally difficult and several tools were developed in [2, 3, 4] to
facilitate computations.

This paper is organized as follows: In Section 2, basic notions of digital
topology and the homologies $dH_{q}(X)$ and $H_{q}^{c_{1}}(X)$ are described.
In Section 3 we discuss some useful properties of digital cubes. Section 4
gives a classification of non-injective cubes. Section 5 constructs a chain
map $dH_{q}(X)\to H_{q}^{c_{1}}(X)$, and in Section 6, we show that this chain
map is an isomorphism.

##  2 Preliminaries

A digital image $(X,\kappa)$ is a subset $X$ of $\mathbb{Z}^{d}$ with an
adjacency relation, $\kappa$. Various adjacency relations are used in digital
topology for digital images, to give a concept of proximity or closeness among
elements (or pixels) of digital images. Concepts like continuity,
connectivity, homotopy and homology are developed using these adjacency
relations.

In this paper we exclusively focus on _$c_{1}$ -adjacency_, in which two
points of $\mathbb{Z}^{n}$ are regarded as adjacent if and only if their
coordinates differ by 1 in one dimension, and are equal in all other
dimensions. In $\mathbb{Z}^{2}$, this is referred to as “4-adjacency”, since
each point of $\mathbb{Z}^{2}$ is $c_{1}$-adjacent to exactly 4 other points.
In $\mathbb{Z}^{3}$ this is sometimes called “6-adjacency”.

Since we exclusively use $c_{1}$ adjacency in this paper, we will almost never
need to specify the particular adjacency relation, so we will write a digital
image $(X,\kappa)$ or $(X,c_{1})$ simply as $X$.

For digital images $X$ and $Y$, the function $f:X\to Y$ is continuous, if
every pair of adjacent points in $X$ maps to equal or adjacent points in $Y$.

We quote the definition of (digital) homotopy from [6]. For integers, $a,b$
with $a<b$, the set $[a,b]_{\mathbb{Z}}=\\{a,a+1,\ldots,b\\}$ is called
digital interval. Let $f,g:X\to Y$ be continuous maps between digital images
$X$ and $Y$. If there is an integer $k>0$ and a function
$H:X\times[0,k]_{\mathbb{Z}}\to Y$ satisfying:

  * •

for all $x\in X$, we have $H(x,0)=f(x)$ and $H(x,k)=g(x)$;

  * •

for all $x\in X$, the induced function $H_{x}:[0,k]_{\mathbb{Z}}\to Y$ defined
by $H_{x}(t)=H(x,t)$ is continuous;

  * •

for all $t\in[0,k]_{\mathbb{Z}}$, the induced function $H_{t}:X\to Y$ defined
by $H_{t}(x)=H(x,t)$ is continuous.

Then $H$ is a _homotopy_ from $f$ to $g$, and $f$ and $g$ are _homotopic_.

We say that the digital images $X$ and $Y$ are _homotopy equivalent_ or have
the same _homotopy type_ , if there is a continuous function $f:X\to Y$ and a
continuous function $g:Y\to X$, such that $g\circ f$ is homotopic to the
identity map on $X$ and $f\circ g$ is homotopic to the identity map on $Y$.

We also quote the definitions of digital analogues of retraction, deformation
retraction from [7] and contractibility. Let $X$ be a digital image and
$A\subset X$. A continuous function $r:X\to A$ is called a _retraction_ if
$r(a)=(a)$ for all $a\in A$. If the identity map on $X$ is homotopic to the
$\kappa$-retraction $r$, then we say that $A$ is a _deformation retract_ of
$X$ and the homotopy from the identity map to $r$ is called a _deformation
retraction_ of $X$ onto $A$. If $A$ is a deformation retract of $X$, then $A$
and $X$ are homotopically equivalent [7]. If $A$ is a single point, then $H$
is called a _contraction_ and $X$ is said to be _contractible_.

###  2.1 Digital cubical singular homology, $dH_{q}(X)$

We give below the definition and some results on digital cubical singular
homology developed in [9]. Consider a digital image $(X,\kappa)$ and digital
interval $I=[0,1]_{Z}$. For an integer $q$, define $dQ_{q}(X)$ to be the free
abelian group generated by the set $\\{\sigma:I^{q}\rightarrow
X\mid\sigma\text{ is continuous}\\}$. We say a digital $q$-cube $\sigma$ in
$dQ_{q}(X)$ is degenerate if there is some coordinate $t_{i},1\leq i\leq q$
such that $\sigma(t_{1},\ldots,t_{q})$ does not depend on $t_{i}$. Let
$dD_{q}(X)$ be a free abelian group generated by the set of all degenerate
$q$-cubes. Define the quotient group $dC_{q}(X)=dQ_{q}(X)/dD_{q}(X)$. It can
be shown that $dC_{q}(X)$ is a free abelian group generated by the set of all
nondegenerate $q$-cubes. Define “face operators” $A_{i},B_{i}:I^{q-1}\to
I^{q}$ as
$A_{i}(t_{1},t_{2},\ldots,t_{q-1})=(t_{1},\ldots,t_{i-1},0,t_{i},\ldots,t_{q-1})$
and
$B_{i}(t_{1},t_{2},\ldots,t_{q-1})=(t_{1},\ldots,t_{i-1},1,t_{i},\ldots,t_{q-1})$,
and a homomorphism $\partial_{q}:dC_{q}(X)\rightarrow dC_{q-1}(X)$ as
$\sum_{i=1}^{q}(-1)^{i}(A_{i}\sigma-B_{i}\sigma)$, where:
$A_{i}\sigma=\sigma\circ A_{i}$ and $B_{i}\sigma=\sigma\circ B_{i}$. It can be
shown that $\partial_{q-1}\circ\partial_{q}=0$. Therefore,
$(dC_{q}(X),\partial_{q})$ is a chain complex, with homology $dH_{q}(X)$.

Elements of $dC_{q}(X)$, $ker\partial_{q}$, $Im\partial_{q+1}$ and $dH_{q}(X)$
are called $q$-chains, $q$-cycles, $q$-boundaries and homology classes,
respectively. The digital $(q-1)$-cubes $A_{i}\sigma$ and $B_{i}\sigma$ are
called front and back $i$-faces of $\sigma$, respectively.

Using [9, Proposition 3.1 and 3.2], the following can be stated.

######  Proposition 2.1.

Let $(X,\kappa)$ be a digital image consisting of $d$ isolated points. Then
$dH_{0}(X)=\mathbb{Z}^{d}$ and $dH_{q}(X)=0$ for $q>0$.

######  Proposition 2.2.

Let $(X,\kappa)$ be a digital image consisting of $d$ components. Then
$dH_{0}(X)=\mathbb{Z}^{d}$.

We restate Corollary 3.8 and Theorem 5.10 of [9] below.

######  Proposition 2.3.

Let $(X,\kappa)$ and $(Y,\lambda)$ be digital images of the same homotopy
type. Then $dH_{q}(X)\cong dH_{q}(Y)$.

###  2.2 $c_{1}$-cubical homology, $H^{c_{1}}_{q}(X)$

A homology for digital images with $c_{1}$-adacency, called $c_{1}$-cubical
homology, was given in [11]. Basic definitions from [11] are given below: For
any integer $a$, a set of the form $[a,a+1]_{\mathbb{Z}}=\\{a,a+1\\}$ or
$[a,a]_{\mathbb{Z}}=\\{a\\}$ is called an elementary interval. The elementary
interval $[a]_{\mathbb{Z}}$ is called degenerate, while the elementary
interval $[a,a+1]_{\mathbb{Z}}$ is called nondegenerate. An elementary cube
$Q$ is a cartesian product of $n$ elementary intervals $J_{i}$, for
$i\in\\{1,2,\ldots,n\\}$, i.e. $Q=J_{1}\times\cdots\times J_{n}$. The
dimension of an elementary cube $Q$ is the number of nondegenerate elementary
intervals $J_{i}$. If the dimension of elementary cube is $q$, we call it an
_elementary $q$-cube_. For a digital image $(X,c_{1})$ and $q\geq 0$, let
$C_{q}^{c_{1}}(X)$ be the free abelian group generated by the set of all
elementary $q$-cubes in $X$. The face maps
$A_{i}^{c_{1}},B_{i}^{c_{1}}:C_{q}^{c_{1}}(X)\to C_{q-1}^{c_{1}}(X)$ are
defined as follows:

$A_{i}^{c_{1}}Q=J_{1}\times\cdots\times J_{i-1}\times[\min{J_{i}}]\times
J_{i+1}\times\cdots\times J_{n}$ $B_{i}^{c_{1}}Q=J_{1}\times\cdots\times
J_{i-1}\times[\max{J_{i}}]\times J_{i+1}\times\cdots\times J_{n}$

Note that $A_{i}^{c_{1}}Q$ and $B_{i}^{c_{1}}Q$ are distinct only if $J_{i}$
is nondegenerate. Let $\\{j_{1},\ldots,j_{q}\\}\subset\\{1,\ldots,n\\}$ be the
set of all indices such that $J_{j_{i}}$ is nondegenerate. The boundary
operator is given as:
$\partial_{q}^{c_{1}}Q=\sum_{i=1}^{q}(-1)^{i}(A_{j_{i}}^{c_{1}}Q-B_{j_{i}}^{c_{1}}Q)$.
Thus we have a chain complex $(C_{q}^{c_{1}}(X),\partial_{q}^{c_{1}})$ and the
associated homology $H_{q}^{c_{1}}(X)$ is called the _$c_{1}$ -cubical
homology_ of the digital image $X$. The following results for $c_{1}$-cubical
homology from [11] are used in this paper:

######  Proposition 2.4 [11].

Let $(X,c_{1})$ be a digital image consisting of $d$ isolated points. Then
$H_{0}^{c_{1}}(X)=\mathbb{Z}^{d}$ and $H_{q}^{c_{1}}(X)=0$ for $q>0$.

######  Proposition 2.5 [11].

Let $(X,c_{1})$ be a digital image consisting of $d$ components. Then
$H_{0}^{c_{1}}(X)=\mathbb{Z}^{d}$.

##  3 Properties of digital $q$-cubes

In this section we give some properties of digital $q$-cubes, ending with a
complete classification of injective $q$-cubes.

###  3.1 Properties of $c_{1}$-neighborhoods

For a digital image $(X,c_{1})$, and $x\in X$, define $c_{1}$-neighborhood and
deleted $c_{1}$-neighborhood of $x$, respectively as:

$\displaystyle\overline{N}_{c_{1}}^{X}(x)$ $\displaystyle=\\{y\in X\mid
x\text{ is }c_{1}\text{-adjacent or equal to }y\\}$ $\displaystyle
N_{c_{1}}^{X}(x)$ $\displaystyle=\\{y\in X\mid x\text{ is
}c_{1}\text{-adjacent to }y\\}.$

Let $\overline{N}_{c_{1}}^{X}(x_{1},\ldots,x_{n})$ denote the set
$\overline{N}_{c_{1}}^{X}(x_{1})\cap\cdots\cap\overline{N}_{c_{1}}^{X}(x_{n})$
of common $c_{1}$-neighbors of elements $x_{1},\ldots,x_{n}$ of $(X,c_{1})$,
and similarly for $N_{c_{1}}^{X}$.

######  Proposition 3.1.

For distinct elements $x,y$ in $\mathbb{Z}^{n}$, the set
$N_{c_{1}}^{\mathbb{Z}^{n}}(x,y)$ is nonempty if and only if one of the
following is true:

  * •

$x$ and $y$ differ in exactly $2$ coordinates by $1$ unit each.

  * •

$x$ and $y$ differ in exactly $1$ coordinate by $2$ units.

###### Proof.

Let $z\in N_{c_{1}}^{\mathbb{Z}^{n}}(x,y)$. There are indices
$j,k\in\\{1,2,\ldots,n\\}$ such that

$z_{i}=x_{i},\forall i\neq j\text{ and }x_{j}\in\\{z_{j}\pm 1\\}$
$z_{i}=y_{i},\forall i\neq k\text{ and }y_{k}\in\\{z_{k}\pm 1\\}$

Therefore $y_{i}=x_{i},\forall i\notin\\{j,k\\}$. Furthermore, if $j\neq k$
then $x$ and $y$ differ by 1 unit in coordinates $j$ and $k$, while if $j=k$
then $x$ and $y$ differ by 2 units in coordinate $j$. ∎

We can conclude from Proposition 3.1 that whenever $c_{1}$-neighborhoods of
two distinct elements intersect, they intersect in at most two elements. Also,
the deleted $c_{1}$-neighborhoods of two $c_{1}$-adjacent elements do not
intersect.

