@ Appl. Gen. Topol. 22, no. 2 (2021), 223-250doi:10.4995/agt.2021.13154
© AGT, UPV, 2021

Digital homotopy relations and digital

homology theories

P. Christopher Staecker

Department of Mathematics, Fairfield University, USA (cstaecker@fairfield.edu)

Communicated by S. Romaguera

Abstract

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 homol-

ogy for digital images with c1-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 homomor-

phisms in the cubical homology theory, and strong homotopic maps

additionally have the same induced homomorphisms in the simplicial

theory.

2010 MSC: 55P10; 68R10.

Keywords: digital topology; digital homotopy; homology; cubical homology.

Received 17 February 2020 – Accepted 06 May 2021

http://dx.doi.org/10.4995/agt.2021.13154


P. C. Staecker

1. Introduction

A digital image is a set of points X, with some adjacency relation κ, which is
symmetric and antireflexive. The standard notation for such a digital image is
(X,κ). Typically in digital topology the set X is a finite subset of Zn, and the
adjacency relation is based on some notion of adjacency of points in the integer
lattice. This style of digital topology has its origins in the work of Rosenfeld
and others, see [15] for an early work.

We will use the notation x ↔κ y when x is adjacent to y by the adjacency κ,
and x -κ y when x is adjacent or equal to y. The particular adjacency relation
will usually be clear from context, and in this case we will omit the subscript.

The results of this paper hold generally, without any specific reference to the
embedding of the images in Zn. Thus we will often simply consider a digital
image X as an abstract simple graph, where the vertices are points of X, and
an edge connects two vertices x,x′ ∈ X whenever x ↔ x′.

Definition 1.1. Let (X,κ),(Y,λ) be digital images. A function f : X → Y is
(κ,λ)-continuous iff whenever x ↔κ y then f(x) -λ f(y).

When f is a continuous bijection with continuous inverse, we say f is an
isomorphism.

For simplicity of notation, we will generally not need to reference the adja-
cency relation specifically. Thus we typically will denote a digital image simply
by X, and when the appropriate adjacency relations are clear we simply call
a function between digital images “continuous”. We will often refer to digital
images as simply “images”.

For any digital image X the identity map idX : X → X is always continuous.
The topological theory of digital images has, to a large part, been character-

ized by taking ideas from classical topology and “discretizing” them. Typically
Rn is replaced by Zn, and so on. Viewing Z as a digital image, it makes sense
to say a and b are adjacent if and only if |a − b| = 1. This adjacency relation
corresponds to connectivity in the standard topology of R.

When n > 1 there is no canonical “standard adjacency” to use in Zn which
corresponds naturally to the standard topology of Rn. In the case of Z2, for
example, at least two different adjacency relations seem reasonable: we can
view Z2 as a rectangular lattice connected by the coordinate grid, so that
each point is adjacent to 4 neighbors; or we can additionally allow diagonal
adjacencies so that each point is adjacent to 8 neighbors. This is formalized
in the following definition from [7] (though these adjacencies had been studied
for many years earlier):

Definition 1.2. Let k,n be positive integers with k ≤ n. Then define an
adjacency relation ck on Z

n as follows: two points x,y ∈ Zn are ck-adjacent if
their coordinates differ by at most 1 in at most k positions, and are equal in
all other positions.

We will also make use of the notion of connectedness. For a,b ∈ Z and
a < b, let [a,b]Z denote the set {a,a + 1, . . . ,b}. This set is called the digital

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Digital homotopy and homology

interval from a to b. Given two points x,y ∈ (X,κ), a κ-path from x to y is a
(c1,κ)-continuous function p : [0,n]Z → X with p(0) = x and p(n) = y. When
the adjacency relation is understood, a κ-path is called simply a path.

A digital image (X,κ) is connected when any two points of X can be joined
by a path. A connected component of X is a maximal connected subset of X.

The following is the standard definition of digital homotopy. For a,b ∈ Z
with a ≤ b, let [a,b]Z be the digital interval [a,b]Z = {a,a + 1, . . . ,b}.

Definition 1.3 ([2]). Let (X,κ) and (Y,λ) be digital images. Let f,g : X → Y
be (κ,λ)-continuous functions. Suppose there is a positive integer m and a
function H : X × [0,k]Z → Y such that:

• for all x ∈ X, H(x,0) = f(x) and H(x,k) = g(x);
• for all x ∈ X, the induced function Hx : [0,k]Z → Y defined by

Hx(t) = H(x,t) for all t ∈ [0,k]Z

is (c1,λ)−continuous.
• for all t ∈ [0,k]Z, the induced function Ht : X → Y defined by

Ht(x) = H(x,t) for all x ∈ X

is (κ,λ)−continuous.

Then H is a [digital] homotopy between f and g, and f and g are [digitally]
homotopic, denoted f ≃ g.

If k = 1, then f and g are homotopic in one step.

Homotopy in one step can easily be expressed in terms of individual adja-
cencies:

Proposition 1.4. Let f,g : X → Y be continuous. Then f is homotopic to g
in one step if and only if for every x ∈ X, we have f(x) - g(x).

Proof. Assume that f is homotopic to g in one step. The homotopy is simply
defined by H(x,0) = f(x) and H(x,1) = g(x). Then by continuity in the
second coordinate of H, we have f(x) = H(x,0) - H(x,1) = g(x) as desired.

Now assume that f(x) - g(x) for each x. Define H : X × [0,1]Z → Y by
H(x,0) = f(x) and H(x,1) = g(x). Clearly H(x,t) is continuous in x for each
fixed t, since f and g are continuous. Also H(x,t) is continuous in t for fixed
x because H(x,0) = f(x) - g(x) = H(x,1). Thus H is a one step homotopy
from f to g as desired. �

Definition 1.3 is inspired by the concept of homotopy from classical topology,
but the classical definition is simpler because it can use the product topology.
In classical topology the continuity of the two types of “induced function” is
expressed simply by saying that H : X × [0,1] → Y is continuous with respect
to the product topology on X × [0,1].

Given two digital images A and B, we can consider the product A × B as
a digital image, but there are several choices for the adjacency to be used.
The most natural adjacencies are the normal product adjacencies, which were

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P. C. Staecker

defined by Boxer in [3]. This generalizes an earlier product construction from
[7], which is equivalent to NP2.

Definition 1.5 ([3]). For each i ∈ {1, . . . ,n}, let (Xi,κi) be digital images.
Then for some u ∈ {1, . . . ,n}, the normal product adjacency NPu(κ1, . . . ,κn)
is the adjacency relation on

∏n
i=1 Xi defined by: (x1, . . . ,xn) and (x

′

1, . . . ,x
′

n)
are adjacent if and only if their coordinates are adjacent in at most u positions,
and equal in all other positions.

When the underlying adjacencies are clear, we abbreviate NPu(κ1, . . . ,κn)
as simply NPu.

The normal product adjacency is inspired by the various standard adjacen-
cies typically used on Zn. On Z1, as mentioned above, the standard adjacency
is c1. Viewing Z

2 as the product Z2 = Z × Z, it is easy to see that 4-adjacency
is the same as NP1(c1,c1), and 8-adjacency is NP2(c1,c1). The typical adja-
cencies used in Z3 are 6-, 18-, and 26-adjacency, depending on which types of
diagonal adjacencies are allowed. These adjacencies are exactly NP1(c1,c1,c1),
NP2(c1,c1,c1), and NP3(c1,c1,c1).

Boxer showed that the definition of homotopy can be rephrased in terms of
an NP1 product adjacency:

Theorem 1.6 ([3, Theorem 3.6]). Let (X,κ) and (Y,λ) be digital images.
Then H : X × [0,k]Z → Y is a homotopy if and only if H is (NP1(κ,c1),λ)-
continuous.

Once we have a notion of homotopy, it is natural to define homotopy equiv-
alence:

Definition 1.7. Digital images X and Y are homotopy equivalent when there
exist continuous functions f : X → Y and g : Y → X with g ◦ f ≃ idX and
f ◦ g ≃ idY , where idX and idY denote the identity functions on X and Y . In
this case f and g are called homotopy equivalences.

The structure of this paper is as follows: in Section 2 we define strong
homotopy and give some of its basic properties. In Section 3 we show that
a homotopy is strong if and only if it can be made “punctuated”, that is,
changing by only one point at a time. In Section 4 we describe three different
digital homology theories existing in the literature. In Section 5 we describe
a new theory, defined only for images in Zn with c1-adjacency, which we call
c1-cubical homology, similar to a construction described in [11]. In Section 6
we show that the simplicial and singular theories are isomorphic. In Section 7
we prove homotopy and strong homotopy invariance properties for the various
theories. In Section 8 we describe some relationships between the simplicial
and cubical theories, and in Section 9 we describe some specific examples.

The author would like to thank Samira Jamil and Danish Ali for helpful
conversations concerning early drafts of this paper.

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2. Strong homotopy

Theorem 1.6 states that a homotopy is a function H : X ×[0,k]Z → Y which
is continuous when we use the NP1 product adjacency in the domain. We will
explore the following definition, which simply uses NP2 in place of NP1. As we
will see this imposes extra restrictions on the homotopy H.

