Digital k-Contractibility of an n-Times Iterated Connected Sum of Simple Closed k-Surfaces and Almost Fixed Point Property

The paper firstly establishes the so-called n-times iterated connected sum of a simple closed k-surface in Z 3 , denoted by C k n , k ∈ { 6 , 18 , 26 } . Secondly, for a simple closed 18-surface M S S 18 , we prove that there are only two types of connected sums of it up to 18-isomorphism. Besides, given a simple closed 6-surface M S S 6 , we prove that only one type of M S S 6 ♯ M S S 6 exists up to 6-isomorphism, where ♯ means the digital connected sum operator. Thirdly, we prove the digital k-contractibility of C k n : = M S S k ♯ ⋯ ♯ M S S k ︷ n - times , k ∈ { 18 , 26 } , which leads to the simply k-connectedness of C k n , k ∈ { 18 , 26 } , n ∈ N . Fourthly, we prove that C 6 2 and C k n do not have the almost fixed point property (AFPP, for short), k ∈ { 18 , 26 } . Finally, assume a closed k-surface S k ( ⊂ Z 3 ) which is ( k , k ¯ ) -isomorphic to ( X , k ) in the picture ( Z 3 , k , k ¯ , X ) and the set X is symmetric according to each of x y -, y z -, and x z -planes of R 3 . Then we prove that S k does not have the AFPP. In this paper given a digital image ( X , k ) is assumed to be k-connected and its cardinality | X | ≥ 2 .


Introduction
In Z 3 , the concept of closed k-surface was introduced in [1-3] and its digital topological characterizations were also studied in many papers including [4][5][6][7][8][9][10]. Many explorations of various properties of closed k-surfaces have been proceeded from the viewpoints of digital topology, digital geometry, and fixed point theory [1,2,[4][5][6][9][10][11][12][13][14][15][16]. Despite the studies of the earlier works [5][6][7]17,18], given (digital) closed k-surfaces, we need to further study both the digital k-contractibility of n-times iterated connected sums of closed k-surfaces and the non-almost fixed point property of them. Besides, we need to find a condition determining if a digital image (X, k) in Z n has the AFPP. This approach facilitates the studies of digital geometry and fixed point theory.
So far, there were several kinds of approaches to establish a digital k-surface [3,[5][6][7]9]. In the present paper we will often use the symbol " :=" to define a new term, and given a digital image (X, k) is assumed to be k-connected and its cardinality | X | ≥ 2. Since the digital surface theory is related to computer science, the present paper mainly deals with digital k-surfaces X in Z 3 . Hence, we need to consider a binary digital image structure (X, k,k) in Z 3 , denoted by P := (Z 3 , k,k, X), where the k-adjacency is concerned with the set Z 3 \ X. To be precise, in the case of the study of a closed k-surface X ⊂ Z 3 , we should assume X in the binary digital picture P. For instance, P ∈ {(Z 3 , k,k, X) | (k,k) ∈ { (6,26), (18,6), (26,6)}}. (1) Let us now study a (digital) closed k-surface X with one of the above frames P of (1). Given two closed k-surfaces S k and S k in Z n , the concept of digital connected sum of them was firstly introduced in [5,7] by using several types of simple closed k-curves in Z 2 , k ∈ {4, 8} (see Section 4). Hereafter, we denote by S k a (simple) closed k-surface in Z 3 (for the details, see Definition 5). Indeed, when studying various properties of closed k-surfaces, some digital k-homotopic features of S k such as the k-contractibility are very important in digital surface theory.
For convenience, let MSS 6 (resp. MSS 18 ) be the minimal simple closed 6-surface (resp. the minimal simple closed 18-surface) [6]. The present paper deals with the following queries.
(Q1) We may ask if it is possible to propose the simple closed 6-surface MSS 6 in the picture (Z 3 , 6, 18, MSS 6 ) instead of (Z 3 , 6, 26, MSS 6 ). Hereafter, the operator " " means the digital connected sum (see Section 4 for the details). (Q2) How many types of MSS 6 MSS 6 exist ? Let C n 6 := n-times MSS 6 · · · MSS 6 . Then we have the following queries: (Q3) How can we formulate C n 6 , n ∈ N \ {1} ? Given an MSS 18 , we may raise the following query. The rest of the paper is organized as follows: Section 2 refers to some notions involving a digital k-surface and a connected sum of two digital k-surfaces. Section 3 stresses some utilities of the minimal simple closed surfaces MSS 6 , MSS 18 , MSS 18 , and MSS 26 from the viewpoints of digital curve and digital surface theory. Section 4 shows several types of n-times iterated connected sums of the minimal simple closed 6-surfaces, e.g., C 3 6 := MSS 6 MSS 6 MSS 6 . Section 5 proves that there are only two types of connected sums MSS 18 MSS 18 up to 18-isomorphism. Besides, in the case of MSS 18 MSS 18 = MSS 18 , we prove that only one type of C 3 18 := MSS 18 MSS 18 MSS 18 exists up to 18-isomorphism. Section 6 intensively explores the 18-contractibility of an n-times iterated connected sum of simple closed 18-surfaces C n 18 := n-times MSS 18 · · · MSS 18 . Section 7 proves that both C 2 6 and C n k do not have the almost fixed point property, k ∈ {18, 26}, n ∈ N. Thus, these approaches play important roles in digital topology, digital geometry, fixed point theory, and so on. Section 8 concludes the paper with some remarks.