######  Corollary 3.2.

The following is true for distinct elements $x,y$ in $\mathbb{Z}^{n}$:

  1. _(i)_

$\left|\overline{N}_{c_{1}}^{\mathbb{Z}^{n}}(x,y)\right|\leq 2$.

  2. _(ii)_

If $x$ and $y$ are $c_{1}$-neighbors then
$\overline{N}_{c_{1}}^{\mathbb{Z}^{n}}(x,y)=\\{x,y\\}$ or
$N_{c_{1}}^{\mathbb{Z}^{n}}(x,y)=\emptyset$.

The following corollary asserts that $c_{1}$-neighborhoods of more than $3$
points intersect in at most $1$ point.

######  Corollary 3.3.

Let $i$ be an integer in $\\{1,2,\ldots,k\\},k\geq 3$. For distinct elements
$x_{i}$ in $\mathbb{Z}^{n}$,
$\left|N_{c_{1}}^{\mathbb{Z}^{n}}(x_{1},\ldots,x_{k})\right|\leq 1$. If
$k>2n$, then $N_{c_{1}}^{\mathbb{Z}^{n}}(x_{1},\ldots,x_{k})=\emptyset$.

###### Proof.

For $k=3$, we have $3$ distinct points $x,y,z$ in $\mathbb{Z}^{n}$. If
$N_{c_{1}}^{\mathbb{Z}^{n}}(x,y,z)$ is nonempty then let $w\in
N_{c_{1}}^{\mathbb{Z}^{n}}(x,y,z)$, then there exist indices
$i_{1},i_{2},i_{3}$ in $\\{1,2,\ldots,n\\}$, such that

$w_{i}=x_{i},\forall i\neq i_{1}\text{ and }w_{i_{1}}\in\\{x_{i_{1}}\pm 1\\},$
$w_{i}=y_{i},\forall i\neq i_{2}\text{ and }w_{i_{2}}\in\\{y_{i_{2}}\pm 1\\},$
$w_{i}=z_{i},\forall i\neq i_{3}\text{ and }w_{i_{3}}\in\\{z_{i_{3}}\pm 1\\}.$

It can be seen that each coordinate of $w$ is equal to the corresponding
coordinate of $x,y$ or $z$, therefore, $w$ is unique. For $k>3$, the result
follows from $N_{c_{1}}^{\mathbb{Z}^{n}}(x_{1},\ldots,x_{k})\subset
N_{c_{1}}^{\mathbb{Z}^{n}}(x_{1},x_{2},x_{3})$. Common neigborhood of more
than $2n$ elements is empty, because an element in $\mathbb{Z}^{n}$ has at
most $2n$ $c_{1}$-neighbors. ∎

###  3.2 Properties of the elementary cube $I^{q}$ and digital $q$-cube

Since elements of $I^{q}$ are $q$-tuples of $0$ and $1$, we can use logical
operators componentwise. For $x=(x_{i})_{i=1}^{q}$ and $y=(y_{i})_{i=1}^{q}$
in $I^{q}$, $x\vee y=(x_{i}\vee y_{i})_{i=1}^{q}$. Partition the set $I^{q}$
into sets
$S_{n}=\\{(t_{1},t_{2},\ldots,t_{q})|\sum_{j=1}^{q}t_{j}=n\\},n\in\\{0,1,\ldots,q\\}$.
Clearly $S_{0}$ and $S_{1}$ are singletons. We develop below some notation for
elements of $I^{q}$.

######  Notation 3.4 ($\theta$-notation for elements of $I^{q}$).

For elementary $q$-cube $I^{q}$, let $\mathit{0}$ denote the only element of
$S_{0}$. For a set $\\{i_{1},\ldots,i_{n}\\}$ of $n$ distinct indices in
$\\{1,\ldots,q\\}$, let $\theta^{i_{1}i_{2}\cdots i_{n}}$ denote the element
of $S_{n}$, with $1$s in coordinates $i_{1},\ldots,i_{n}$, and $0$s everywhere
else.

The elements $\theta^{i_{1}i_{2}\cdots i_{n}}$ and $\theta^{l_{1}l_{2}\cdots
l_{n}}$ are equal if the sets $\\{i_{1},i_{2},\ldots,i_{n}\\}$ and
$\\{l_{1},l_{2},\ldots,l_{n}\\}$ are equal. Therefore the order of
superscripts does not make any difference. For $1\leq n\leq q-1$, each element
in $S_{n}$ has $c_{1}$-neighbors only in $S_{n-1}$ and $S_{n+1}$.

######  Proposition 3.5.

For $n\in\\{2,3,\ldots,q\\},q\geq 2$, every $u\in S_{n}$ is a $c_{1}$-neighbor
of exactly $n$ elements $v^{(1)},\ldots,v^{(n)}$ of $S_{n-1}$, such that
$u=v^{(1)}\vee\cdots\vee v^{(n)}.$

###### Proof.

Let $u=\theta^{i_{1}i_{2}\cdots i_{n}}$ and let
$\theta^{i_{1}i_{2}\cdots\widehat{i_{j}}\cdots i_{n}}$ represent the element
$\theta^{i_{1}i_{2}\cdots i_{j-1}i_{j+1}\cdots i_{n}}$ of $S_{n-1}$, then

$\theta^{i_{1}i_{2}\cdots
i_{n}}\in\bigcap_{j=1}^{n}N_{c_{1}}^{I^{q}}\left(\theta^{i_{1}i_{2}\cdots\widehat{i_{j}}\cdots
i_{n}}\right),$

and also,

$\theta^{i_{1}i_{2}\cdots
i_{n}}=\bigvee_{j=1}^{n}\theta^{i_{1}i_{2}\cdots\widehat{i_{j}}\cdots
i_{n}}.\vspace{-20pt}$

∎

######  Remark 3.6.

Let $\sigma:I^{q}\to X$ be a digital $q$-cube, $q\geq 2$ in digital image $X$.
Since $\sigma$ is $(c_{1},c_{1})$-continuous, for any
$\theta^{i_{1}i_{2}\cdots i_{n}},n\geq 2$,

$\sigma\left(\theta^{i_{1}i_{2}\cdots
i_{n}}\right)\in\bigcap_{j=1}^{n}\overline{N}_{c_{1}}^{X}\left(\sigma(\theta^{i_{1}i_{2}\cdots\widehat{i_{j}}\cdots
i_{n}})\right).$

We prove below that if the digital $q$-cube $\sigma$ is injective when
restricted to a certain subset of $I^{q}$, then $\sigma$ is injective.

######  Lemma 3.7.

Let $\sigma:I^{q}\to X$ be a digital $q$-cube, $q\geq 2$ in digital image
$(X,c_{1})$. If the restriction of $\sigma$ to $S_{0}\cup S_{1}\cup S_{2}$ is
injective, then $\sigma$ is injective.

###### Proof.

The statement is trivially true for $q=2$. It suffices to show that if
$\sigma|_{S_{0}\cup\cdots\cup S_{n-1}}$ is injective, then
$\sigma|_{S_{0}\cup\cdots\cup S_{n}}$ is injective, for $3\leq n\leq q$. For
any $\theta^{i_{1}i_{2}\cdots i_{n}}\in S_{n}$,
$\sigma\left(\theta^{i_{1}i_{2}\cdots i_{n}}\right)$ is a common neighbor of
more than $3$ distinct elements from Remark 3.6. Therefore from Corollary 3.3,
$\sigma\left(\theta^{i_{1}i_{2}\cdots i_{n}}\right)$ is unique for every set
$\\{i_{1},i_{2},\ldots,i_{n}\\}$ of indices. ∎

Now we show that for an injective cube $\sigma$, the restriction to $S_{0}\cup
S_{1}$ completely determines all other values of $\sigma$.

######  Lemma 3.8.

Let $\sigma$ be a digital $q$-cube with $q\geq 2$. If $\sigma$ is injective,
then all values of $\sigma$ are uniquely determined by its values on
$S_{0}\cup S_{1}$.

###### Proof.

Let $\sigma$ be injective. We prove the Lemma by showing that, if the values
of $\sigma$ on $S_{0}\cup\dots\cup S_{n}$ are given for some $n\geq 1$, then
these values will uniquely determine any value of $\sigma$ on $S_{n+1}$.

Assume for some $n\geq 1$ that the values of $\sigma$ on $S_{0}\cup\dots\cup
S_{n}$ are specified, and take some point $x=\theta^{i_{1}\dots
i_{n}i_{n+1}}\in S_{n+1}$. Then $\sigma(x)$ is a common neighbor of every
element of the form $\sigma(\theta^{i_{1}\dots\hat{i}_{k}\dots i_{n+1}})$.
Since $\sigma$ is injective, $\sigma(x)$ is a common neighbor of $n+1\geq 2$
points and so the value $\sigma(x)$ is uniquely determined. ∎

Injective cubes are closely related to embeddings:

######  Definition 3.9.

For a digital $q$-cube $\sigma:I^{q}\to X$, we say $\sigma$ is an _embedding_
if $\sigma(I^{q})$ is an elementary $q$-cube in $X$.

The term “embedding” is used because an embedding is an isomorphism onto its
image.

######  Theorem 3.10.

If $\sigma:I^{q}\to X$ is an embedding, then $\sigma:I^{q}\to\sigma(I^{q})$ is
an isomorphism of digital images.

###### Proof.

If $\sigma:I^{q}\to X$ is an embedding, then $\sigma(I^{q})$ is an elementary
$q$-cube, and so $\sigma(I^{q})$ is naturally isomorphic (as a digital image)
to $I^{q}$. Let $i:\sigma(I^{q})\to I^{q}$ be this isomorphism. Then
$\sigma\circ i:I^{q}\to I^{q}$ is continuous and injective (and thus
bijective). By Lemma 2.3 of [8], $\sigma\circ i$ is an isomorphism, and thus
$\sigma$ is an isomorphism. ∎

Then we can easily obtain:

######  Theorem 3.11.

If $\sigma:I^{q}\to X$ is injective, then $\sigma$ is an embedding.

###### Proof.

Let $\sigma$ be injective. Then $\sigma$ is uniquely determined by its values
on the set $V=S_{0}\cup S_{1}=\\{0,\theta^{1},\dots,\theta^{q}\\}$. There is
an embedding which agrees with $\sigma$ on $V$: the affine linear map which
sends $V$ to $\sigma(V)$. By Lemma 3.8, $\sigma$ is uniquely determined by its
values on $V$, and so we conclude that $\sigma$ is this affine linear map,
which is an embedding. ∎

###  3.3 Operators on digital $q$-cubes

We define below some operators for digital $q$-cubes.

######  Definition 3.12.

Define $F_{j},C_{i,j},R_{i,j},S_{i,j}:I^{q}\to I^{q}$, for
$i,j\in\\{1,\ldots,q\\}$, $q\geq 1$

  1. _(i)_

$F_{j}(t_{1},\ldots,t_{q})=(t_{1},\ldots,t_{j-1},1-t_{j},t_{j+1},\ldots,t_{q})$

  2. _(ii)_

$C_{i,j}(t_{1},\ldots,t_{q})=\begin{cases}(t_{1},\ldots
t_{i-1},t_{j},t_{i+1},\ldots,t_{j-1},t_{i},t_{j+1},\ldots,t_{q}),&\text{if
}i<j\\\ (t_{1},\ldots
t_{j-1},t_{i},t_{j+1},\ldots,t_{i-1},t_{J},t_{i+1},\ldots,t_{q}),&\text{if
}j<i\\\ (t_{1},\ldots,t_{q}),&\text{if }i=j\end{cases}$

  3. _(iii)_

$S_{i,j}(t_{1},\ldots,t_{q})=\begin{cases}(t_{1},\ldots
t_{i-1},t_{j},t_{i},\ldots,t_{j-1},t_{j+1},\ldots,t_{q})&\text{if }i<j\\\
(t_{1},\ldots
t_{j-1},t_{j+1},\ldots,t_{i},t_{j},t_{i+1},\ldots,t_{q})&\text{if }i>j\\\
(t_{1},\ldots,t_{q}),&\text{if }i=j\end{cases}$

  4. _(iv)_

$R_{i,j}=F_{j}\circ C_{i,j}$

$F_{j}$ is a ‘flip’ of the cube $I^{q}$ in coordinate $j$, while $C_{i,j}$
‘swaps’ coordinates $i$ and $j$. $S_{i,j}$ ‘shifts’ the coordinate $j$ to
position of the coordinate $i$. For $i\neq j$, $R_{i,j}$ is the ‘rotation’ of
$I^{q}$ by a quarter turn in the plane formed by coordinates $i$ and $j$. For
$q=2$, $R_{1,2}$ is counterclockwise and $R_{2,1}$ is clockwise rotation.