Definition 2.1. Let (X,κ) and (Y,λ) be digital images. We say H : X ×
[0,k]Z → Y is a strong homotopy when H is (NP2(κ,c1),λ)-continuous.

If there is a strong homotopy H between f and g, we say f and g are
strongly homotopic, and we write f ∼= g. If additionally k = 1, we say f and g
are strongly homotopic in one step.

Since we have exchanged NP1 for NP2 in the definition above, we may say
informally that strong homotopy and digital homotopy provide two different
but equally natural “digitizations” of the classical topological idea of homotopy.
The difference between homotopy and strong homotopy of continuous functions
is analogous to the difference between 4-adjacency and 8-adjacency of points
in the plane. The strong homotopy relation matches one used in recent work
[13] to build a digital homotopy theory in many respect matching the classical
one.

It is clear from the definition of the normal product adjacency that if two
points are NP1-adjacent, then they are NP2-adjacent. Thus any function
f : A × B → C which is continuous when using NP2 in the domain will
automatically be continuous when using NP1 in the domain. Thus we obtain
the following, which justifies the use of the word “strong.”

Theorem 2.2. Let H : X × [0,k]Z → Y be a strong homotopy. Then H is a
homotopy.

A standard argument shows that strong homotopy is an equivalence relation.

Theorem 2.3. Strong homotopy is an equivalence relation.

Strong homotopy in one step can be expressed in terms of adjacencies as in
Proposition 1.4.

Theorem 2.4. Let f,g : X → Y be continuous. Then f is strongly homotopic
to g in one step if and only if for every x,x′ ∈ X with x ↔ x′, we have
f(x) - g(x′).

Proof. First assume that f is homotopic to g in one step. The homotopy is
simply defined by H(x,0) = f(x) and H(x,1) = g(x). Then if x ↔ x′, we
will have (x,0) ↔NP2 (x

′,1), and thus since H is NP2-continuous we have
f(x) = H(x,0) - H(x′,1) = g(x′) as desired.

Now assume that f(x) - g(x′) for each x. Define H : X × [0,1]Z → Y
by H(x,0) = f(x) and H(x,1) = g(x), and we must show that H is NP2-
continuous. Take (x,t),(x′, t′) ∈ X × [0,1]Z with (x,t) ↔NP2 (x

′, t′), and we
will show that H(x,t) - H(x′, t′). We have a few cases based on the values of
t,t′ ∈ {0,1}.

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P. C. Staecker

If t = t′, without loss of generality say t = t′ = 0. Since (x,t) ↔NP2 (x
′, t′),

we have x - x′, and so we have

H(x,t) = H(x,0) = f(x) - f(x′) = H(x′,0) = H(x′, t′)

since f is continuous. Thus H(x,t) - H(x′, t′) as desired.
Finally, if t 6= t′, without loss of generality assume t = 0 and t′ = 1. Since

(x,t) ↔NP2 (x
′, t′), we have x - x′, and so we have

H(x,t) = H(x,0) = f(x) - g(x′) = H(x′,1) = H(x′, t′)

and so H(x,t) - H(x′, t′) as desired. �

By Theorem 2.2, if f ∼= g then automatically we have f ≃ g. The converse
is not true, however, as the following example shows.

One important source of examples in the study of digital images is the digital
cycle Cn = {c0, . . . ,cn−1}, with adjacency given by ci ↔ ci+1 for each i, where
for convenience we always read the subscripts modulo n. Thus Cn is a digital
image of n points which is in many ways topologically analogous to the circle.

Example 2.5. It is well known that all selfmaps on C4 are homotopic to one
another. But we will show that the identity map idC4 : C4 → C4 is not strongly
homotopic to any map f with #f(C4) < 4. It will suffice to show that idC4 is
not strongly homotopic in 1 step to any such map.

Without loss of generality assume that f(C4) ⊆ {c0,c1,c2}, and assume for
the sake of a contradiction that f is strongly homotopic in one step to idC4.
Then by Theorem 2.4, since c2 ↔ c3 and c0 ↔ c3, we will have c2 - f(c3)
and also c0 - f(c3). Thus f(c3) is adjacent to both c0 and c2, but cannot
equal c3 since c3 6∈ f(C4). We conclude that f(c3) = c1. By Theorem 1.4, this
contradicts the fact that f is homotopic to the identity in 1 step.

3. Punctuated homotopy

Given a homotopy H(x,t), we say that H is punctuated if, for each t, there
is some xt such that H(x,t) = H(x,t + 1) for all x 6= xt. That is, from each
stage of the homotopy to the next, the induced map Ht(x) is changing by at
most one point at a time.

Theorem 3.1. Any punctuated homotopy is a strong homotopy.

Proof. Let H be a punctuated homotopy, and let (x,t) ↔NP2 (x
′, t′). We must

show that H(x,t) - H(x′, t′). Since (x,t) ↔NP2 (x
′, t′) we have x - x′ and

t - t′. If t = t′, then we have (x,t) ↔NP1 (x
′, t′) and so H(x,t) - H(x′, t′)

since H is a homotopy. It remains to consider when t ↔ t′ and t 6= t′. In this
case, without loss of generality assume t′ = t + 1.

Since H is a punctuated homotopy, there is some xt such that H(x,t) =
H(x,t + 1) for all x 6= xt. When x 6= xt, we have

H(x,t) = H(x,t + 1) - H(x′, t + 1) = H(x′, t′)

since H is a homotopy. Similarly we have H(x,t) - H(x′, t′) when x′ 6= xt.

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Digital homotopy and homology

Thus it remains only to consider when x = x′ = xt. That is, we must
show that H(xt, t) - H(xt, t

′) = H(xt, t + 1), and this is true because H is a
homotopy. �

We also have a sort of converse to the above. While not every strong homo-
topy is punctuated, any two strongly homotopic maps can be connected by a
punctuated homotopy, provided that the domain is finite.

Theorem 3.2. Let X be a finite digital image, and let f,g : X → Y be strongly
homotopic. Then f and g are homotopic by a punctuated homotopy.

Proof. By induction, it suffices to show that if f and g are strongly homotopic
in one step, then they are homotopic by a punctuated homotopy. Enumerate
the points of X as X = {x0, . . . ,xn}, and define H : X × [0,n + 1]Z → Y by:

H(xi, t) =

{
f(xi) if i ≥ t,

g(xi) if i < t.

Then H moves one point at a time, so we need only to show that it has the
appropriate continuity properties to be a homotopy.

First we show that H(x,t) is continuous in x for fixed t. Take x ↔ x′,
then H(x,t) ∈ {f(x),g(x)} and H(x′, t) ∈ {f(x′),g(x′)}. Since f and g are
homotopic in one step we have f(x) - f(x′) and g(x) - g(x′), and since f and
g are strongly homotopic in one step we have f(x) - g(x′) and g(x) - f(x′).
Thus in any case we have H(x,t) - H(x′, t) as desired.

Now we show that H(x,t) is continuous in t for fixed x. It suffices to show
that H(x,t) - H(x,t + 1) for any t. We have H(x,t) ∈ {f(x),g(x)} and also
H(x,t + 1) ∈ {f(x),g(x)}. Since f and g are homotopic in one step we have
f(x) - g(x), and thus H(x,t) - H(x,t + 1) as desired. �

The finiteness assumption above is necessary, as the following example shows.

Example 3.3. Let X = Z × {0,1} ⊂ Z2, with 8-adjacency, and let f(x,y) =
(x,0). Then f is strongly homotopic to idX in one step. But f(x,y) and id(x,y)
differ for infinitely many values of (x,y) ∈ X. Since a punctuated homotopy
has finite time interval, and can only change one value at a time, there can be
no punctuated homotopy from f to idX.

Combining the two theorems above gives us a nice characterization of strong
homotopy.

Corollary 3.4. If X is finite, two continuous maps f,g : X → Y are strongly
homotopic if and only if they are homotopic by a punctuated homotopy.

As an application, we show that the identity map on the n-cycle for n ≥ 4
is not strongly homotopic to any other map:

Example 3.5. Let n ≥ 4, and assume for the sake of a contradiction that
there is some continuous f : Cn → Cn with f ∼= idCn and f 6= idCn. Without
loss of generality, assume that the homotopy from f to idCn is punctuated and

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P. C. Staecker

in one step. Since the homotopy is punctuated, f moves one point, so without
loss of generality we may assume that f(c0) = c1 and f(ci) = ci for all i 6= 0.
This contradicts the continuity of f, however, since we will have c0 ↔ cn−1 but
c1 = f(c0) 6- f(cn−1) = cn−1 since n ≥ 4.

4. Digital homology theories in the literature

Digital topological invariants are typically inspired by classical topology,
though most are not literally topological in nature. For example, when X and
Y are digital images, a digitally continuous function f : X → Y is not actually
continuous in the classical sense with respect to any topologies on X and Y .
The digital fundamental group π1(X) is not actually the fundamental group of
X with respect to any topology on X, etc.