######  Proposition 3.13.

For indices $i,j\in\\{1,\ldots,q\\}$, and operators $F_{j},C_{i,j},R_{i,j}$
and $S_{i,j}$, we have

  * •

$F_{j},C_{i,j},R_{i,j}$ and $S_{i,j}$ are all $(c_{1},c_{1})$-continuous
bijections.

  * •

$S_{i,j}$ can be broken into $j-i$ factors:

$\displaystyle S_{i,j}$ $\displaystyle=\begin{cases}C_{i,i+1}\circ
C_{i+1,i+2}\circ\cdots\circ C_{j-2,j-1}\circ C_{j-1,j}&\text{if }i\leq j\\\
C_{i-1,i}\circ C_{i-2,i-1}\circ\cdots\circ C_{j+1,j+2}\circ C_{j,j+1}&\text{if
}i\geq j\end{cases}$

  * •

$C_{i,j}=C_{j,i}$ and $C_{i,j}\circ F_{i}=F_{j}\circ C_{i,j}$

  * •

$C_{i,i}=S_{i,i}=F_{j}\circ F_{j}=C_{i,j}\circ C_{i,j}=R_{i,j}\circ
R_{j,i}=S_{i,j}\circ S_{j,i}$ are the identity map $\operatorname{id}$ on
$I^{q}$ and $R_{i,i}=F_{i}$.

The operators $F_{i}$, for $i\in\\{1,\ldots,q\\}$ commute in a composition,
but do not commute with the operator $C_{j,k}$, if $i\in\\{j,k\\}$. We state
this as the proposition below.

######  Proposition 3.14.

For indices $i,j,k\in\\{1,\ldots,q\\}$, and operators $F_{j}$ and $C_{i,j}$,
we have

  * •

$F_{i}\circ F_{j}=F_{j}\circ F_{i}$

  * •

$F_{i}\circ C_{j,k}=C_{j,k}\circ F_{i}$, if $i\notin\\{j,k\\}$

It can be shown that

$\displaystyle A_{k}(\sigma\circ F_{j})=\begin{cases}A_{k}\sigma\circ
F_{j-1},&k<j,\\\ B_{k}\sigma,&k=j,\\\ A_{k}\sigma\circ
F_{j},&k>j,\end{cases}~{}~{}~{}~{}~{}~{}~{}B_{k}(\sigma\circ
F_{j})=\begin{cases}B_{k}\sigma\circ F_{j-1},&k<j,\\\ A_{k}\sigma,&k=j,\\\
B_{k}\sigma\circ F_{j},&k>j,\end{cases}$

and therefore,

$\displaystyle\partial(\sigma\circ F_{j})=$
$\displaystyle\sum_{k=1}^{j-1}(-1)^{k}(A_{k}\sigma\circ
F_{j-1}-B_{k}\sigma\circ F_{j-1})+(-1)^{j}(B_{j}\sigma-A_{j}\sigma)$
$\displaystyle+\sum_{k=j+1}^{q}(-1)^{k}(A_{k}\sigma\circ
F_{j}-B_{k}\sigma\circ F_{j}).$

We also have for $i<j$:

$\displaystyle A_{k}(\sigma\circ C_{i,j})=\begin{cases}A_{k}\sigma\circ
C_{i-1,j-1},&k<i<j\\\ A_{j}\sigma\circ S_{i,j-1},&k=i<j\\\ A_{k}\sigma\circ
C_{i,j-1},&i<k<j\\\ A_{i}\sigma\circ S_{j-1,i},&i<k=j\\\ A_{k}\sigma\circ
C_{i,j},&i<j<k\end{cases}$ (1)

and similarly,

$\displaystyle B_{k}(\sigma\circ C_{i,j})=\begin{cases}B_{k}\sigma\circ
C_{i-1,j-1},&k<i<j\\\ B_{j}\sigma\circ S_{i,j-1},&k=i<j\\\ B_{k}\sigma\circ
C_{i,j-1},&i<k<j\\\ B_{i}\sigma\circ S_{j-1,i},&i<k=j\\\ B_{k}\sigma\circ
C_{i,j},&i<j<k\end{cases}$ (2)

Therefore, for $i<j$ we obtain:

$\displaystyle\partial(\sigma\circ C_{i,j})=$
$\displaystyle\sum_{k=1}^{i-1}(-1)^{k}(A_{k}\sigma\circ
C_{i-1,j-1}-B_{k}\sigma\circ C_{i-1,j-1})$
$\displaystyle+(-1)^{i}(A_{j}\sigma\circ S_{i,j-1}-B_{j}\sigma\circ
S_{i,j-1})$ $\displaystyle+\sum_{k=i+1}^{j-1}(A_{k}\sigma\circ
C_{i,j-1}-B_{k}\sigma\circ C_{i,j-1})$
$\displaystyle+(-1)^{j}(A_{i}\sigma\circ S_{j-1,i}-B_{i}\sigma\circ
S_{j-1,i})$ $\displaystyle+\sum_{j+1}^{q}(-1)^{k}(A_{k}\sigma\circ
C_{i,j}-B_{k}\sigma\circ C_{i,j}).$

We can ‘append’ two digital $q$-cubes $\sigma$ and $\gamma$ to get a function
$I^{q+1}\to X$. This function is $(c_{1},c_{1})$-continuous and thus a digital
$(q+1)$-cube, provided that $\sigma$ and $\gamma$ are ‘compatible’. We define
below the append operation and the compatibility condition.

######  Definition 3.15.

Let $\sigma,\gamma$ be digital $q$-cubes in a digital image $(X,c_{1})$.

  * •

We say that $\sigma$ and $\gamma$ are compatible if
$\sigma(t_{1},\ldots,t_{q})$ is $c_{1}$-adjacent or equal to
$\gamma(t_{1},\ldots,t_{q})$, for all $t_{i}\in\\{0,1\\}$.

  * •

Define $\sigma\boxminus\gamma:I^{q+1}\to X$ as follows:

$\displaystyle\sigma\boxminus\gamma(t_{1},\ldots,t_{q+1})=\begin{cases}\sigma(t_{1},\ldots,t_{q}),&\text{if
}t_{q+1}=0\\\ \gamma(t_{1},\ldots,t_{q}),&\text{if }t_{q+1}=1\end{cases}$

Note that if $\sigma$ and $\gamma$ are compatible, then
$\sigma\boxminus\gamma$ is a digital ($q+1$)-cube with

$A_{q+1}(\sigma\boxminus\gamma)=\sigma,\qquad
B_{q+1}(\sigma\boxminus\gamma)=\gamma.$

The boundary of $\sigma\boxminus\gamma$ is given as:

$\partial(\sigma\boxminus\gamma)=\sum_{k=1}^{q}(-1)^{k}(A_{k}\sigma\boxminus
A_{k}\gamma-B_{k}\sigma\boxminus B_{k}\gamma)+(-1)^{q+1}(\sigma-\gamma).$ (3)

###  3.4 Properties of injective digital $q$-cubes

Now we will prove some descriptions of injective $q$-cubes in terms of the
swaps $C_{i,j}$ and flips $F_{i}$. First we show that any injection of $I^{q}$
to itself is a composition of swaps and flips.

######  Lemma 3.16.

Let $\sigma:I^{q}\to I^{q}$ be an injective digital $q$-cube. Then $\sigma$ is
a composition of swaps and flips.

###### Proof.

Let $\sigma(0)=\theta^{s_{1}\dots s_{m}}$. Each point $\sigma(\theta^{i})$
will differ in one coordinate from $\sigma(0)$, and so we will have

$\sigma(\theta^{i})=\theta^{s_{1}\dots s_{m}}+\epsilon_{i}\theta^{k_{i}}$

where $\epsilon_{i}=1$ if $k_{i}\not\in\\{s_{1},\dots,s_{m}\\}$ and
$\epsilon_{i}=-1$ if $k_{i}\in\\{s_{1},\dots,s_{m}\\}$. We will show that:

$\sigma=F_{s_{1}}\circ\dots\circ F_{s_{m}}\circ C_{1,k_{1}}\circ\dots\circ
C_{m,k_{m}}.$ (4)

Let $\rho$ be the right-hand side of (4), and it suffices to show that
$\rho(x)=\sigma(x)$ for each $x\in I^{q}$. In fact by Lemma 3.8, it suffices
to show that $\rho(0)=\sigma(0)$ and $\rho(\theta^{i})=\sigma(\theta^{i})$ for
each $i$.

We have:

$\displaystyle\rho(0)$ $\displaystyle=F_{s_{1}}\circ\dots\circ F_{s_{m}}\circ
C_{1,k_{1}}\circ\dots\circ C_{m,k_{m}}(0)$
$\displaystyle=F_{s_{1}}\circ\dots\circ F_{s_{m}}(0)=\theta^{s_{1}\dots
s_{m}}=\sigma(0),$

and

$\displaystyle\rho(\theta^{i})$ $\displaystyle=F_{s_{1}}\circ\dots\circ
F_{s_{m}}\circ C_{1,k_{1}}\circ\dots\circ C_{m,k_{m}}(\theta^{i})$
$\displaystyle=F_{s_{1}}\circ\dots\circ
F_{s_{m}}(\theta^{k_{i}})=\theta^{s_{1}\dots
s_{m}}+\epsilon_{i}\theta^{k_{i}}=\sigma(\theta^{i})$

as desired. ∎

Now we show that when two injective cubes have the same image, they differ by
swaps and flips.

######  Lemma 3.17.

Let $\sigma,\phi:I^{q}\to X$ be injective digital $q$-cubes in $X$ with
$\sigma(I^{q})=\phi(I^{q})$. Then:

$\phi=\sigma\circ G,$

where $G:I^{q}\to I^{q}$ is a composition of swaps and flips.

###### Proof.

Let $Q=\sigma(I^{q})=\phi(I^{q})$, and since $\sigma$ and $\phi$ are
injective, $Q$ is an elementary $q$-cube in $X$. There is a canonical
isomorphism $\tau:Q\to I^{q}$, and then we have injections
$\tau\circ\sigma,\tau\circ\phi:I^{q}\to I^{q}$ such that:

$\phi=\sigma\circ(\tau\circ\sigma)^{-1}\circ(\tau\circ\phi).$

By Lemma 3.16, each of $\tau\circ\sigma$ and $\tau\circ\phi$ is a composition
of swaps and flips, which gives the desired result. ∎

When two injective cubes are compatible, they can differ only by very specific
types of swaps and flips.

######  Lemma 3.18.

Let $\sigma,\phi$ be injective $q$-cubes which are compatible, with
$\sigma(I^{q})=\phi(I^{q})$. Then one of the following holds:

  1. 1.

$\phi=\sigma$

  2. 2.

$\phi=\sigma\circ F_{j}$ for some $j$

  3. 3.

$\phi=\sigma\circ F_{j}\circ C_{i,j}$ for some distinct $i$ and $j$

###### Proof.

By Lemma 3.17, we have $\phi=\sigma\circ G$, where $G$ is a composition of
swaps and flips. In fact, we can commute the swaps and flips of $G$ so that:

$\phi=\sigma\circ F_{i_{1}}\circ\dots\circ F_{i_{n}}\circ C$

where $C$ is a (possibly trivial) composition of swaps, and possibly $n=0$.