Digital homology, however, does fit neatly into the classical theory of homo-
logical algebra. Each of the homology theories described in [4, 12, 9] is indeed a
homology theory of a classical chain complex. Though it is not always done in
these references, we will make use of results from classical homological algebra
whenever possible to avoid the need for specialized definitions and proofs of
basic results.

We will review the basic homological algebra that will be useful, see e.g. [8].
A chain complex is a sequence of abelian groups C0,C1, . . . and homomor-

phisms ∂q : Cq → Cq−1 satisfying ∂q−1 ◦ ∂q = 0 for all q. Given some chain
complex C = (Cq,∂q), for each q we define the cycle and boundary subgroups
Zq and Bq of Cq as: Zq = ker∂q and Bq = im∂p+1. The dimension q homology
group of the chain complex is defined as Hq = Zq/Bq. When we are discussing
the homology groups of various different chain complexes, we will write Hq(C)
for the dimension q homology of the chain complex C.

Given two chain complexes A = (Aq,∂
A
q ) and B = (Bq,∂

B
q ), a sequence of

homomorphisms fq : Aq → Bq is a chain map from A to B when fq−1 ◦ ∂
A
q =

∂Bq ◦ fq for each q. Every such chain map induces a well-defined sequence of
homomorphisms f∗,q : Hq(A) → Hq(B). Furthermore, this correspondence is
functorial in the sense that (f ◦ g)∗,q = f∗,q ◦ g∗,q, and the induced homo-
morphism of the identity function is the identity homomorphism. When the
dimensions are clear, we will omit the subscript q.

Given three chain complexes A, B, C, and an exact sequence of chain maps:

0 → A
f
−→ B

g
−→ C → 0,

for each q there is a connecting homomorphism δq : Hq(C) → Hq−1(A) such
that the following sequence is exact:
(4.1)

· · · → Hq+1(C)
δq+1
−−−→ Hq(A)

f∗,q
−−→ Hq(B)

g∗,q
−−→ Hq(C)

δq
−→ Hq−1(A) → . . .

The “long exact sequence” above is the fundamental tool of relative homology
theory.

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Digital homotopy and homology

4.1. Simplicial homology. Digital simplicial homology theory was first de-
fined by Arslan, Karaca, and Öztel in [1], in Turkish. This material was ex-
tended and published in English in [4]. We review the definitions as presented
in [4].

For a digital image X and some q ≥ 0, a q-simplex is defined to be any set of
q+1 mutually adjacent points of X. For some ordered list of mutually adjacent
points x0, . . . ,xq, the associated ordered q-simplex is denoted 〈x0, . . . ,xq〉.

The chain group Cq(X) is defined to be the abelian group generated by
the set of all ordered q-simplices, where if ρ : {0, . . . ,q} → {0, . . . ,q} is a
permutation, then in Cq(X) we identify

(4.2) 〈x0, . . . ,xq〉 = (−1)
ρ〈xρ(0), . . . ,xρ(q)〉,

where (−1)ρ = 1 when ρ is an even permutation, and (−1)ρ = −1 when ρ is
an odd permutation.

The boundary homomorphism ∂q : Cq(X) → Cq−1(X) is the homomorphism
induced by defining:

(4.3) ∂q(〈x0, . . . ,xq〉) =

q∑

i=0

(−1)i〈x0, . . . , x̂i, . . .xq〉,

where x̂i indicates omission of the xi coordinate.
It can be verified that ∂q−1 ◦ ∂q = 0, and so (Cq(X),∂q) forms a chain com-

plex, and the dimension q homology group Hq(X) = Zq(X)/Bq(X) is defined
to be the homology of this chain complex.

Any continuous function f : X → Y induces a homomorphism f#,q :
Cq(X) → Cq(Y ) defined by

f#,q(〈x0, . . . ,xq〉) = 〈f(x0), . . . ,f(xq)〉,

where the right side is interpreted as 0 if the set {f(x0), . . . ,f(xq)} has cardinal-
ity less than q. When the value of q is understood, we simply write f#,q = f#.

It is easy to check that this f#,q is a chain map, and thus induces a homo-
morphism f∗,q : Hq(X) → Hq(X). Again, we typically write f∗,q = f∗ when
the q is understood.

We will remark that the constructions above match exactly the homology
of the clique complex of X when viewed as a graph. The clique complex is the
simplicial complex built from the complete subgraphs of a given graph, and
the homology of this simplicial complex is the same as the digital homology
defined above. The free mathematics software SageMath has built-in func-
tions to compute the clique complex of a graph, and further to compute the
homology of any finite complex. Thus it is easy to implement algorithms to
compute the simplicial homology groups of a digital image. Source code for
computing simplicial homology groups is available at the author’s website for
experimentation.1

These definitions and results are all exactly as expected from the classical
homology theory of a simplicial complex. As an example, we compute the

1http://faculty.fairfield.edu/cstaecker

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P. C. Staecker

homology groups of the cycle Cn for n ≥ 4. The case n = 4 appears as
Theorem 3.17 of [4].

Theorem 4.1. If n ≥ 4, we have:

Hq(Cn) =

{
Z if q ∈ {0,1},

0 if q > 1.

Proof. First we prove the case q = 0. The chain group C0(Cn) is generated by
n different 0-simplices 〈c0〉, . . . ,〈cn−1〉. Since ∂0 is a trivial homomorphism, we
have Z0(Cn) = C0(Cn). Note that for each i, we have

〈ci〉 = (〈ci〉 − 〈ci+1〉) + 〈ci+1〉 = ∂〈ci+1,ci〉 + 〈ci+1〉,

and thus 〈ci〉 − 〈ci+1〉 ∈ B0(Cn). Thus 〈ci〉 and 〈ci+1〉 are equal in H0(Cn) for
every i. That is, H0(Cn) is the group generated by 〈c0〉, and so H0(Cn) = Z
as desired.

Now for q = 1, first we note that there are no 2-simplices in Cn (because
n ≥ 4), so C2(Cn) is trivial, and thus B1(Cn) is trivial. Thus H1(Cn) will
be isomorphic to Z1(Cn). To determine Z1(Cn), we must determine which
α ∈ C1(Cn) satisfy ∂α = 0. Any α ∈ C1(Cn) can be expressed as:

α = w1〈c0,c1〉 + · · · + wn〈cn−1,c0〉

for wi ∈ Z, and then ∂α = 0 if and only if:

0 = ∂α = w1(〈c1〉 − 〈c0〉) + · · · + wn(〈c0〉 − 〈cn−1〉)

= (wn − w1)〈c0〉 + (w1 − w2)〈c1〉 + · · · + (wn−1 − wn)〈cn−1〉

and thus we have w1 = w2 = · · · = wn since the 〈ci〉 are linearly independent
in C1(Cn). Then we have shown that H1(Cn) = Z1(Cn) is generated by the
single element

σ =

n−1∑

i=0

〈ci,ci+1〉,

and thus H1(Cn) = Z.
For q > 1, there are no q-simplices and so Cq(Cn) is trivial, and thus Hq(Cn)

is trivial. �

In the case of Cn, we can make a full computation of the induced homomor-
phisms for any selfmap, using results from [5]. Let c : Cn → Cn be the constant
map c(ci) = c0, let l : Cn → Cn be the “flip map” l(ci) = c−i, and for some
integer d, let rd : Cn → Cn be the rotation rd(ci) = ci+d. Theorem 9.3 of [5]
shows that there are exactly 3 homotopy classes of selfmaps on Cn: any map
f : Cn → Cn is either strongly homotopic to a constant, or is homotopic to the
identity and equals rd for some d, or is homotopic to the flip map and equals
rd ◦ l for some d. (The strongness of the homotopy to the constant was not
mentioned in [5], but the homotopy demonstrated in the proof in that paper is
easily made punctuated and therefore strong.)

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Digital homotopy and homology

Theorem 4.2. Let n > 4, and let f : Cn → Cn be continuous. Then for all
q > 1, the induced homomorphism f∗,q : Hq(Cn) → Hq(Cn) is trivial, for q = 0
the induced homomorphism f∗,0 = id, and for q = 1 we have:

f∗,1 =





id if f ≃ idCn,

− id if f ≃ l,

0 if f ≃ c.

Proof. For q > 1 we have already seen that Hq(Cn) is a trivial group, so we will
have f∗ = 0 automatically. When f is homotopic to a constant, as mentioned
above, in fact f ∼= c and thus f∗ = c∗ for all q, and so f∗,0 is the identity and
f∗,q is trivial for q > 0.

Now we consider when f is homotopic to the identity or the flip map, for
q ∈ {0,1}. First we consider q = 0 and f ≃ idCn. Since H0(Cn) is generated
by 〈c0〉, it suffices to show that f∗(〈c0〉) = 〈c0〉. Let d be some integer with
f = rd, and we have:

f∗(〈c0〉) = 〈f(c0)〉 = 〈cd〉 = 〈c0〉

as desired. Exactly the same argument applies for f ≃ l, since we will still have
〈f(c0)〉 = 〈cd〉 for some d.