First we show that $n\in\\{0,1\\}$. Let $x=C^{-1}(\theta^{i_{1}\dots i_{n}})$.
Then:

$\phi(x)=\sigma\circ F_{i_{1}}\circ\dots\circ F_{i_{n}}(\theta^{i_{1}\dots
i_{n}})=\sigma(0)$

Since $\sigma$ and $\phi$ are compatible, $\sigma(x)$ is $c_{1}$-adjacent to
$\phi(x)=\sigma(0)$. Since $\sigma$ is an embedding, this means $0$ is
$c_{1}$-adjacent to $x$, and so $x$ can have at most 1 nonzero coordinate.
Since $x=C^{-1}(\theta^{i_{1}\dots i_{n}})$ and $C$ is a permutation of
coordinates, $x$ will have exactly $n$ nonzero coordinates, and so $n\leq 1$
as desired.

Thus far we have shown that

$\phi=\sigma\circ F\circ C,$

where $F$ is either a single flip, or the identity.

Next we will show that $C(\theta^{k})=\theta^{k}$ for all except at most 3
values of $k$. To obtain a contradiction, assume that we have 4 distinct
values $\theta^{i},\theta^{j},\theta^{p},\theta^{q}$ which are not fixed by
$C$. Since $C$ is a bijection, the image of a non-fixed point is also not
fixed, so we can assume that $C(\theta^{i})=\theta^{j}$ and
$C(\theta^{p})=\theta^{q}$.

Since $\phi$ and $\sigma$ are compatible and $C(\theta^{i})=\theta^{j}$, we
see that $\sigma(\theta^{i})$ is $c_{1}$-adjacent to
$\phi(\theta^{i})=\sigma(F(\theta^{j}))$. Since $\sigma$ is an embedding, this
means that $\theta^{i}$ is $c_{1}$-adjacent to $F(\theta^{j})$. This is
possible only when $F=F_{k}$ for some $k\in\\{i,j\\}$.

Applying the same reasoning as the above paragraph to
$C(\theta^{p})=\theta^{q}$ gives $F=F_{k}$ for $k\in\\{p,q\\}$, which is a
contradiction since $\\{i,j\\}$ is disjoint from $\\{p,q\\}$. Thus we conclude
that $C(\theta^{k})=\theta^{k}$ for all except at most 3 values of $k$. This
means that when we express $C$ as a composition of swaps, $C$ must either be
identity, or $C=C_{i,j}$ for some $i\neq j$, or $C=C_{i,j}\circ C_{j,k}$ for
some distinct $i,j,k$. We finish the proof by considering these cases
individually, showing that the third is impossible.

If $C$ is the identity, then $\phi$ has one of the forms numbered 1 or 2 in
the Lemma.

If $C=C_{i,j}$, then $C(\theta^{i})=\theta^{j}$ and arguments above show that
$F=F_{k}$ for some $k\in\\{i,j\\}$. If $k=j$, then $\phi=F_{j}\circ C_{i,j}$
which is form 3, and if $k=i$ then $\phi=F_{i}\circ C_{i,j}=F_{i}\circ
C_{j,i}$ which is also form 3.

We will show that the remaining case, that $C=C_{i,j}\circ C_{j,k}$ for
distinct $i,j,k$, is impossible. In this case we have
$C(\theta^{i})=\theta^{j}$ and arguments above show that $F=F_{k}$ for some
$k\in\\{i,j\\}$. Also we have $C(\theta^{j})=\theta^{k}$, so the same
arguments show $k\in\\{j,k\\}$. Since $i,j,k$ are distinct, we must have
$k=j$, and so $\phi$ has the form:

$\phi=\sigma\circ F_{j}\circ C_{i,j}\circ C_{j,k}.$

We will show that the form above leads to a contradiction, rendering this case
impossible.

We have:

$\phi(\theta^{ij})=\sigma\circ F_{j}\circ C_{i,j}\circ
C_{j,k}(\theta^{ij})=\sigma\circ F_{j}\circ C_{i,j}(\theta^{ik})=\sigma\circ
F_{j}(\theta^{jk})=\sigma(\theta^{k})$

Since $\phi$ and $\sigma$ are compatible, $\sigma(\theta^{ij})$ is
$c_{1}$-adjacent to $\phi(\theta^{ij})=\sigma(\theta^{k})$. Since $\sigma$ is
an embedding, $\sigma(\theta^{ij})$ being $c_{1}$-adjacent to
$\sigma(\theta^{k})$ implies that $\theta^{ij}$ is $c_{1}$-adjacent to
$\theta^{k}$, which is a contradiction. ∎

##  4 Classification of degree-$(q-1)$ digital $q$-cubes

In this section we focus on noninjective cubes. We first define degree of
injectivity or simply degree of a digital $q$-cube. Informally speaking, the
degree of injectivity of $\sigma$ is the maximal dimension of any injective
subcube of $\sigma$. So an injective $q$-cube has degree $q$, but if $\sigma$
is noninjective and some face $A_{i}\sigma$ is injective, then $\sigma$ has
degree $q-1$. If $\sigma$ and all faces $A_{i}\sigma$ and $B_{i}\sigma$ are
noninjective, but some $A_{i}B_{j}\sigma$ is injective, then $\sigma$ has
degree $q-2$, etc.

######  Definition 4.1.

For a digital $q$-cube $\sigma$, we define the _degree of injectivity_ , or
simply _degree_ , of $\sigma$ inductively as follows: if $\sigma$ is
injective, then the degree of $\sigma$ is $q$. If $\sigma$ is noninjective,
then the degree of $\sigma$ is the maximal degree of any face $A_{i}\sigma$ or
$B_{i}\sigma$.

Clearly, a constant digital $q$-cube has degree of injectivity $0$ and an
injective digital $q$-cube has degree $q$, while a noninjective digital
$q$-cube, with at least one face injective, has degree $q-1$.

The goal for this section is the following classification. If $\sigma$ be a
digital $q$-cube with degree $q-1$ and $q\geq 2$, then $\sigma$ has one of the
following types:

  * Type-1:

$\sigma$ has exactly four injective faces:
$A_{i}\sigma,B_{i}\sigma,A_{j}\sigma,B_{j}\sigma$ for some $i\neq j$

  * Type-2:

$\sigma$ has exactly two injective faces, one in each coordinate $i$ and $j$
for some $i\neq j$

  * Type-3:

$\sigma$ has exactly two injective faces, both in the same coordinate $i$

We begin with two lemmas about a noninjective cube in which one face is
injective but has different image from the opposite face.

######  Lemma 4.2.

Let $\sigma$ be a digital $q$-cube with $q\geq 2$, such that some $i$-face is
injective, but the image of this face is not equal to the image of the
opposite face.

Then there is some $j\neq i$ such that $\sigma\circ F=\sigma\circ F\circ
C_{i,j}$, where $F\in\\{\operatorname{id},F_{j}\\}$.

###### Proof.

Our strategy is to first show that $\sigma\circ F=\sigma\circ F\circ C_{i,j}$,
where $F$ is some (possibly trivial) composition of flips, and then to show
that in fact we must have $F=\operatorname{id}$ or $F=F_{j}$.

By replacing $\sigma$ with $\sigma\circ F_{i}$ if necessary, we assume without
loss of generality that $A_{i}\sigma$ is injective, and
$A_{i}\sigma(I^{q-1})\neq B_{i}\sigma(I^{q-1})$. This means that there is some
$x\in I^{q}$ such that $\sigma(x)\in A_{i}\sigma(I^{q-1})$ but
$\sigma(x)\notin B_{i}\sigma(I^{q-1})$. In fact,
$\sigma(x)\notin\sigma(I^{q}-\\{x\\})$, because $A_{i}\sigma$ is injective.

Let $x=\theta^{i_{1}\cdots i_{n}}$ be the minimal element (lexicographically)
with $\sigma(x)\notin\sigma(I^{q}-\\{x\\})$. Note that
$i\notin\\{i_{1},\ldots,i_{n}\\}$. Let $F=F_{i_{1}}\circ\dots\circ F_{i_{n}}$,
and $\overline{\sigma}=\sigma\circ F$. Since
$i\notin\\{i_{1},\ldots,i_{n}\\}$, we have $A_{i}\overline{\sigma}$ injective
and $B_{i}\overline{\sigma}$ noninjective. We also have
$\overline{\sigma}(\mathit{0})=\sigma(x)$, which means that
$\overline{\sigma}(\mathit{0})\notin\overline{\sigma}(I^{q}-\\{\mathit{0}\\})$.

We will show that there is some index $j\neq i$, with
$\overline{\sigma}=\overline{\sigma}\circ C_{i,j}$ to get the desired result.

It will suffice to show that there is some $j$ with

$\overline{\sigma}\left(\theta^{ik_{1}\cdots
k_{n}}\right)=\overline{\sigma}\left(\theta^{jk_{1}\cdots k_{n}}\right),$ (5)

for all indices $k_{1},\dots,k_{n}$ different from $i$ and $j$. We prove (5)
by induction on $n$.

For $n=0$, we must show that
$\overline{\sigma}\left(\theta^{i}\right)=\overline{\sigma}\left(\theta^{j}\right)$
for some $j$. To obtain a contradiction, assume that
$\overline{\sigma}(\theta^{i})\neq\overline{\sigma}(\theta^{j})$ for all
$i,j$. Note that the element $\overline{\sigma}\left(\theta^{i}\right)$ is not
equal to $\overline{\sigma}(\mathit{0})$ because
$\overline{\sigma}(\mathit{0})\notin\overline{\sigma}(I^{q}-\\{\mathit{0}\\})$
and so from injectivity of $A_{i}\overline{\sigma}$,
$\overline{\sigma}|_{S_{0}\cup S_{1}}$ is injective.

Now for any $k\neq i$, the point $\overline{\sigma}(\theta^{ik})$ is a common
neighbor of $\overline{\sigma}(\theta^{i})$ and
$\overline{\sigma}(\theta^{k})$. Since
$\overline{\sigma}(\theta^{i})\neq\overline{\sigma}(\theta^{k})$ and
$\overline{\sigma}(\theta^{ik})\neq\overline{\sigma}(\mathit{0})$, this point
$\overline{\sigma}(\theta^{ik})$ is unique for each $k$. And so
$\overline{\sigma}|_{S_{0}\cup S_{1}\cup S_{2}}$ is injective. But then, by
Lemma 3.7 $\overline{\sigma}$ is injective. This contradicts our assumption
that $\sigma$ has degree $q-1$, because
$\sigma(I^{q})=\overline{\sigma}(I^{q})$.

Therefore there is an index $j$ with
$\overline{\sigma}\left(\theta^{i}\right)=\overline{\sigma}\left(\theta^{j}\right)$,
and the $n=0$ case of our inductive argument for (5) is proved.

For $n=1$, we show that for any index $k$ different from $i$ and $j$,
$\overline{\sigma}(\theta^{ik})=\overline{\sigma}(\theta^{jk})$:
$\overline{\sigma}\left(\theta^{ik}\right)$ is a common neighbor of
$\overline{\sigma}(\theta^{i})=\overline{\sigma}(\theta^{j})$ and
$\overline{\sigma}(\theta^{k})$. By injectivity of $A_{i}\overline{\sigma}$
and Lemma 3.8, the only common neighbors of $\overline{\sigma}(\theta^{j})$
and $\overline{\sigma}(\theta^{k})$ are $\overline{\sigma}(\mathit{0})$ and
$\overline{\sigma}(\theta^{jk})$, which means
$\overline{\sigma}\left(\theta^{ik}\right)=\overline{\sigma}\left(\theta^{jk}\right)$,
because
$\overline{\sigma}(\mathit{0})\notin\overline{\sigma}(I^{q}-\\{\mathit{0}\\})$.

For $n>1$, note that $\overline{\sigma}(\theta^{ik_{1}\cdots k_{n}})$ is a
common neighbor of at least three elements:
$\overline{\sigma}(\theta^{k_{1}\cdots k_{n}})$, and any elements of the form
$\overline{\sigma}(\theta^{ik_{1}\cdots\hat{k_{l}}\cdots k_{n}})$ for
$l\in\\{1,\ldots,n\\}$. By induction
$\overline{\sigma}(\theta^{ik_{1}\cdots\hat{k_{l}}\cdots
k_{n}})=\overline{\sigma}(\theta^{jk_{1}\cdots\hat{k_{l}}\cdots k_{n}})$, and
by Corollary 3.3 and injectivity of $A_{i}\sigma$, the only common neighbor of
$\overline{\sigma}(\theta^{k_{1}\cdots k_{n}})$ and all
$\overline{\sigma}(\theta^{jk_{1}\cdots\hat{k_{l}}\cdots k_{n}})$ for
$l\in\\{1,\ldots,n\\}$ is $\overline{\sigma}(\theta^{jk_{1}\cdots k_{n}})$.
Therefore, $\overline{\sigma}(\theta^{ik_{1}\cdots
k_{n}})=\overline{\sigma}(\theta^{jk_{1}\cdots k_{n}})$, establishing (5).