Now for q = 1, and f ≃ idCn we must show f∗(σ) = σ, where σ =∑n−1
i=0 〈ci,ci+1〉 as in the proof of Theorem 4.1. Let f = rd, and we have:

f(σ) =

n−1∑

i=0

〈f(ci),f(ci+1)〉 =

n−1∑

i=0

〈ci+d,ci+1+d〉 = σ

as desired.
Finally we consider q = 1 and f ≃ l, and we must show f∗(σ) = −σ. Let

f = rd ◦ l, so f(ci) = cd−i, and we have:

f(σ) =
n−1∑

i=0

〈f(ci),f(ci+1)〉 =
n−1∑

i=0

〈cd−i,cd−i−1〉 =
n−1∑

i=0

〈c−i+1,c−i〉

=
n−1∑

i=0

〈ci+1,ci〉 =
n−1∑

i=0

−〈ci,ci+1〉 = −σ

as desired. �

The three cases of Theorem 4.2 suffice to compute f∗ for any selfmap of
Cn, and we note that in each of the three homotopy classes, the set of induced
homomorphisms is different. We obtain a sort of Hopf theorem for digital
cycles:

Corollary 4.3. Let n > 4, and let f,g : Cn → Cn be continuous. Then
f∗,q = g∗,q for each q if and only if f ≃ g.

One major difference between the digital theory and classical homology is
that the induced homomorphism f∗ is not always a digital homotopy invariant.

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P. C. Staecker

Example 4.4. By Theorem 4.1, the homology group H1(C4) is isomorphic to
Z. Because all maps on C4 are homotopic, the identity map is homotopic to a
constant map c. But id∗ : H1(C4) → H1(C4) is the identity homomorphism of
Z, while c∗ : H1(C4) → H1(C4) is the trivial homomorphism. Thus id ≃ c but
id∗ 6= c∗.

The lack of a homotopy-invariant induced homorphism is a major deficiency
in the homology theory of digital images. Lacking this homotopy invariance,
the homology groups are not well-behaved with respect to typical topological
constructions. For example two homotopy equivalent digital images may have
different homology groups. As a consequence the digital Euler characteristic
is not a digital homotopy type invariant. Example 7.2 of [6] shows that the
Hurewicz theorem also fails: that is, that H1(X) may not be isomorphic to the
abelianization of the fundamental group of X as defined in [2].

The homology group in dimension zero is easy to predict:

Theorem 4.5. Let X be any digital image with d connected components. Then
H0(X) ∼= Z

d.

Proof. The chain group C0(X) (which equals the group Z0(X) of 0-cycles) has
basis given by the points of X. It is easy to see that two points x,y ∈ Z0(X)
are homologous if and only if x and y are in the same connected component of
X. �

The simplicial homology is also easy to compute when X consists of finitely
many isolated points, that is, points which are not adjacent to any other points.

Theorem 4.6. Let X be any digital image consisting of k isolated points. Then
H0(X) ∼= Z

k and Hq(X) = 0 for q > 0.

Proof. The statement concerning H0(X) follows immediately from 4.5. Since
X has no adjacencies, it contains no q-simplices when q > 0. Thus Hq(X) = 0
when q > 0 as desired. �

4.2. Singular homology. Singular homology for digital images was defined
by D.W. Lee in [12]. We will review Lee’s definitions, using some different
notations to fit more cleanly with the other homology theories.

For any natural number q, let ∆q be the standard q-simplex, the digital
image consisting of q + 1 mutually adjacent points. Viewed as a graph, ∆q

is the complete graph of q + 1 vertices. The points of ∆q will be labeled and
ordered as ∆q = (e0, . . . ,eq).

Definition 4.7. Let X be a digital image. A singular q-simplex in X is a
continuous function

φ : ∆q → X.

We will write such a singular q-simplex as the ordered list [φ(e0), . . . ,φ(eq)].

For any q ≥ 0, the group of singular q-chains, denoted Čq(X), is the free
abelian group whose basis is the set of all singular q-simplices of X.

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Digital homotopy and homology

The singular boundary operator ∂q : Čq(X) → Čq−1(X) is defined as follows:

(4.4) ∂q[x0, . . . ,xq] =

q∑

i=0

(−1)i[x0, . . . , x̂i, . . . ,xq],

where as usual x̂i denotes omission of the ith element. When φ is a q-simplex,
∂qφ will be a singular (q−1)-chain. As before, we will often omit the subscript q.
Note that we are using the same notation to denote the boundary operators in
both simplicial and singular homology. In practice this will not cause confusion.

The singular q-simplex [x0, . . . ,xq] is very similar to the ordered q-simplex
〈x0, . . . ,xq〉. The main difference algebraically is that, in the singular chain
group, we do not identify permutations of the listings. For example we have
〈x0,x1〉 = −〈x1,x0〉 in C1(X), but in Č1(X) the basis elements [x0,x1] and
[x1,x0] are linearly independent. We also will always have 〈x,x〉 = −〈x,x〉 = 0
in C1(X), while [x,x] will be nontrivial in Č1(X). This means that in particular
Čq(X) has nonzero elements which, when written as lists of points, include
repetitions. Such elements are always 0 in Cq(X) because of (4.2).

Theorem 3.9 of [12] shows that ∂q−1 ◦ ∂q = 0, and thus (Čq(X),∂q) is a
chain complex, and the dimension q singular homology group is defined as
Ȟq(X) = Žq(X)/B̌q(X).

If f : X → Y is continuous, then there is a homomorphism f# : Čq(X) →

Čq(Y ) defined on singular chains by f#(φ) = f ◦ φ. This is easily shown to
be a chain map, and thus we obtain the induced homomorphism on singular
homology f∗ : Ȟq(X) → Ȟq(Y ).

We will require an analogue of Theorem 4.5 for singular homology. The
proof is the same as that of Theorem 4.5.

Theorem 4.8. Let X be any digital image with d connected components. Then
Ȟ0(X) ∼= Z

d.

Lee’s work provides an analogue of Theorem 4.6 for singular homology. The
following is a consequence of Theorems 3.16 and 3.20 of [12]:

Theorem 4.9. Let X be a digital image which consists of k isolated points.
Then Ȟ0(X) ∼= Z

k and Ȟq(X) = 0 for q > 0.

4.3. Cubical homology. Cubical homology was introduced by Jamil & Ali
in [9], with definitions mimicking the classical cubical homology as presented
in [14]. We will review the definition and basic results.

Let I = [0,1]Z = {0,1}, and we consider I
n ⊂ Zn as a digital image with

c1-adjacency for each n ≥ 1. When n = 0, we define I
0 to be a single point.

Definition 4.10. Let (X,κ) be a digital image. Then a q-cube in X is a
(c1,κ)-continuous function

σ : Iq → X.

A q-cube is degenerate if there is some i such that the function σ(t1, . . . , tq) does
not depend on the coordinate ti. Let Qq(X) be the free abelian group whose
basis is the set of all q-cubes in X, and let Dq(X) be the subgroup generated

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P. C. Staecker

by the degenerate q-cubes. Then the group of cubical q-chains, denoted C̄q(X),
is the quotient C̄q(X) = Qq(X)/Dq(X).

The cubical boundary operator is defined in terms of cubical face operators.
For some q-cube σ and some i ∈ {1, . . . ,q}, define (q − 1)-cubes Aiσ and Biσ
as:

(Aiσ)(t1, . . . , tq−1) = σ(t1, . . . , ti−1,0, ti, . . . , tq−1),

(Biσ)(t1, . . . , tq−1) = σ(t1, . . . , ti−1,1, ti, . . . , tq−1),

These Ai and Bi give the “front face” and “back face” of the cube in each of
its q dimensions.

The boundary operator ∂q : C̄q(X) → C̄q−1(X) is defined on cubes by the
formula:

(4.5) ∂q(σ) =

q∑

i=1

(−1)i(Aiσ − Biσ),

and extended to chains by linearity. We will sometimes omit the subscript q.
A routine calculation shows that ∂q−1 ◦ ∂q = 0, and thus (C̄q(X),∂q) is a

chain complex. We obtain cycle and boundary groups Z̄q(X) and B̄q(X) and
the homology groups H̄q(X) = Z̄q(X)/B̄q(X).

Exactly as in the singular theory, if f : X → Y is continuous, then f# :
C̄q(X) → C̄q(Y ) is defined on cubical chains by f#(σ) = f ◦σ, and this defines
the induced homomorphism on cubical homology f∗ : H̄q(X) → H̄q(Y ).

The most important property of cubical homology which distinguishes it
from simplicial homology is the following, which is Theorem 3.7 of [9]:

Theorem 4.11 ([9, Theorem 3.7]). Let X and Y be any digital images and
f,g : X → Y with induced homomorphisms f∗,g∗ : H̄q(X) → H̄q(X). If f ≃ g,
then f∗ = g∗.

Immediate corollaries include:

Corollary 4.12 ([9, Corollary 3.8]). If X and Y are homotopy equivalent, then
H̄q(X) ∼= H̄q(Y ) for all q.

Corollary 4.13 ([9, Example 3.9]). If X is contractible (i.e. X is homotopy
equivalent to a point), then

H̄q(X) =

{
Z if q = 0,

0 if q 6= 0.