Thus far, we have shown that $\sigma\circ F=\sigma\circ F\circ C_{i,j}$ where
$F$ is some (possibly trivial) composition of flips.

By Proposition 3.14, flips commute in a composition of flips and so $F$ can be
broken into the composition $F^{\prime}\circ G$, where $F^{\prime}$ is a
composition of flips in dimensions $i$ and $j$ and $G$ is composition of flips
in dimensions other than $i$ and $j$. This means that
$F^{\prime}\in\\{\operatorname{id},F_{i},F_{j},F_{i}\circ F_{j}\\}$.

We have $\sigma\circ F^{\prime}\circ G=\sigma\circ F^{\prime}\circ G\circ
C_{i,j}.$ But $G$ commutes with $C_{i,j}$, by Proposition 3.14, and so we can
write

$\sigma\circ F^{\prime}\circ G=\sigma\circ F^{\prime}\circ C_{i,j}\circ G.$

Since $G$ is a bijection and a self inverse, we may cancel $G$ to obtain

$\sigma\circ F^{\prime}=\sigma\circ F^{\prime}\circ C_{i,j},$

where $F^{\prime}\in\\{\operatorname{id},F_{i},F_{j},F_{i}\circ F_{j}\\}$. To
finish the proof, it remains only to handle the cases where $F^{\prime}=F_{i}$
and $F^{\prime}=F_{i}\circ F_{j}$.

When $F^{\prime}=F_{i}$, we have $\sigma\circ F_{i}=\sigma\circ F_{i}\circ
C_{i,j}$, then composing with $F_{i}\circ F_{j}$ on both sides and using
$F_{i}\circ C_{i,j}=C_{i,j}\circ F_{j}$, we get $\sigma\circ F_{j}=\sigma\circ
F_{j}\circ C_{i,j}$ as desired.

Similarly, when $F^{\prime}=F_{i}\circ F_{j}$, we have $\sigma\circ F_{i}\circ
F_{j}=\sigma\circ F_{i}\circ F_{j}\circ C_{i,j}$, and composition with
$F_{i}\circ F_{j}$ will give $\sigma=\sigma\circ C_{i,j}$ as desired. ∎

######  Lemma 4.3.

Let $\sigma$ be a degree-$(q-1)$ digital $q$-cube with $q\geq 2$ such that
$\sigma\circ F=\sigma\circ F\circ C_{i,j}$ where $i\neq j$ and
$F\in\\{\operatorname{id},F_{j}\\}$.

Then the faces $A_{k}\sigma$ and $B_{k}\sigma$, for index $k\notin\\{i,j\\}$
are noninjective.

###### Proof.

We have either $\sigma=\sigma\circ C_{i,j}$ or $\sigma\circ F_{j}=\sigma\circ
F_{j}\circ C_{i,j}$.

From $\sigma=\sigma\circ C_{i,j}$, we have
$\sigma(\theta^{i})=\sigma(\theta^{j})$ and
$\sigma(\theta^{ik})=\sigma(\theta^{jk}),$ $k\notin\\{i,j\\}$, which show that
the faces $A_{k}\sigma$ and $B_{k}\sigma$ are noninjective .

From $\sigma\circ F_{j}=\sigma\circ F_{j}\circ C_{i,j}$, we easily get
$\sigma(\mathit{0})=\sigma(\theta^{ij})$ and
$\sigma(\theta^{k})=\sigma(\theta^{ijk}),$ $k\notin\\{i,j\\}$, which show that
the faces $A_{k}\sigma$ and $B_{k}\sigma$ are noninjective. ∎

In the case of the lemmas above, we can be very specific about the $i$ and $j$
faces of $\sigma$.

######  Lemma 4.4.

Let $\sigma$ be a degree-$(q-1)$ digital $q$-cube with $q\geq 2$.

If $\sigma=\sigma\circ C_{i,j},i<j$, then:

  1. _(i)_

$A_{i}\sigma=A_{j}\sigma\circ S_{i,j-1}$

  2. _(ii)_

$B_{i}\sigma=B_{j}\sigma\circ S_{i,j-1}$

  3. _(iii)_

$A_{i}B_{j}\sigma=B_{i}A_{j}\sigma$

If $\sigma\circ F_{j}=\sigma\circ F_{j}\circ C_{i,j},i<j$, then:

  1. _(iv)_

$A_{i}\sigma=B_{j}\sigma\circ S_{i,j-1}\circ F_{j-1}$

  2. _(v)_

$B_{i}\sigma=A_{j}\sigma\circ S_{i,j-1}\circ F_{j-1}$

  3. _(vi)_

$A_{i}A_{j}\sigma=B_{i}B_{j}\sigma$

###### Proof.

Items (i) and (ii) easily follow from $\sigma=\sigma\circ C_{i,j}$ and
equations (1) and (2) for faces of $\sigma\circ C_{i,j}$. Item (iii) also
follows from $\sigma=\sigma\circ C_{i,j}$. Similarly, the rest of the items
follow from $\sigma\circ F_{j}=\sigma\circ F_{j}\circ C_{i,j}$. ∎

Now we are ready to prove the classification outlined at the beginning of the
section.

######  Lemma 4.5.

Let $\sigma$ be a degree-$(q-1)$ digital $q$-cube, $q\geq 2$, such that both
$i$-faces are injective but $A_{i}\sigma(I^{q-1})\neq B_{i}\sigma(I^{q-1})$.
Then $\sigma$ is Type-1.

###### Proof.

Our cube $\sigma$ will satisfy the hypotheses of Lemma 4.2, and so we have
$\sigma\circ F=\sigma\circ F\circ C_{i,j}$ for some $j\neq i$ and some
$F\in\\{\operatorname{id},F_{j}\\}$. By Lemma 4.3, all the faces in
coordinates other than $i$ and $j$ are noninjective. So it remains to show
that both $j$-faces are injective.

By Lemma 4.4, the two faces in coordinate $i$ are obtained from the two faces
in coordinate $j$ by composing with an isomorphism. Thus, since both $i$-faces
are injective, we conclude that both $j$-faces are injective. ∎

######  Lemma 4.6.

Let $\sigma$ be a degree-$(q-1)$ digital $q$-cube, $q\geq 2$, such that one
$i$-face is injective and the other is noninjective. Then $\sigma$ is Type-2.

###### Proof.

We follow the same argument as in Lemma 4.5. By Lemma 4.2, and Lemma 4.3, all
faces in coordinates other than $i$ and $j$ are noninjective. So it remains to
show that one $j$-face is injective while the other is noninjective.

By Lemma 4.4, the two faces in coordinate $i$ are obtained from the two faces
in coordinate $j$ by composing with an isomorphism. Thus, since one $i$-face
is injective and the other is not, we conclude that one $j$-face is injective
while the other is not. ∎

######  Lemma 4.7.

Let $\sigma$ be a degree-$(q-1)$ digital $q$-cube, $q\geq 2$, such that both
$i$-faces are injective and each one is a flip of the other. Then $\sigma$ is
a Type-1 digital $q$-cube.

###### Proof.

Let $A_{i}\sigma=B_{i}(\sigma\circ F_{j})$, which means that $A_{i}\sigma$ and
$B_{i}\sigma$ are flips of each other in dimension $j$ or $j-1$, depending on
whether $j<i$ or $i<j$.

First we show that $\sigma=\sigma\circ C_{i,j}$. It can be verified that
$A_{i}\sigma=B_{i}(\sigma\circ F_{j})$ implies $\sigma=\sigma\circ F_{i}\circ
F_{j}$. For $t\in I^{q}$, with $t_{i}=t_{j}$, it is obvious that
$\sigma(t)=\sigma\circ C_{i,j}(t)$, while for $t_{i}\neq t_{j}$, this equation
can be verified by using $\sigma=\sigma\circ F_{i}\circ F_{j}$ and replacing
$1-t_{i}$ with $t_{j}$ and $1-t_{j}$ with $t_{i}$.

To show that $\sigma$ is Type-1, recall that we have assumed that both
$i$-faces are injective. Since $\sigma=\sigma\circ C_{i,j}$, by Lemma 4.4
parts (i) and (ii) we see that $A_{j}\sigma$ and $B_{j}\sigma$ are also
injective. By Lemma 4.3 all other faces are noninjective, and so $\sigma$ is
Type-2. ∎

######  Lemma 4.8.

Let $\sigma$ be a degree-$(q-1)$ digital $q$-cube, $q\geq 2$, such that both
$i$-faces are injective and each one is a rotation of the other. Then $\sigma$
is Type-3.

###### Proof.

It suffices to show that $A_{k}\sigma$ and $B_{k}\sigma$ are noninjective for
every $k\neq i$.

Suppose that $A_{i}\sigma=B_{i}\sigma\circ F_{j}\circ C_{j,m}$, for $j<m<i$.
We easily get $\sigma(\mathit{0})=\sigma(\theta^{ij})$ and
$\sigma(\theta^{m})=\sigma(\theta^{i})$, which means that the faces
$A_{k}\sigma$, $k\neq i$ are noninjective. We also have
$\sigma(\theta^{j})=\sigma(\theta^{ijm})$, and
$\sigma(\theta^{k})=\sigma(\theta^{ijk}),$ $k\notin\\{i,j\\}$, which means
that $B_{k}\sigma$ for $k\neq i$ are all noninjective. ∎

The following theorem shows that a nondegenerate degree-$(q-1)$ digital
$q$-cube belongs to either of the three types 1,2 or 3. This classification
will play an important role in the proof of Theorem 5.5.

######  Theorem 4.9.

Let $\sigma$ be a nondegenerate digital $q$-cube in $(X,c_{1})$ with $q\geq 2$
and degree of injectivity $q-1$. Then $\sigma$ is a Type-1, Type-2 or a Type-3
digital $q$-cube.

###### Proof.

We suppose that for some index $i$, an $i$-face of $\sigma$ is injective. If
the opposite $i$-face is not injective, then $\sigma$ is Type-2 by Lemma 4.5.

It remains to consider the case where $A_{i}\sigma$ and $B_{i}\sigma$ are both
injective. If the images of $A_{i}\sigma$ and $B_{i}\sigma$ are not equal,
then $\sigma$ is Type-1 by Lemma 4.6.

Now we consider the case where $A_{i}\sigma$ and $B_{i}\sigma$ are both
injective and $A_{i}\sigma(I^{q-1})=B_{i}\sigma(I^{q-1})$. In this case,
$A_{i}\sigma$ and $B_{i}\sigma$ are compatible digital $(q-1)$-cubes with
equal images, and so by Lemma 3.18, $A_{i}\sigma$ is either flip or rotation
of $B_{i}\sigma$, because $A_{i}\sigma=B_{i}\sigma$ means that $\sigma$ is
degenerate which is not true.

If $A_{i}\sigma$ is a flip of $B_{i}\sigma$, then by Lemma 4.7, $\sigma$ is a
Type-1 digital $q$-cube, while if $A_{i}\sigma$ is a rotation of $B_{i}\sigma$
then by Lemma 4.8, $\sigma$ is a Type-3 digital $q$-cube. ∎

##  5 A chain map from $dC_{q}(X)$ to $C^{c_{1}}_{q}(X)$

Our goal for this section is to define a chain map $\beta_{q}:dC_{q}(X)\to
C_{q}^{c_{1}}(X)$. We will define $\beta_{q}(\sigma)$ to be zero when $\sigma$
is not injective. When $\sigma$ is injective (and thus an embedding),
$\beta_{q}(\sigma)$ will be defined in terms of the orientation with which
$\sigma$ maps $I^{q}$ into $X$.

We begin with a detailed discussion of orientations. If $\sigma:I^{q}\to X$ is
an embedding, then $\sigma(I^{q})\subset X$ is an elementary $q$-cube in $X$,
and so we can express $\sigma$ as an affine linear map:

$\sigma(x)=\sigma(0)+M^{\sigma}x$

where $x\in I^{q}\subset\mathbb{Z}^{n}$ and $M^{\sigma}$ is a $n\times q$
matrix with entries in $\\{-1,0,1\\}$. Since $\sigma(I^{q})$ is an elementary
$q$-cube, the matrix $M^{\sigma}$ will have $n-q$ rows of zeros corresponding
to the coordinates where the elementary cube $\sigma(I^{q})$ is degenerate.