Jamil & Ali also prove a Hurewicz theorem, that H̄1(X) is isomorphic to
the abelianization of the fundamental group of X. They also prove results
concerning connected components and single points. The following theorems
follow from [9, Propositions 3.1, 3.2] and Corollary 4.13.

Theorem 4.14. Let X be a digital image with d connected components. Then
H̄0(X) ∼= Z

d.

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Digital homotopy and homology

Theorem 4.15. Let X be a digital image consisting of k isolated points. Then
H̄0(X) ∼= Z

k and H̄q(X) ∼= 0 for q > 0.

5. A new cubical homology theory for images with c1-adjacency

Ege & Karaca in [11] describe another type of cubical homology theory based
on classical constructions from [10]. Their construction is not generally well-
defined for any digital image, but only a digital image which is a “cubical set”,
that is, a finite union of “elementary cubes.” We will describe the construc-
tion, sometimes using different terminology that is more convenient for making
comparisons with the other theories in this paper.

Ege & Karaca’s focus on “cubical sets” requires that the digital image X be
a subset of Zn, and that we always use c1 as the adjacency relation. This is a
significant restriction, but still allows many useful examples and results.

All of the following definitions appear in [11]: an elementary interval is a
set of the form [a,a + 1]Z = {a,a + 1} or [a,a]Z = {a}. An elementary interval
of 1 point is called degenerate, and one of 2 points is called nondegenerate. An
elementary cube is any set:

Q = J1 × · · · × Jq ⊂ Z
q

where each Ji is an elementary interval. The dimension of Q is the number
of nondegenerate factors. An elementary cube of dimension q will be called an
elementary q-cube.

When X ⊂ Zn is a digital image with c1-adjacency, it has a unique maximal
expression as a union of elementary cubes. For each q ≥ 0, let C̄c1q (X) be the
free abelian group generated by the set of all elementary q-cubes in X.

We will give a definition for a boundary operator which differs from the one
used in [11], but is rephrased to more closely resemble the boundary operator
from the cubical theory.

Define face operators Ai and Bi as follows: for an elementary cube Q =
J1 × · · · × Jn, let:

AiQ = J1 × · · · × Ji−1 × {minJi} × Ji+1 × · · · × Jn,

BiQ = J1 × · · · × Ji−1 × {maxJi} × Ji+1 × · · · × Jn.

Note that when Ji is a degenerate interval, we have AiQ = BiQ. When Ji is
nondegenerate, AiQ and BiQ are distinct elementary cubes of dimension one
less than the dimension of Q.

Now we define the boundary operator: given an elementary q-cube Q = J1 ×
· · ·×Jn, let (j1, . . . ,jq) be the sequence of indices for which Jji is nondegenerate.
Then we define:

∂qQ =

q∑

i=1

(−1)i(AjiQ − BjiQ).

It can be verified that ∂q−1◦∂q = 0, and thus (C̄
c1
q (X),∂q) is a chain complex.

The homology groups are H̄c1q (X) = Z̄
c1
q (X)/B̄q(X).

We immediately have c1 analogues of Theorems 4.14 and 4.15.

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P. C. Staecker

Theorem 5.1. Let X ⊆ Zn be a digital image with c1-adjacency having d
components. Then H̄c10 (X)

∼= Zd.

Proof. The chain group C̄c10 (X) is generated by the points of X, and it is easy
to see that two such points are homologous in H̄c10 (X) if and only if there is
a path connecting them. (The two points will be the boundary of the chain
formed by the path.) Thus H̄c10 (X) has a generator for each component of
X. �

Theorem 5.2. Let X ⊆ Zn be a digital image with c1-adjacency which consists
of a set of k isolated points. Then H̄c10 (X)

∼= Zk and H̄c1q (X)
∼= 0 for q > 0.

Proof. The first part follows immediately from Theorem 5.1. For the second
part, observe that if X consists only of isolated points, then the chain group
C̄c1q (X) is trivial for all q > 0. �

Karaca & Ege’s presentation in [11] is different from our c1-cubical theory in
some important ways. The theory in [11] starts with some digital image (X,κ)
endowed with some specified cubical structure. Then based on this structure,
a homology theory is defined. The work in [11] gives no general system for
defining a cubical structure on X, and thus the resulting homology groups are
not intrinsic to the digital image (X,κ), but rather depend on the choice of
cubical structure.

For example the digital image X = [0,1]3
Z
⊂ Z3 is considered. There is an

obvious cubical structure on X as a single elementary 3-cube together with its
faces, and we would expect the cubical homology to be Z in dimension 0 and
trivial in all other dimensions. But instead the calculation in [11, Theorem
4.5] gives Z in dimensions 0 and 2 and trivial in all other dimensions. This
is because the calculation in that theorem (without explicit mention) uses the
cubical structure consisting of six 2-cubes together with their faces, but without
the “solid” 3-cube. We will see in Example 9.5 that H̄c1q (X) is indeed Z in
dimension 0, and trivial in other dimensions.

The author has implemented an algorithm with the free mathematical pack-
age SageMath to compute the c1-cubical homology groups of any digital image.
Source code is available for experimentation at the author’s website.2

The induced homomorphism in this c1-cubical homology theory is nontrivial
to develop (there was no effort to define the concept in [11]). Given two digital
images X and Y , both with c1-adjacency, and some continuous f : X → Y , we
wish to define an induced homomorphism f̄# : C̄

c1
q (X) → C̄

c1
q (Y ) which is a

chain map.
Given an elementary q-cube Q ⊂ X, we say f is an embedding on Q if f(Q)

is an elementary q-cube in Y . In this case let ǫf,Q ∈ {−1,1} be the orientation
with with f maps Q onto f(Q). (This orientation could be defined as the
determinant of the affine linear map describing f’s restriction to Q.) When f
is not an embedding on Q, we let ǫf,Q = 0.

2http://faculty.fairfield.edu/cstaecker

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 238

http://faculty.fairfield.edu/cstaecker


Digital homotopy and homology

Then we define f̄#(Q) = ǫf,Q(f(Q)), where the right side is interpreted as
the coefficient ǫf,Q times the elementary q-cube f(Q) ⊂ Y . Extending linearly
gives the induced homomorphism f̄# : C̄

c1
q (X) → C̄

c1
q (Y ).

Now to obtain an induced homomorphism in homology, we must show that
f̄# is a chain map. We have been unable to prove this analytically, although in
low dimensions we can show that f̄# is a chain map by computer enumerations.

Theorem 5.3. Let X ⊂ Zn and Y ⊂ Zm be digital images with c1-adjacency
and n ≤ 4, and let f : X → Y be continuous. Then f# : C

c1
q (X) → C

c1
q (Y ) is

a chain map.

Proof. It suffices to show that f#(∂Q) = ∂f#(Q) for any elementary q-cube
Q. We may assume that q ≤ n, since there are no q-cubes in X of dimension
greater than n. For simplicity, by relabeling points (translating to the origin),
we may assume that Q = Iq = [0,1]

q
Z
, and also that f(0, . . . ,0) = (0, . . . ,0).

Then there are only finitely many possibilities for the behavior of f on Q, and
we can simply check that f#(∂Q) = ∂f#(Q) in all cases.

We can narrow down the number of cases to check as follows: The set f(Q)
is contained in a q-dimensional subspace of Zm, and so it suffices to consider
Y ⊂ Zq. Also note that Q is a set of diameter q, and so since f is continuous,
f(Q) has diameter at most q. Since f(Q) has diameter q and maps the origin
to the origin, we may assume that f(Q) ⊂ [−q, . . . ,q]

q
Z
.

Thus the enumeration must only construct all possible continuous functions
from Iq to [−q, . . . ,q]

q
Z
. To further narrow down the search, we assign an

ordering to the points of Iq and assume without loss of generality that the first
nonzero value of f is the point (1,0, . . . ,0).

With these filters in place, the computation becomes tractable for q = n ≤ 4.
When q = 0 there is only 1 function to check, for q = 1 there are only 2, for
q = 2 there are 16, and for q = 3 there are 2128. For q = 4 there are 23,943,296
functions, requiring 4 days to complete the enumeration, which meets the limit
of the author’s patience. �

Obviously it would be preferable to have a human readable proof of the
above, using arguments which suffice in any dimension. We state this as a
conjecture.

Conjecture 5.4. Let X ⊂ Zn and Y ⊂ Zm be digital images with c1-adjacency,
and let f : X → Y be continuous. Then f# : Cq(X) → Cq(Y ) is a chain map.

6. Equivalence of simplicial and singular homologies

In this section we show that the simplicial and singular homology theories are
equivalent. We will follow the argument used in classical algebraic topology to
show that the simplicial and singular homology theories of a simplicial complex
are equivalent. We will give a complete proof here, following the general idea
used in [8, Theorem 2.27].

There is an obvious homomorphism α : Čq(X) → Cq(X) defined by:

α[x0, . . . ,xq] = 〈x0, . . . ,xq〉,

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P. C. Staecker

and it is easy to check that this is a chain map. Thus we obtain an induced
homomorphism α∗ : Ȟq(X) → Hq(X). In this section we show that α∗ is an
isomorphism for each q.

For each k ≥ 0, let Cq(X
k) be the free abelian group generated by the set of

q-simplices of dimension k or less. (This group will always either be trivial or
equal to Cq(X), so it is not very interesting in its own right, but we introduce
the notation in order to make an inductive argument below.) For any k, the
sequence (Cq(X

k),∂) is a chain complex, a subcomplex of (Cq(X),∂), and we
will have Cq(X

k) ≤ Cq(X
k+1) ≤ Cq(X), where ≤ indicates a subgroup.

There are natural inclusion and projection maps which make the following
sequence exact:

0 → Cq(X
k−1) → Cq(X

k) → Cq(X
k)/Cq(X

k−1) → 0,

and so we may construct a long exact relative homology sequence as in (4.1):

· · · → Hq+1(X
k,Xk−1) → Hq(X

k−1) → Hq(X
k) → Hq(X

k,Xk−1) → Hq−1(X
k−1) → . . .

Similarly for each k ≥ 0, let Čq(X
k) be the free abelian group generated by

the set of singular q-simplices φ such that φ(∆q) is a simplex of dimension k
or less. The group Čq(X

k) is trivial when q < k but nontrivial for q ≥ k.

For any k, the sequence (Čq(X
k),∂) is a chain complex, a subcomplex of

(Čq(X),∂), and we will have Čq(X
k) ≤ Čq(X

k+1) ≤ Čq(X).
Again from (4.1) we have a long exact relative homology sequence:

· · · → Ȟq+1(X
k,Xk−1) → Ȟq(X

k−1) → Ȟq(X
k) → Ȟq(X

k,Xk−1) → Ȟq−1(X
k−1) → . . .

The same map α above will restrict to a chain map of Čq(X
k) → Cq(X

k) and

induce a chain map of Čq(X
k,Xk−1) → Cq(X

k,Xk−1), and thus we obtain the
following commutative diagram using the relative homology sequence, where
the vertical arrows are induced by α.
(6.1)

Ȟq+1(X
k,Xk−1) Ȟq(X

k−1) Ȟq(X
k) Ȟq(X

k,Xk−1) Ȟq−1(X
k−1)

Hq+1(X
k,Xk−1) Hq(X

k−1) Hq(X
k) Hq(X

k,Xk−1) Hq−1(X
k−1)

Lemma 6.1. For each q, the function ᾱq : Ȟq(X
k,Xk−1) → Hq(X

k,Xk−1)
induced by α is an isomorphism.

Proof. It is clear from its definition that α is surjective, so ᾱ will be surjective,
and it is enough to show that ᾱ is injective.

The chain group Čq(X
k,Xk−1) = Čq(X

k)/Čq(X
k−1) is the free abelian

group generated by all singular q-simplices φ of Xk such that φ(∆q) is a simplex
of dimension k. When q < k this group is trivial, and so Ȟq(X

k,Xk−1) is
trivial, and thus ᾱ is injective for q < k.

It remains to consider the case q ≥ k. Let ᾱq : Čq(X
k,Xk−1) → Cq(X

k,Xk−1)
be the homomorphism induced by α. To show that ker ᾱ∗,q is trivial, it suffices

to show that ker ᾱq ⊂ B̌q(X
k,Xk−1). By the definition of α, we see that kerᾱq

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 240



Digital homotopy and homology

is generated by all sums φ + ψ where φ,ψ ∈ Čq(X
k,Xk−1) and ψ is an odd

permutation of φ. Any odd permutation can be expressed as a composition of
an odd number of transpositions, and so to show that kerαq ⊂ B̌q(X

k,Xk−1),

it suffices to show φ + ψ ∈ B̌q(X
k,Xk−1) whenever ψ is a transposition of φ.

That is, given φ = [x0, . . . ,xq] and any i ∈ {0, . . . ,q − 1}, we must show that:

(6.2) [x0, . . . ,xi,xi+1, . . . ,xq] + [x0, . . . ,xi+1,xi, . . . ,xq] ∈ B̌k(X
q,Xq−1).

The assumption that φ ∈ Čq(X
k,Xk−1) means that the set {x0, . . . ,xq}

consists of exactly k distinct points. Let σ = [x0, . . . ,xi+1,xi,xi+1, . . . ,xq] ∈
Cq+1(X

k,Xk−1), and we compute:

∂q+1σ =[x1, . . . ,xi+1,xi,xi+1, . . . ,xq] + · · · + (−1)
i[x0, . . . ,xi,xi+1, . . . ,xq]

+ (−1)i+1[x0, . . . ,xi+1,xi+1, . . . ,xq] + (−1)
i+2[x0, . . . ,xi+1,xi, . . . ,xq]

+ (−1)q+1[x0, . . . ,xi+1,xi,xi+1, . . . ,xq−1]

Any term above listing fewer than k distinct points will be zero in Čq(X
k,Xk−1),

and so most of the terms above are zero. We are left with:

∂q+1σ = (−1)
i([x0, . . . ,xi,xi+1, . . . ,xq] + [x0, . . . ,xi+1,xi, . . . ,xq])

which establishes (6.2). �

Our main theorem requires a very weak finiteness condition on X. Any
finite digital image will satisfy the condition, as will any image X ⊂ Zn with ck-
adjacency for any k. The main work of the proof has already been accomplished
by Lemma 6.1, the proof below is simply a homological argument.

Theorem 6.2. Let (X,κ) be a digital image, and assume there is some dimen-
sion k for which X contains no simplicies of dimension greater than k. Then
for each q, we have Hq(X) ∼= Ȟq(X).

Proof. We prove the theorem by induction on q and k. For q = 0, we have
H0(X) ∼= Ȟ0(X), since by Theorems 4.5 and 4.8 these homology groups are
both Zd where d is the number of components of X.

When k = 0 then X simply consists of a set of isolated points, and so
Hq(X) ∼= Ȟq(X) by Theorems 4.6 and 4.9.

Now we consider the inductive case. Since X contains only simplices of
dimension k or less, we will have Cq(X) = Cq(X

k) and Čq(X) = Čq(X
k) for

all q. Thus Hq(X) = Hq(X
k) and Ȟq(X) = Ȟq(X

k) for all q, and it suffices to
show that the vertical arrow in the middle of (6.1) is an isomorphism.

By the Five Lemma (see [8]), it suffices to show that the other 4 verti-
cal arrows are isomorphisms. By induction in k, the second vertical arrow
Ȟq(X

k−1) → Hq(X
k−1) is an isomorphism, and by induction in k and q the

last vertical arrow Ȟq−1(X
k−1) → Hq−1(X

k−1) is an isomorphism. By Lemma
6.1, the first and fourth arrows are isomorphisms. �

The author’s original intention was to additionally prove a cubical version
of Theorem 6.2, that is, if X ⊂ Zn is a digital image with c1-adjacency, then
H̄q(X) ∼= H̄

c1
q (X) for each q.

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P. C. Staecker

This turned out to be much harder than anticipated: even constructing a
chain map β : C̄q(X) → C̄

c1
q (X) is difficult. We believe it should be possible,

but we will simply state it as a conjecture:

Conjecture 6.3. Let X ⊂ Zn be a digital image with c1-adjacency. Then H̄q(X)
and H̄c1q (X) are isomorphic.

It seems likely that the conjecture could be verified in low dimensions by
computer enumerations, but we have not attempted this.

7. Homotopy invariance in simplicial and cubical homology

In this section we discuss the homotopy invariance of the induced homomor-
phism on the various homology groups. In [9] it is shown that, when f and g
are homotopic, their induced maps on cubical homology groups are the same.

We will prove the same property holds for c1-cubical homology, and that
a similar result holds for simplicial homology, but in this case the homotopy
must be strong.

Theorem 7.1. If X is finite and f,g : X → Y are strongly homotopic, then
the induced homomorphisms f∗,g∗ : Hq(X) → Hq(Y ) are equal for each q.

Proof. By induction and Theorem 3.2, it suffices to prove the result when f
and g are homotopic by a punctuated homotopy in one step. We mimic the
proof for this result in classical homology theory, see for example Proposition
2.10 of [8]. For each q, we define the “prism operator” P : Cq(X) → Cq+1(Y )
as follows: For σ ∈ Cq(X) with σ = 〈x0, . . . ,xq〉, let:

P(σ) =

q∑

j=0

(−1)j〈f(x0), . . . ,f(xj),g(xj), . . . ,g(xq)〉,

where the term 〈f(x0), . . . ,f(xj),g(xj), . . . ,g(xq)〉 is interpreted as 0 if any of
these points are equal.

Note that the definition of P only makes sense if {f(x0), . . . ,f(xj),
g(xj), . . . ,g(xq)} is indeed a set of q + 2 points that are mutually adjacent
or equal. This is ensured because f and g are strongly homotopic in 1 step,
and thus by Theorem 2.4, since {x0, . . . ,xq} are mutually adjacent, the points
of {f(x0), . . . ,f(xq),g(x0), . . . ,g(xq)} are mutually adjacent or equal. This is
the point at which the proof will fail if the homotopy is not strong.

The bulk of the proof consists of proving the following formula:

(7.1) ∂(P(σ)) = g#(σ) − f#(σ) − P(∂σ).