For each $i$, we have
$\sigma(\theta^{i})=\sigma(0)+o^{\sigma}_{i}\theta^{k_{i}}$ for some
$k_{i}\in\\{1,\dots n\\}$ and some sign $o^{\sigma}_{i}\in\\{-1,+1\\}$ which
represents the orientation with which $\sigma$ carries the edge from $0$ to
$\theta^{i}$ onto its image. Thus column $i$ of $M^{\sigma}$ consists of
$o^{\sigma}_{i}$ in row $k_{i}$, and zero everywhere else. The set of nonzero
rows of $M^{\sigma}$ is exactly the set of these $k_{i}$ for various $i$. Let
$\overline{M}^{\sigma}$ be the $q\times q$ matrix obtained by removing the
rows of zeros from $M^{\sigma}$.

######  Definition 5.1.

Let $\sigma$ be an injective digital cube. Then we define the _orientation of
$\sigma$_ as $o^{\sigma}=\det(\overline{M}^{\sigma})$.

Now we consider the relationship between the orientation of $\sigma$ and its
faces.

The matrix $M^{A_{i}\sigma}$ will be the matrix $M^{\sigma}$ with column $i$
deleted, and thus $\overline{M}^{A_{i}\sigma}$ is $\overline{M}^{\sigma}$ with
column $i$ deleted. Let $j_{i}$ be the row of $\overline{M}^{\sigma}$ which
corresponds to the nonzero row $k_{i}$ of $M^{\sigma}$. Then column $i$ of
$\overline{M}^{\sigma}$ consists of $o^{\sigma}_{i}$ in row $j_{i}$, and zero
everywhere else. Thus, computing $\det(\overline{M}^{\sigma})$ by cofactor
expansion along column $i$ gives:

$\det(\overline{M}^{\sigma})=(-1)^{i+k_{i}}o^{\sigma}_{i}\det(\overline{M}^{A_{i}\sigma})$

Thus we have

$o^{A_{i}\sigma}=(-1)^{i+k_{i}}o^{\sigma}_{i}o^{\sigma}.$ (6)

Since $B_{i}\sigma$ is an affine translation of $A_{i}\sigma$, we have
$M^{A_{i}\sigma}=M^{B_{i}\sigma}$ and thus $o^{B_{i}\sigma}=o^{A_{i}\sigma}$.

These orientations $o^{\sigma}_{i}$ are also well-behaved with respect to the
elementary face operators $A^{c_{1}}_{i}$ and $B^{c_{1}}_{i}$. The specific
orientation $o^{\sigma}_{i}$ tells us whether or not the edge from $0$ to
$\theta^{i}$ is preserved or reversed in orientation. In particular, if
$o^{\sigma}_{i}=1$, then $\sigma$ maps the front $i$-face of $I^{q}$ to the
front $k_{i}$-face of $\sigma(I^{q})$, and if $o^{\sigma}_{i}=-1$ then
$\sigma$ maps the front $i$-face of $I^{q}$ to the back $k_{i}$-face of
$\sigma(I^{q})$. In other words, we have:

$A_{i}\sigma(I^{q-1})=\begin{cases}A^{c_{1}}_{k_{i}}(\sigma(I^{q}))&\text{ if
}o^{\sigma}_{i}=1,\\\ B^{c_{1}}_{k_{i}}(\sigma(I^{q}))&\text{ if
}o^{\sigma}_{i}=-1,\end{cases}$

and similarly for $B^{c_{1}}_{i}\sigma(I^{q-1})$. Thus we will have:

$o_{i}^{\sigma}(A_{i}\sigma(I^{q-1})-B_{i}\sigma(I^{q-1}))=A^{c_{1}}_{k_{i}}(\sigma(I^{q}))-B^{c_{1}}_{k_{i}}(\sigma(I^{q})).$
(7)

######  Definition 5.2.

For each $q$, we define a homomorphism $\beta_{q}:dC_{q}(X)\to
C_{q}^{c_{1}}(X)$ on generators as follows: Let $\sigma$ be a digital
$q$-cube, and we define

$\beta_{q}(\sigma)=\begin{cases}0&\text{ if $\sigma$ is noninjective},\\\
o^{\sigma}\cdot\sigma(I^{q})&\text{ if $\sigma$ is injective}.\end{cases}$

The formulation of orientations in terms of determinants lets us easily prove
some statements about the cubical operators of Section 3.3.

######  Theorem 5.3.

For a digital $q$-cube $\sigma$ with $q\geq 1$, and any $i,j$, we have

$\beta_{q}(\sigma\circ F_{j})=\beta_{q}(\sigma\circ
C_{i,j})=-\beta_{q}(\sigma).$

###### Proof.

If $\sigma$ is noninjective, then $\sigma\circ F_{j}$ and $\sigma\circ
C_{i,j}$ are also noninjective, and so we have $\beta_{q}(\sigma\circ
F_{j})=\beta_{q}(\sigma\circ C_{i,j})=-\beta_{q}(\sigma)=0$.

Now assume that $\sigma$ is injective. Since $(\sigma\circ
F_{j})(I^{q})=(\sigma\circ C_{i,j})(I^{q})=\sigma(I^{q})$, it suffices to show
that $o^{\sigma\circ F_{j}}=o^{\sigma\circ C_{i,j}}=-o^{\sigma}$.

For $\sigma\circ F_{j}$, observe that we will have:

$o^{\sigma\circ F_{j}}_{i}=\begin{cases}o^{\sigma}_{i}&\text{ if }j\neq i\\\
-o^{\sigma}_{i}&\text{ if }j=i\end{cases}$

Thus the matrix $\overline{M}^{\sigma\circ F_{j}}$ is the same as the matrix
$\overline{M}^{\sigma}$, but with column $i$ negated. These matrices have
opposite determinants, so $o^{\sigma\circ F_{j}}=-o^{\sigma}$ as desired.

For $\sigma\circ C_{i,j}$ we note that the matrix $\overline{M}^{\sigma\circ
C_{i,j}}$ is the same as the matrix $\overline{M}^{\sigma}$, but with columns
$i$ and $j$ interchanged. Again these matrices have opposite determinants, so
$o^{\sigma\circ C_{i,j}}=-o^{\sigma}$ as desired. ∎

For the other operators, the following result easily follows using Remark
3.13.

######  Corollary 5.4.

For any digital $q$-cube $\sigma$ and $i<j$,

$\beta(\sigma\circ R_{i,j})=\beta\sigma.$

Furthermore,

$\beta(\sigma\circ S_{i,j})=(-1)^{j-i}\beta\sigma.$

Now we are ready for our main result of the section.

######  Theorem 5.5.

$\beta_{q}:C_{q}(X)\to C_{q}^{c_{1}}(X)$ is a chain map.

###### Proof.

For any digital $q$-cube $\sigma$, we need to show that
$\beta\partial\sigma=\partial^{c_{1}}\beta\sigma$.

First we consider the case when $\sigma$ is injective. Then we use (7) and (6)
to obtain:

$\displaystyle\beta(\partial\sigma)$
$\displaystyle=\beta\left(\sum_{i}(-1)^{i}(A_{i}\sigma-B_{i}\sigma)\right)$
$\displaystyle=\sum_{i}(-1)^{i}(o^{A_{i}\sigma}(A_{i}\sigma(I^{q-1}))-o^{B_{i}\sigma}(B_{i}\sigma(I^{q-1})))$
$\displaystyle=\sum_{i}(-1)^{i}(-1)^{i+k_{i}}o_{i}^{\sigma}o^{\sigma}(A_{i}\sigma(I^{q-1})-B_{i}\sigma(I^{q-1}))$
$\displaystyle=o^{\sigma}\sum_{i}(-1)^{k_{i}}(A^{c_{1}}_{k_{i}}\sigma(I^{q})-B^{c_{1}}_{k_{i}}\sigma(I^{q}))$
$\displaystyle=o^{\sigma}\partial^{c_{1}}(\sigma(I^{q}))=\partial^{c_{1}}(o^{\sigma}\sigma(I^{q}))=\partial^{c_{1}}(\beta\sigma).$

If $\sigma$ is noninjective, and all faces of $\sigma$ are also noninjective,
then we have $\beta\partial\sigma=0=\partial^{c_{1}}\beta\sigma$.

It remains to consider the case when $\sigma$ is noninjective and at least one
face of $\sigma$ is injective. In this case we automatically have
$\beta\sigma=0$, and thus $\partial^{c_{1}}(\beta\sigma)=0$, and so we must
show that $\beta(\partial\sigma)=0$.

Such a digital $q$-cube is either degenerate or belongs to any of the types:
Types 1,2 and 3, from Theorem 4.9. We consider these types below.

If $\sigma$ is type-1 or type-2, then the faces $A_{k}\sigma$ and
$B_{k}\sigma$, for $k\notin\\{i,j\\}$, are all noninjective, and the $i$-faces
and $j$-faces are related through equations given in Lemma 4.4. If $\sigma$ is
type-2, with $A_{i}\sigma$ injective and $\sigma=\sigma\circ C_{i,j}$, then

$\displaystyle\beta\partial\sigma$
$\displaystyle=\beta\left((-1)^{i}A_{i}\sigma+(-1)^{j}A_{j}\sigma\right)$
$\displaystyle=\begin{cases}(-1)^{i}\beta
A_{i}\sigma+(-1)^{j}\beta\left(A_{i}\sigma\circ S_{i,j-1}\right),&i<j\\\
(-1)^{i}\beta A_{i}\sigma+(-1)^{j}\beta\left(A_{i}\sigma\circ
S_{j,i-1}\right),&i>j\end{cases}$ $\displaystyle=\begin{cases}(-1)^{i}\beta
A_{i}\sigma+(-1)^{j+j-1-i}\beta A_{i}\sigma,&i<j\\\ (-1)^{i}\beta
A_{i}\sigma+(-1)^{j+i-1-j}\beta A_{i}\sigma,&i>j\end{cases}~{}~{}=~{}~{}0$

Similar calculations can be carried out for the cases when $B_{i}\sigma$ is
injective, or $\sigma\circ F_{j}=\sigma\circ F_{j}\circ C_{i,j}$ or when
$\sigma$ is a type-1 digital $q$-cube, to show $\beta\partial\sigma=0$.

Whereas for a Type-3 digital $q$-cube, we get $\beta\partial\sigma=0$ from
equation $A_{i}\sigma=B_{i}\sigma\circ R_{j,m}$ and Corollary 5.4, because all
the faces of $\sigma$ in coordinates other than $i$ are noninjective. ∎

For a $(c_{1},c_{1})$-continuous map $f:X\to Y$ between digital images
$(X,c_{1})$ and $(Y,c_{1})$, the existence of a chain map
$f_{\\#}^{c_{1}}:C_{q}^{c_{1}}(X)\to C_{q}^{c_{1}}(Y)$ was conjectured in
[11]. This conjecture was proven in low dimensions using computer
enumerations, but we provide a complete analytical proof below.

For an elementary $q$-cube $Q\subset X$ and some continuous function $f:X\to
Y$, if $f$ is injective on $Q$ then we can define an orientation $o^{f,Q}$ in
the same was as we have defined $o^{\sigma}$ for a digital $q$-cube $\sigma$:
we let $o^{f,Q}$ be the determinant of the affine linear map with which $f$
maps $Q$ onto $f(Q)$. Then for any elementary $q$-cube $Q\subset X$, define
$f_{q}^{c_{1}}(Q)$ as follows:

$f^{c_{1}}_{q}(Q)=\begin{cases}0&\text{ if $f$ is noninjective on $Q$,}\\\
o^{f,Q}f(Q)&\text{ if $f$ is injective on $Q$.}\end{cases}$

######  Theorem 5.6.

Let $f:X\to Y$ be a $(c_{1},c_{1})$-continuous function between digital images
$(X,c_{1})$ and $(Y,c_{1})$. Then $f^{c_{1}}_{q}:C_{q}^{c_{1}}(X)\to
C_{q}^{c_{1}}(Y)$ is a chain map, satisfying $(f\circ
g)^{c_{1}}_{q}=f^{c_{1}}_{q}\circ g^{c_{1}}_{q}$ and $(1_{X})_{q}^{c_{1}}$ is
the identity homomorphism, where $1_{X}$ denotes the identity function on $X$.