Since f and g are homotopic in one step by punctuated homotopy, there
is some x′ ∈ X such that f(x) = g(x) for all x 6= x′. If σ = 〈x0, . . . ,xq〉
with xi 6= x

′ for all i, then f(xi) = g(xi) for each i. Thus P(σ) = 0, since
〈f(x0), . . . ,f(xj),g(xj), . . . ,g(xq)〉 will repeat the point f(xj) = g(xj). Thus,
whenever σ does not use x′, we have P(σ) = 0.

Formula (7.1) is easy to prove when σ does not use the vertex x′: in that
case P(σ) = 0 and thus the left side of (7.1) is 0. For the right side, note that

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Digital homotopy and homology

∂σ also does not use the point x′, and so we have P(∂σ) = 0. Also since σ
does not use x′, we will have f#(σ) = g#(σ), and thus the right side of (7.1)
is also 0, and we have proved (7.1).

Now we prove (7.1) in the case when σ does use the point x′. Without loss
of generality assume σ = 〈x′,x1, . . . ,xq〉. In this case we have

P(σ) = 〈f(x′),g(x′),g(x1), . . . ,g(xq)〉

+

q∑

j=1

(−1)j〈f(x′),f(x1), . . . ,f(xj),g(xj), . . . ,g(xq)〉

= 〈f(x′),g(x′),g(x1), . . . ,g(xq)〉

where most of the terms above are 0 because they repeat the point f(xj) =
g(xj). Then the left side of (7.1) is

∂(P(σ)) = ∂(〈f(x′),g(x′),g(x1), . . . ,g(xq)〉)

= 〈g(x′),g(x1), . . . ,g(xq)〉 − 〈f(x
′),g(x1), . . . ,g(xq)〉

+

q∑

i=1

(−1)i+1〈f(x′),g(x′),g(x1), . . . , ĝ(xi), . . . ,g(xq)〉

Since g(xi) = f(xi), the above simplifies to:

∂(P(σ)) = g#(σ)−f#(σ)+

q∑

i=1

(−1)i+1〈f(x′),g(x′),g(x1), . . . , ĝ(xi), . . . ,g(xq)〉.

To prove (7.1), it suffices to show that the summation above equals −P(∂(σ)).
We have:

P(∂(σ)) = P

(
〈x1, . . . ,xq〉 +

q∑

i=1

(−1)i〈x′,x1, . . . , x̂i, . . . ,xq〉

)

= P

(
q∑

i=1

(−1)i〈x′,x1, . . . , x̂i, . . . ,xq〉

)

where P(〈x1, . . . ,xq〉) = 0 since this simplex does not use x
′. Now when we

apply P above, the only nonzero terms are those with j = 0 in the definition
of P . All others will repeat some point f(xi) = g(xi). Thus we have:

P(∂(σ)) =

q∑

i=1

(−1)i〈f(x′),g(x′),g(x1), . . . , ĝ(xi), . . . ,g(xn)〉

= −

q∑

i=1

(−1)i+1〈f(x′),g(x′),g(x1), . . . , ĝ(xi), . . . ,g(xn)〉

and we have proved (7.1).
The formula (7.1) holds when σ is any simplex, and so by linearity it will

hold for any chain. Now let α ∈ Zq(X) be a q-cycle, so ∂(α) = 0 and thus

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P. C. Staecker

P(∂(α)) = 0. Then by (7.1) we have:

g#(α) − f#(α) = ∂P(α) ∈ Bq(Y ).

Thus g#(α) and f#(α) differ by a q-boundary, that is, g∗(α) = f∗(α) ∈ Hq(Y ).
�

Now we prove a homotopy invariance property for c1-cubical homology.
Since the result concerns the induced homomorphism in c1-cubical homology,
we must require that f# : C̄

c1
q (X) → C̄

c1
q (Y ) is a chain map. Barring a proof

of Conjecture 5.4, we can only demonstrate the theorem in low dimensions.

Theorem 7.2. Let X ⊆ Zm and Y ⊆ Zn be digital images both with c1-
adjacency, with m ≤ 3, and let f,g : X → Y be homotopic. Then the induced
homomorphisms f̄∗, ḡ∗ : H̄

c1
q (X) → H̄

c1
q (Y ) are equal for each q. If Conjecture

5.4 is true, then this holds for any m.

Proof. As in the proof of Theorem 7.1, we may assume that f is homotopic
to g in 1 step, that is, that f(x) - g(x) for each x. By the same homological
argument at the end of the proof of Theorem 7.1, it will suffice to define a
homomorphism P : C̄c1q (X) → C̄

c1
q+1(Y ) which satisfies:

(7.2) ∂P(Q) = ḡ#(Q) − f̄#(Q) − P(∂Q).

For an elementary q-cube Q ⊂ X, we define P(Q) ∈ C̄c1q+1(Y ) as follows: if

the set f(Q) ∪g(Q) is an elementary (q + 1)-cube, then P(Q) = f(Q) ∪g(Q) ∈
C̄c1q+1(Y ). Otherwise we define P(Q) = 0. Extending the definition linearly

defines a homomorphism P : C̄c1q (X) → C̄
c1
q+1(Y ).

Let T : I × X → Y be defined as T(0,x) = f(x) and T(1,x) = g(x). Since
f(x) - g(x) for each x, this function T is continuous (using NP1 adjacency in
the product I×X). The induced homomorphism T̄# : Cq+1(I×X) → Cq+1(Y )
is closely related to P . In particular P(Q) = T̄#(I × Q), and when i > 1 we
have:

T̄#(Ai(I × Q)) = P(Ai−1Q), T̄#(Bi(I × Q)) = P(Bi−1Q).

When i = 1, we instead have T̄#(A1(I × Q)) = T̄#({0} × Q) = f#(Q) and
similarly T̄#(B1(I × Q) = g#(Q).

Since X ⊂ Z3 we may consider I × X ⊂ Z4, and so by Theorem 5.3 we will
have T̄#(∂σ) = ∂T̄#(σ) for any chain σ. Thus we have:

∂P(Q) = ∂T̄#(I × Q) = T̄#(∂(I × Q))

=

q+1∑

i=1

(−1)i(T#(Ai(I × Q)) − T#(Bi(I × Q)))

= (g#(Q) − f#(Q)) +

(
q+1∑

i=2

(−1)i(P(Ai−1Q) − P(Bi−1Q))

)

= g#(Q) − f#(Q) − P(∂Q)

which establishes (7.2). �

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Digital homotopy and homology

Theorem 7.2 will imply that, if X and Y are homotopy equivalent, then they
have the same c1-cubical homology groups. We immediately obtain:

Corollary 7.3. If X ⊂ Zn is a contractible digital image with c1-adjacency,
and assume n ≤ 3 or that Conjecture 5.4 is true. Then:

H̄c1q (X) =

{
Z if q = 0,

0 if q > 0.

8. Relationships between cubical and simplicial homology

So far we have exhibited four homology theories: simplicial, singular, cubical,
and c1-cubical. The simplicial and singular theories are isomorphic, and we
have conjectured that the two cubical theories are isomorphic. Thus, subject
to the conjecture, there are two different types of homology under discussion:
simplicial and cubical. The simplicial and cubical theories are not isomorphic,
as we will see in the next section. In this section we consider some relationships
that do exist between the simplicial and cubical theories.

We have already seen results that imply that the simplicial and cubical
theories are the same in dimension 0. Theorems 4.5, 4.14, and 5.1 give:

Theorem 8.1. Let X be any digital image. Then:

H0(X) = H̄0(X) = Z
d

where d > 0 is the number of connected components of X. If X ⊂ Zn with
c1-adjacency, then also H̄

c1
0 (X) = Z

d.

In dimension 1, there is also a relationship between simplicial and cubical
homology. The chain group Č1(X) is generated by the set of all maps φ : ∆

1 →
X, while C̄1(X) is generated by the set of all maps σ : I

1 → X. But ∆1 and I1

are the same digital image– so we may identify chain groups Č1(X) = C̄1(X),
and the singular and cubical boundary maps ∂1 are the same. Thus we may
also identify the groups of cycles Ž1(X) = Z̄1(X).

This identification induces a natural homomorphism Ȟ1(X) → H̄1(X) which
simply regards a singular 1-cycle as a cubical 1-cycle.

In exactly the same way, when X ⊂ Zn with c1-adjacency there is a natural
homomorphism H1(X) → H̄

c1
1 (X) induced by the chain map which regards

each 1-simplex as an elementary 1-cube.

Theorem 8.2. For any digital image X, the natural map Ȟ1(X) → H̄1(X) is
surjective. When X ⊂ Zn with c1-adjacency then the map H1(X) → H̄

c1
1 (X)

is surjective.

Proof. We have Ȟ1(X) = Ž1(X)/B̌1(X) and H̄1(X) = Z̄1(X)/B̄1(X). Thus
it suffices to show that, when we identify Ž1(X) = Z̄1(X), we have B̌1(X) ≤
B̄1(X).