###### Proof.

Defining $f^{c_{1}}_{q}$ as above will clearly satisfy the desired properties
concerning compositions and identity, so we need only show that it is a chain
map. Let $Q\subset X$ be an elementary $q$-cube, and we will show that
$\partial^{c_{1}}_{q}(f_{q}^{c_{1}}(Q))=f_{q-1}^{c_{1}}(\partial^{c_{1}}_{q}(Q))$.

The elementary $q$-cube $Q$ is naturally isomorphic to $I^{q}$ by an
isomorphism $\iota=\tau\circ\Theta:I^{q}\to Q$, where $\tau$ is a translation
moving the origin to $\min(Q)$ (the lexicographical minimum point), and
$\Theta$ is a map satisfying $\Theta(\theta^{i})=\theta^{k_{i}}$ for some
increasing sequence $k_{1},\dots,k_{q}$ indicating the nondegenerate
coordinates of $Q$. Let $\sigma:I^{q}\to Y$ be defined by
$\sigma=f\circ\iota$, and $\sigma$ will be a digital $q$-cube in $Y$
satisfying $\beta_{q}(\sigma)=f^{c_{1}}_{q}(Q)$ and also
$\beta_{q-1}(A_{i}\sigma)=f_{q-1}^{c_{1}}(A^{c_{1}}_{k_{i}}Q)$ and
$\beta_{q-1}(B_{i}\sigma)=f_{q-1}^{c_{1}}(B^{c_{1}}_{k_{i}}Q)$. The last two
equalities mean that
$\beta_{q-1}(\partial_{q}\sigma)=f_{q-1}^{c_{1}}(\partial^{c_{1}}_{q}(Q))$.
Then since $\beta$ is a chain map, we have:

$\partial^{c_{1}}_{q}(f_{q}^{c_{1}}(Q))=\partial_{q}^{c_{1}}(\beta_{q}(\sigma))=\beta_{q-1}(\partial_{q}(\sigma))=f_{q-1}^{c_{1}}(\partial^{c_{1}}_{q}(Q)$

as desired. ∎

Using Theorem 7.2 of [11] and the theorem above, we get the following result.

######  Corollary 5.7.

Let $(X,c_{1})$ and $(Y,c_{1})$ be digital images of the same homotopy type.
Then $H_{q}^{c_{1}}(X)\cong H_{q}^{c_{1}}(Y)$.

##  6 $dH_{q}(X)$ is isomorphic to $H^{c_{1}}_{q}(X)$

From the previous section, $\beta_{q}:dC_{q}(X)\to C^{c_{1}}_{q}(X)$ is a
chain map, and it is clear that $\beta_{q}$ is surjective. Denote the quotient
chain complex $dC_{q}(X)/\ker\beta_{q}$ as $\overline{C}_{q}(X)$, and its
homology as $\overline{H}_{q}(X)$. Then $\beta_{q}$ gives an isomorphism
$\overline{C}_{q}(X)\cong C_{q}^{c_{1}}(X)$ and thus $\overline{H}_{q}(X)\cong
H_{q}^{c_{1}}(X)$.

The goal of this section is to show that $dH_{q}(X)\cong H_{q}^{c_{1}}(X)$,
and by the above paragraph it will suffice to show that
$dH_{q}(X)\cong\overline{H}_{q}(X)$ for each $q$.

We define below a filtration on chain complexes $dC_{q}(X)$ and
$\overline{C}_{q}(X)$ based on the degree of injectivity of digital $q$-cubes.

For integers $k\geq 0$ and a fixed $q$, let $dC_{q}^{k}(X)$ be a free abelian
group generated by nondegenerate digital $q$-cubes with degree of injectivity
less than or equal to $k$. Clearly, $dC_{q}^{k-1}(X)\subset dC_{q}^{k}(X)$ for
$k\geq 1$, and $\partial(dC_{q}^{k}(X))\subset dC_{q-1}^{k}(X)$ for for $k\geq
0$. Therefore, we have a filtration of chain complexes

$dC_{q}^{k-1}(X)\subset dC_{q}^{k}(X)\subset dC_{q}(X),$

for $k\geq 0$. Note that for $k\geq q$, we have
$dC_{q}^{k}(X)=dC_{q}^{k+1}(X)=dC_{q}(X)$.

Letting $dC_{q}^{k,k-1}(X)=dC_{q}^{k}(X)/dC_{q}^{k-1}(X)$, we have the
following short exact sequence of chain complexes:

$\displaystyle 0\to dC_{q}^{k-1}(X)\to dC_{q}^{k}(X)\to dC^{k,k-1}_{q}(X)\to
0.$ (8)

Let $H_{q}^{k}(X)$ be the homology of $dC_{q}^{k}(X)$, and $H_{q}^{k,k-1}(X)$
be the homology of $dC^{k,k-1}_{q}(X)$, and we have the following long exact
sequence of homology groups:

$\displaystyle\dots\to H_{q+1}^{k,k-1}(X)\to H_{q}^{k-1}(X)\to H_{q}^{k}(X)\to
H_{q-1}^{k-1}(X)\to\dots$

We can also make similar constructions starting with the chain complex
$\overline{C}_{q}(X)$. We have a filtration
$\overline{C}^{k-1}_{q}(X)\subset\overline{C}^{k}_{q}(X)\subset\overline{C}_{q}(X),k\geq
0$, where for some $q\geq 0$, $\overline{C}^{k}_{q}(X)$ is the free abelian
group generated by nondegenerate digital $q$-cubes of $\overline{C}_{q}(X)$
with degree of injectivity less than or equal to $k$, modulo the kernel of
$\beta_{q}$.

Note that, since every generator of $\overline{C}_{q}(X)$ is injective,
$\overline{C}^{k}_{q}(X)=\overline{C}_{q}(X)$ if $k\geq q$ and
$\overline{C}_{q}^{k}(X)$ is trivial for $k<q$.

Copying notations as above, let
$\overline{C}^{k,k-1}_{q}(X)=\overline{C}_{q}^{k}(X)/\overline{C}_{q}^{k-1}(X)$,
and we have the short exact sequence

$\displaystyle
0\to\overline{C}_{q}^{k-1}(X)\to\overline{C}_{q}^{k}(X)\to\overline{C}^{k,k-1}_{q}(X)\to
0,$ (9)

and taking homology groups gives the long exact sequence

$\displaystyle\dots\to\overline{H}_{q+1}^{k,k-1}(X)\to\overline{H}_{q}^{k-1}(X)\to\overline{H}_{q}^{k}(X)\to\overline{H}_{q-1}^{k,k-1}(X)\to\dots$

Recall that our goal for the section is to show that $dH_{q}(X)\cong
H_{q}^{c_{1}}(X)$ for all $q$. We will use the classification of injective
cubes according to flips and swaps. The following lemmas describe the
relationships between a digital $q$-cube $\sigma$ and its compositions with
flip and swap maps in the quotient group $dC_{q}^{q,q-1}(X)$.

######  Lemma 6.1.

Let $\sigma$ be a digital $q$-cube in $dC_{q}^{q,q-1}(X)$. Then
$\sigma+\sigma\circ F_{i}$ is a boundary in $dC_{q}^{q,q-1}(X)$.

###### Proof.

Let $\rho\in dC_{q+1}^{q,q-1}(X)$ be the digital cube given by:

$\rho(t_{1},\dots,t_{q+1})=\begin{cases}\sigma(t_{1},\dots,t_{i-1},t_{i+1},t_{i+2},\dots,t_{q+1})&\text{
if }t_{i}=0\\\ \sigma(t_{1},\dots,t_{i-1},0,t_{i+2},\dots,t_{q+1})&\text{ if
}t_{i}=1\end{cases}$

We will show that $(-1)^{i}\partial\rho=\sigma+\sigma\circ F_{i}\in
dC_{q}^{q,q-1}(X)$, which will complete the proof. We must compute the various
faces of $\rho$.

First we compute $A_{i}\rho$ and $B_{i}\rho$:

$\displaystyle A_{i}\rho(t_{1},\dots,t_{q})$
$\displaystyle=\rho(t_{1},\dots,t_{i-1},0,t_{i},\dots,t_{q})=\sigma(t_{1},\dots,t_{i-1},t_{i},\dots,t_{q})$
$\displaystyle B_{i}\rho(t_{1},\dots,t_{q})$
$\displaystyle=\rho(t_{1},\dots,t_{i-1},1,t_{i},\dots,t_{q})=\sigma(t_{1},\dots,t_{i-1},0,\dots,t_{q})$

Here we see that $A_{i}\rho=\sigma$, and $B_{i}\rho$ is degenerate since it
does not depend on coordinate $t_{i}$.

Next we compute $A_{i+1}\rho$ and $B_{i+1}\rho$:

$\displaystyle A_{i+1}\rho(t_{1},\dots,t_{q})$
$\displaystyle=\rho(t_{1},\dots,t_{i},0,t_{i+1},\dots,t_{q})=\sigma(t_{1},\dots,t_{i-1},0,t_{i+2},\dots,t_{q+1})$
$\displaystyle B_{i+1}\rho(t_{1},\dots,t_{q})$
$\displaystyle=\rho(t_{1},\dots,t_{i},1,t_{i+1},\dots,t_{q})$
$\displaystyle=\begin{cases}\sigma(t_{1},\dots,t_{i-1},1,t_{i+1},t_{i+2},\dots,t_{q})&\text{
if }t_{i}=0\\\ \sigma(t_{1},\dots,t_{i-1},0,t_{i+1},\dots,t_{q})&\text{ if
}t_{i}=1\end{cases}$ $\displaystyle=\sigma\circ F_{i}(t_{1},\dots,t_{q})$

Thus $A_{i+1}\rho$ is degenerate, and $B_{i+1}\rho=\sigma\circ F_{i}$.

Now we compute $A_{j}\rho$ and $B_{j}\rho$ for $j\not\in\\{i,i+1\\}$. For
simplicity in the notation below, we will assume that $j<i$:

$\displaystyle A_{j}\rho(t_{1},\dots,t_{q})$
$\displaystyle=\rho(t_{1},\dots,0,t_{j},\dots,t_{q})$
$\displaystyle=\begin{cases}\sigma(t_{1},\dots,0,t_{j},\dots,t_{i-1},t_{i+1},t_{i+2},\dots,t_{q})&\text{
if }t_{i}=0\\\
\sigma(t_{1},\dots,0,t_{j},\dots,t_{i-1},0,t_{i+2},\dots,t_{q})&\text{ if
}t_{i}=1\end{cases}$

From the two cases above, we see that $A_{j}\rho$ can take on at most
$2^{q-1}+2^{q-2}<2^{q}$ values, so $A_{j}\rho$ is not injective, and thus
$A_{j}\rho=0\in dC_{q}^{q,q-1}(X)$. A similar computation shows that
$B_{j}\rho=0\in dC_{q}^{q,q-1}(X)$.

Summing the faces of $\rho$ in $dC_{q}^{q,q-1}(X)$ gives only the two nonzero
terms $A_{i}\rho=\sigma$ and $B_{i+1}\rho=\sigma\circ F_{i}$, and we have

$\partial\rho=\sum_{j}(-1)^{j}(A_{j}\rho-
B_{j}\rho)=(-1)^{i}\sigma-(-1)^{i+1}\sigma\circ
F_{i}=(-1)^{i}(\sigma+\sigma\circ F_{i})$

as desired. ∎

######  Lemma 6.2.

Let $\sigma$ be a digital $q$-cube in $dC_{q}^{q,q-1}(X)$. Then
$\sigma-\sigma\circ R_{i,j}$ is a boundary in $dC_{q}^{q,q-1}(X)$.

###### Proof.

Let $\rho=\sigma\boxminus(\sigma\circ R_{i,j})$. Then by (3) it suffices to
show that $A_{k}\sigma\boxminus A_{k}(\sigma\circ R_{i,j})$ and
$B_{k}\sigma\boxminus B_{k}(\sigma\circ R_{i,j})$ are zero in
$dC_{q}^{q,q-1}(X)$. We will demonstrate this for $A_{i}$, the argument for
$B_{i}$ is similar.

Observe that $A_{k}\sigma(I^{q-1})$ and $A_{k}(\sigma\circ R_{i,j})(I^{q-1})$
have nonempty intersection, since $A_{k}\sigma(I^{q-1})$ is a face of the
elementary cube $\sigma(I^{q})$, and $A_{k}(\sigma\circ R_{i,j})(I^{q-1})$ is
the same face of a quarter-turn rotation of that cube. Thus

$\displaystyle\\#(A_{k}\sigma\boxminus A_{k}(\sigma\circ R_{i,j}))(I^{q})$
$\displaystyle\leq\\#(A_{k}\sigma(I^{q-1})\cup A_{k}(\sigma\circ
R_{i,j})(I^{q-1}))$
$\displaystyle<\\#A_{k}\sigma(I^{q-1})+\\#A_{k}(\sigma\circ R_{i,j})(I^{q-1})$
$\displaystyle\leq 2^{q-1}+2^{q-1}=2^{q}.$

Thus the digital $q$-cube $A_{k}\sigma\boxminus A_{k}(\sigma\circ R_{i,j})$ is
not injective, and so is zero in $dC_{q}^{q,q-1}(X)$. ∎

######  Lemma 6.3.

Let $\sigma$ be a digital $q$-cube in $dC_{q}^{q,q-1}(X)$. Then
$\sigma+\sigma\circ C_{i,j}$ is a boundary in $dC_{q}^{q,q-1}(X)$.

###### Proof.

We must show that $\sigma+\sigma\circ C_{i,j}\in B_{q}^{q,q-1}(X)$. Recall
from Proposition 3.13 that $C_{i,j}=R_{i,j}\circ F_{j}$. We have:

$\sigma+\sigma\circ C_{i,j}=(\sigma-\sigma\circ R_{i,j})+(\sigma\circ
R_{i,j}+(\sigma\circ R_{i,j})\circ F_{j}),$

and the above is a boundary in $dC_{q}^{q,q-1}(X)$ by Lemmas 6.2 and 6.1. ∎

######  Corollary 6.4.

Let $\beta_{q}^{\prime}:dC^{q,q-1}_{q}(X)\to\overline{C}^{q,q-1}_{q}(X)$ be
the map induced by $\beta$. Then the kernel of $\beta^{\prime}_{q}$ is
contained in the boundary subgroup of $dC^{q,q-1}_{q}(X)$.

###### Proof.

The kernel of $\beta_{q}^{\prime}$ is generated by chains of the form
$\sigma+\phi$ where $\sigma,\phi$ are injective with
$\sigma(I^{q})=\phi(I^{q})$ and $o^{\sigma}=-o^{\phi}$, together with chains
of the form $\sigma-\phi$ where $\sigma(I^{q})=\phi(I^{q})$ and
$o^{\sigma}=o^{\phi}$. Equivalently, the kernel of $\beta^{\prime}_{q}$ is
generated by chains of the form $o^{\sigma}\sigma-o^{\phi}\phi$, and so it
suffices to show that any such chain is a boundary in $C_{q}^{q,q-1}(X)$.

By Lemma 3.17 we have $\phi=\sigma\circ G$, where $G$ is a composition of
flips and swaps. From the proof of Theorem 5.3, each flip or swap reverses the
orientation, and so $o^{\phi}=(-1)^{k}o^{\sigma}$

Let:

$\phi=\sigma\circ G_{1}\circ\dots\circ G_{k},$

where each $G$ is a single flip or swap, and let $\phi_{i}=\sigma\circ
G_{1}\circ\dots\circ G_{i}$ for each $i\leq k$, and $\phi_{0}=\sigma$. Then we
have:

$\displaystyle\sum_{i=1}^{k}(-1)^{i-1}(\phi_{i-1}+\phi_{i})$
$\displaystyle=\sigma+(-1)^{k-1}\phi=\sigma-(-1)^{k}\phi$
$\displaystyle=\sigma-o^{\phi}o^{\sigma}\phi=o^{\sigma}(o^{\sigma}\sigma-o^{\phi}\phi)$

and so by Lemma 6.3 we see that $o^{\sigma}\sigma-o^{\phi}\phi$ is a sum of
boundary elements as desired. ∎

Now we are ready to prove one of the main results of the section. The next
theorem shows that the desired isomorphism holds for $q<\dim X$, where $\dim
X$ is the largest integer $k$ such that $X$ contains an elementary cube of
dimension $k$.

######  Theorem 6.5.

For a digital image $(X,c_{1})$, $dH_{q}(X)\cong H_{q}^{c_{1}}(X),$ for
$q<\dim X$.

###### Proof.

By the arguments at the beginning of the section, it suffices to show that
$dH_{q}(X)\cong\overline{H}_{q}(X)$.

Every digital $q$-cube in $(X,c_{1})$ has degree of injectivity less than or
equal to $k$, and thus $dH_{q}^{k}(X)=dH_{q}(X)$ and
$\overline{H}_{q}^{k}(X)=\overline{H}_{q}(X)$ for all $q$. Therefore, we need
to show that the middle vertical arrow of the diagram below is an isomorphism,
for $q\leq k$, where the vertical arrows are induced by the canonical map
$dC_{q}(X)\to\overline{C}_{q}(X)$.

${\cdots}$${dH_{q+1}^{k,k-1}(X)}$${dH_{q}^{k-1}(X)}$${dH_{q}^{k}(X)}$${dH_{q}^{k,k-1}(X)}$${dH_{q-1}^{k-1}(X)}$${\cdots}$${\cdots}$${\overline{H}_{q+1}^{k,k-1}(X)}$${\overline{H}_{q}^{k-1}(X)}$${\overline{H}_{q}^{k}(X)}$${\overline{H}_{q}^{k,k-1}(X)}$${\overline{H}_{q-1}^{k-1}(X)}$${\cdots}$

For this we use the Five Lemma and induction on $k$ and $q$. If $k=0$, then
$X$ consists of isolated points only, and so by Propositions 2.1 and 2.4,
$dH_{q}(X)=H^{c_{1}}_{q}(X)=\overline{H}_{q}(X)$. If $q=0$, then by
Propositions 2.2 and 2.5, $dH_{0}(X)=H_{0}^{c_{1}}(X)=\overline{H}_{0}(X)$.

In order to show that middle vertical arrow is an isomorphism, by the Five
Lemma we need to show that the second and fourth vertical arrows are
isomorphisms, the first is injective, and the fifth is surjective. By
induction on $k$ and $q$, the second and fifth vertical arrows are
isomorphisms.

For the fourth arrow, if $q<k$, we see that $dC_{q}^{k,k-1}(X)$ is trivial
because it is a quotient of $dC_{q}^{k}(X)$ and $dC_{q}^{k-1}(X)$, which are
equal. For the same reason $\overline{d}C_{q}^{k,k-1}(X)$ is trivial Therefore
$H_{q}^{k,k-1}(X)$ and $\overline{H}_{q}^{k,k-1}(X)$ are also trivial, so
$H_{q}^{k,k-1}(X)\to\overline{H}_{q}^{k,k-1}(X)$ is an isomorphism.

If $q=k$, then we need to show
$H_{q}^{q,q-1}(X)\to\overline{H}_{q}^{q,q-1}(X)$ is an isomorphism. It
suffices to show that the kernel of the induced map
$\beta^{\prime}_{q}:dC_{q}^{q,q-1}(X)\to\overline{C}_{q}^{q,q-1}(X)$ is
contained in the boundary subgroup of $dC_{q}^{q,q-1}(X)$, and this was
demonstrated in Corollary 6.4.

∎

######  Theorem 6.6.

For a digital image $(X,c_{1})$, $dH_{q}(X)\cong H_{q}^{c_{1}}(X),$ for all
$q\geq\dim X$.

###### Proof.

Let $q\geq\dim X$, and let $Y=X\times[0,1]^{n}_{\mathbb{Z}}$, where
$n=q+1-\dim X$. Thus $\dim Y=\dim X+n=q+1$. Since $q<\dim Y$, we have
$dH_{q}(Y)\cong H_{q}^{c_{1}}(Y)$ by Theorem 6.5.

Now we show that the digital images $(X,c_{1})$ and $(Y,c_{1})$ have the same
homotopy type. The digital image $[0,1]^{n}_{\mathbb{Z}}$ is
$c_{1}$-contractible as shown in [5, Example 5.4]. Let
$h:[0,1]^{n}_{\mathbb{Z}}\times[0,k]_{\mathbb{Z}}\to[0,1]^{n}_{\mathbb{Z}}$ be
the contraction of $[0,1]^{n}_{\mathbb{Z}}$ to $\\{\mathit{0}\\}$, where
$\mathit{0}\in\mathbb{Z}^{n}$ is the tuple of all zeros. Then the function
$H:Y\times[0,k]_{\mathbb{Z}}\to Y$ defined by $H(x,s,t)=(x,h(s,t))$ for $x\in
X,s\in[0,1]^{n}_{\mathbb{Z}}$ and $t\in[0,k]_{\mathbb{Z}}$ is a deformation
retraction of $Y$ onto $X\times\\{\mathit{0}\\}$. Thus $(Y,c_{1})$ is homotopy
equivalent to $(X\times\\{\mathit{0}\\},c_{1})$, which is clearly isomorphic
to $(X,c_{1})$.

Proposition 2.3 and Corollary 5.7 show that both $dH_{q}$ and $H^{c_{1}}_{q}$
are homotopy-type invariants. Thus since $X$ and $Y$ have the same homotopy
type and $dH_{q}(Y)\cong H^{c_{1}}_{q}(Y)$, we have $dH_{q}(X)\cong
H_{q}^{c_{1}}(X)$ as desired. ∎

Combining the two theorems above gives our main result:

######  Theorem 6.7.

For a digital image $(X,c_{1})$, we have $dH_{q}(X)\cong H_{q}^{c_{1}}(X)$ for
all $q$.

As mentioned earlier, the size of the chain groups of the homology $dH_{q}$
grows superexponentially with $q$, but the theorem above significantly reduces
this complexity for $c_{1}$-digital images. Furthermore, it is difficult to
verify through direct computations that homology $dH_{q}$ vanishes for $q>\dim
X$, for $c_{1}$-digital images with finite dimension. However this is a direct
consequence of the theorem above.

######  Corollary 6.8.

For a digital image $(X,c_{1})$ with $\dim X=n$, we have $dH_{q}(X)\cong 0$
for all $q>n$.

In [9], it was shown that an excision-like property holds for the homology
$dH_{q}$. From Theorem 6.7, the isomorphism between the relative homologies
$dH_{q}$ and $H^{c_{1}}_{q}$ also holds, and thus this excision-like property
holds for the homology $H^{c_{1}}_{q}$ as well, as stated in the next
corollary. Before stating this corollary, we need to define the interior
operator $Int^{i}$ on subsets of digital images.

######  Definition 6.9.

For a subset $A$ of digital image $X$, the interior operator
$\operatorname{Int}^{i}(A)$, for integers $i\geq 0$, is defined as:

$\displaystyle\operatorname{Int}(A)$ $\displaystyle=\\{x\in
A~{}|~{}\overline{N}_{c_{1}}^{X}(x)\subset A\\}$
$\displaystyle\operatorname{Int}^{0}(A)$ $\displaystyle=A$
$\displaystyle\operatorname{Int}^{i}(A)$
$\displaystyle=\operatorname{Int}(\operatorname{Int}^{i-1}(A)),\text{ for
}i\geq 1.$

######  Corollary 6.10.

Consider subsets $A$ and $B$ of a digital image $X$. Suppose that there is a
positive integer $i$, with
$X=\operatorname{Int}^{i}(A)\cup\operatorname{Int}^{i}(B)$, then
$H_{q}^{c_{1}}(X,A)\cong H_{q}^{c_{1}}(B,A\cap B)$, for integers $q<i+1$.

Moreover, due to the isomorphism between the absolute and relative homologies
$dH_{q}$ and $H_{q}^{c_{1}}$, it can be concluded that the sequence of
functors $H_{q}^{c_{1}}$ on the category of $c_{1}$-digital images is a
digital homology theory (see definition 6.2 of [9]).

######  Corollary 6.11.

The $c_{1}$-cubical homology groups $H_{q}^{c_{1}}$ form a digital homology
theory on the category of $c_{1}$-digital images.

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