For clarity, write the singular and cubical boundary operators as ∂̌2 : Č2(X) →
Č1(X) and ∂̄1 : C̄2(X) → C̄1(X). When φ = [x0,x1,x2] is a singular 2-simplex,

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P. C. Staecker

then define L(φ) = σ to be the following 2-cube:

σ(0,0) = x0 σ(0,1) = x1

σ(1,0) = x0 σ(1,1) = x2

and L extends to a map L : Č2(X) → C̄2(X). It is routine to check that
∂̌2(φ) = ∂̄2(L(φ)) for any φ ∈ Č2(X) when we identify Č1(X) = C̄1(X). Thus
we will have B̌1(X) ≤ B̄1(X) as desired.

Now when X ⊂ Zn with c1-adjacency, similarly we must show that B1(X) ≤
B̄c11 (X) but this is obvious because C2(X) is trivial since a digital image with
c1-adjacency contains no 2-simplex. Thus B1(X) = ∂(C2(X)) is trivial and so
is automatically a subgroup. �

Since Ȟ1(X) ∼= H1(X), the above implies that there is always a surjection
H1(X) → H̄1(X). We will see in the next section that H1(X) need not be
isomorphic to H̄1(X), and that there need not be any surjection Hq(X) →
H̄q(X) when q > 1.

9. Examples

In this section we provide some examples comparing the simplicial and cu-
bical homology groups of various images. Most of the examples for simplicial
homology appear already in the literature, but no examples for c1-cubical ho-
mology have been computed.

The simplest examples are the simplicial and cubical homology groups for
single points. We have already seen this result as Theorems 4.6, 4.15, and 5.2.

Theorem 9.1. Let X be an image consisting of d isolated points. Then:

Hq(X) = H̄q(X) = H̄
c1
q (X) =

{
Zd if q = 0,

0 if q > 0.

Now we describe some more interesting examples. A fruitful source of exam-
ple digital images is the digital cycle Cm, the digital image consisting of n points
x1, . . . ,xm where xi ↔ xj only when j = i ± 1, where we read i and j modulo
n. As we will see, the simplicial and cubical homologies for digital cycles (as far
as we can compute them) agree in all cases except H1(C4) = Z 6= 0 = H̄

c1
1 (C4).

We will use a lemma which is straightforward but interesting. For two digital
images (X,κ) and (Y,λ), we say X embeds in Y if X is (κ,λ)-isomorphic to a
subset of Y .

Lemma 9.2. The digital cycle Cm embeds in (Z
n,c1) for some n if and only

if m is even.

Proof. When we view (Zn,c1) as a graph, it is bipartite, with the two parts
given by:

S+ = {(x1, . . . ,xn) | x1 + · · · + xn is even},

S− = {(x1, . . . ,xn) | x1 + · · · + xn is odd}.

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Digital homotopy and homology

Thus Cm can only embed into (Z
n,c1) if it too is bipartite, and this is only

possible when m is even. We have shown that if m embeds in (Zn,c1), then m
is even.

For the converse, let m > 0 be even and we will exhibit a subset of (Zn,c1) for
some n isomorphic to Cm. When m = 2, then we may simply use [0,1]Z ⊂ Z

1,
which is isomorphic to C2. When m = 4, then we may use [0,1]

2 ⊂ Z2, which
is isomorphic to C4. When m = 6, we observe that C6 is isomorphic to the
following subset of Z3, using c1-adjacency:

{(0,0,0),(1,0,0),(1,1,0),(1,1,1),(0,1,1),(0,0,1)}

Finally when m > 6 is even, the cycle Cm is isomorphic to the following
subset of Z2:

([0,m/2 − 1] × {0,2}) ∪ {(0,1),(m/2 − 1,1)}. �

Now we can compute the various homology groups of Cm. Whenever possible
(when m is even), we will consider Cm as embedded in Z

n with c1-adjacency.

Theorem 9.3. For m < 4, we have:

Hq(Cm) = H̄q(Cm) = H̄
c1
q (C2) =

{
Z if q = 0,

0 if q > 0.

For m = 4, we have:

Hq(C4) =





Z if q = 0,

Z if q = 1,

0 if q > 1,

H̄q(C4) = H̄
c1
q (C4) =

{
Z if q = 0,

0 if q > 0.

For m > 4 we have:

Hq(Cm) =





Z if q = 0,

Z if q = 1,

0 if q > 1,

H̄q(Cm) =

{
Z if q = 0,

Z if q = 1.

When m > 4 is even, we have Hq(Cm) = H̄
c1
q (Cm).

Proof. The groups Hq(Cm) were already computed fully in Theorem 4.1, so we
need only prove the statements concerning H̄q(Cm) and H̄

c1
q (Cm).

For H̄q(Cm), we see that H̄0(Cm) = Z by Theorem 8.1, and H̄q(C4) = 0
for q > 0 by Corollary 4.13 since C4 is contractible. In fact Cm is contractible
for all m ≤ 4, and so we have H̄q(Cm) = 0 for q > 1 and m ≤ 4. For m > 4
we have H̄1(Cm) ∼= Z by the Hurewicz Theorem of [9], since π1(Cm) = Z for
m > 4.

For H̄c1q (C2), the required statement is clear because C2 is connected with
no elementary cubical q-cycles for any q > 0.

For H̄c1q (C4), the statement for q = 0 follows from Theorem 8.1. For q = 1
the group of elementary cubical 1-cycles has a single generator, but it is the
boundary of an elementary 2-cube, and so H̄c11 (C4) = 0. For q > 1 there are
no elementary q-cycles, and so H̄c1q (C4) = 0 when q > 1.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 247



P. C. Staecker

It remains to compute H̄c1q (Cm) when m > 4 is even. Since Cm is connected

we have H̄c10 (Cm) = Z by Theorem 8.1. The group of elementary 1-cycles has a
single generator which is not the boundary of any elementary cubical 2-chain,
and so H̄c11 (Cm) = Z. For q > 1 there are no elementary cubical q-cycles, and
so H̄c1q (Cm) = 0 as desired. �

It is obviously to be expected that

Hq(Cm) = H̄q(Cm) =





Z if q = 0,

Z if q = 1,

0 if q > 1,

for all values m > 4 including odd values, though this seems difficult to prove.
The definitions of cubical homology make even the group H̄2(C5) very hard to
compute by hand. So we will state this as a conjecture:

Conjecture 9.4. For m > 4, we have:

Hq(Cm) = H̄q(Cm) =





Z if q = 0,

Z if q = 1,

0 if q > 1,

One example where the simplicial and cubical homologies are quite different
is the standard 3-cube I3 = [0,1]3

Z
taken with c1-adjacency.

Example 9.5. Consider the digital image I3 with c1-adjacency. We have:

Hq(I
3) =





Z if q = 0,

Z5 if q = 1,

Z if q = 2,

0 if q > 2,

H̄q(I
3) = H̄c1q (I

3) =

{
Z if q = 0,

0 if q > 0.

Proof. The computation of the simplicial homology was done in Theorem 3.20
of [4], where the image in question is called MSS′6. The appearance of Z

5 as
the homology group in dimension 1 is surprising. The generators of H1(I

3) are
formed by making 1-cycles around each of the 6 faces of the cube. This produces
six 1-cycles which are not boundaries, but they are not linearly independent–
each one can be obtained by a combination of the other 5, and thus we have
only 5 linearly independent 1-cycles. The details of the computation are given
in [4].

In the case of cubical homology, each of these 1-cycles is indeed a boundary
of the 2-cube which makes the corresponding face of the cube. Indeed I3

is contractible, and so the computations of H̄q(I
3) and H̄c1q (I

3) follow from
Corollary 4.13 and 7.3. �

Example 9.6. Let X = [0,2]3
Z
− {(1,1,1)}, taken with c1-adjacency. This

digital image is called MSS6 in [4], though its homology is not computed,
presumably because H1(X) is very large. By Theorem 8.1 we have H0(X) =
H̄c10 (X) = Z.

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Digital homotopy and homology

In dimension 1 there are 24 different 1-cycles tracing around unit squares.
In H̄c11 (X) these are all trivial since these cycles are boundaries of 2-cubes. In
H1(X) these are all nontrivial, although as in I

3 one of these cycles can be
written as a sum of the other 23. The computer implementations confirm that
H1(X) ∼= Z

23 and H̄c11 (X)
∼= 0.

In dimension 2 there are no 2-simplices, so H2(X) = 0. The above mentioned
24 unit squares, when taken together, form a 2-cycle in Z̄c12 (X) which is not
the boundary of any 3-cycle, and thus H̄c12 (X) 6= 0. This example is tractable
for our computer implementation, which confirms that H̄c12 (X) = Z.

For q > 2 there are no q-simplices or elementary q-cubes, so Hq(X) =
H̄c1q (X) = 0. In summary, we have:

Hq(X) =





Z if q = 0

Z23 if q = 1

0 if q > 1

H̄c1q (X) =





Z if q = 0

0 if q = 1

Z if q = 2

0 if q > 2

Note that H2(X) = 0, while H̄
c1
2 (X)

∼= Z. Thus the example demonstrates
that, in contrast with Theorem 8.2, there is not always a surjective homomor-
phism of Hq(X) → H̄

c1
q (X) when q > 1.

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