Basic Notions Involving Digital k-Surfaces and Connected Sums of Closed k-Surfaces
Let us now recall some terminology from digital curve and digital surface theories. Let N and Z represent the sets of natural numbers and integers, respectively.
We call a set X(⊂ Z n ) with a k-adjacency a digital image, denoted by (X, k) [4,5,7,9,10]. In particular, in digital surface theory, we are absolutely required to consider a closed k-surface (X, k) with a k-adjacency in a binary digital picture (Z n , k,k, X) [19,20], where n ∈ N and thek-adjacency is concerned with the set Z n \ X. In order to study (X, k) in Z n , n ≥ 1, we need the k-adjacency relations of Z n which are generalizations of the commonly used k-adjacency of Z 2 , k ∈ {4, 8}, and k-adjacency of Z 3 , k ∈ {6, 18, 26}. As a generalization of this approach into those of Z n , a paper [17] firstly established the digital k-connectivity of Z n , as follows: We say that distinct points p, q ∈ Z n are k-(or k(t, n)-)adjacent if they satisfy the following property [17] (for the details, see also [21,22]). For a natural number t, 1 ≤ t ≤ n, we say that distinct points p = (p 1 , p 2 , · · · , p n ) and q = (q 1 , q 2 , · · · , q n ) ∈ Z n , are k(t, n)-(k-, for short)adjacent if (2) at most t of their coordinates differs by ± 1, and all the others coincide.
A digital image (X, k) in Z n can indeed be considered to be a set X(⊂ Z n ) with one of the k-adjacency relations of (3). Using the k-adjacency relations of Z n of (3), we say that a digital k-neighborhood of p in Z n is the set [20] N k (p) := {q | p is k-adjacent to q} ∪ {p}.
Furthermore, we often use the notation [19] For a, b ∈ Z with a b, the set [a, b] Z = {n ∈ Z | a ≤ n ≤ b} with 2-adjacency is called a digital interval [19]. Let us now recall some terminology and notions [17,19] which are used in this paper.
• It is natural to say that a digital image (X, k) is k-disconnected if there are nonempty sets X 1 , X 2 ⊂ X such that X = X 1 ∪ X 2 , X 1 ∩ X 2 = ∅ and further, there are no points x 1 ∈ X 1 and x 2 ∈ X 2 such that x 1 and x 2 are k-adjacent.
Owing to this approach, we see that a singleton subset of (X, k) is obviously k-connected. • Given a k-connected digital image (X, k) whose cardinality is greater than 1, the so-called k-path with l + 1 elements in Z n is assumed to be a finite sequence (x i ) i∈[0,l] Z ⊂ Z n such that x i and x j are k-adjacent if | i − j | = 1 [19]. Eventually, in the case that a digital image (X, k) is k-connected, for any distinct points such as x, y in (X, k), we see that there is a k-path (x i ) i∈[0,l] Z ⊂ X such that x = x 0 and y = x l . • For a digital image (X, k), the k-component of x ∈ X is defined to be the maximal k-connected subset of (X, k) containing the point x [19]. • We say that a simple k-path means a finite set (x i ) i∈[0,m] Z ⊂ Z n such that x i and x j are k-adjacent if and only if | i − j | = 1 [19]. In the case of x 0 = x and x m = y, we denote the length of the simple k-path with l k (x, y) := m.
• A simple closed k-curve (or simple k-cycle) with l elements in Z n , denoted by SC n,l k [17,19], l ≥ 4, l ∈ N 0 \ {2}, N 0 is the set of even natural numbers, means the finite set (x i ) i∈[0,l−1] Z ⊂ Z n such that x i and x j are k-adjacent if and only if | i − j | = ±1(mod l).
• For a digital image (X, k), a digital k-neighborhood of x 0 ∈ X with radius ε is defined in X as the following subset [17] of X where l k (x 0 , x) is the length of a shortest simple k-path from x 0 to x and ε ∈ N. For instance, for X ⊂ Z n , we obtain [17] For a digital image (X, k), since X is a subset of Z n , if it is assumed as a subspace of the typical n-dimensional Euclidean topological space, it can naturally be a discrete topological subspace. However, as mentioned above, since a digital image (X, k) with the digital k-connectivity (see (3)) is a kind of a digital graph in Z n , the paper [17] already established another metric for (X, k). Eventually, the sets of (4) and (5) can be represented by using this metric on X derived from (X, k). The important thing is that this metric is different from the typical Euclidean metric. Indeed, a paper [17] firstly established the metric using the "length of a shortest simple k-path from x 0 to x" for two points x 0 , x in (X, k). Owing to the length of a shortest k-path in (4), we prove that a k-connected digital image (X, k) can be considered to be a metric space, as follows: Let us consider the map d k on a k-connected (or k-path connected) digital image (X, k) defined by Owing to (6), we can see that d k (x, x ) ≥ 1 if x = x and further, we obviously see that the function d k satisfies the metric axioms. Thus, we can represent the set N k (x 0 , ε) of (4) in the following way Consequently, we can represent the set of (5), as follows: Rosenfeld [23] defined the notion of digital continuity of a map f : (X, k 0 ) → (Y, k 1 ) by saying that f maps every k 0 -connected subset of (X, k 0 ) into a k 1 -connected subset of (Y, k 1 ).
Motivated by this approach, using the set of (5) or (8), we can represent the digital continuity of a map between digital images by using a digital k-neighborhood (see Proposition 1 below). Due to this approach, we have strong advantages of calculating digital fundamental groups of digital images (X, k) in terms of the unique digital lifting theorem [17], the digital homotopy lifting theorem [24], a radius 2-(k 0 , k 1 )-isomorphism and its applications [24], the study of multiplicative properties for a digital fundamental group [25,26], a Cartesian product of the covering spaces [26], and so on, as follows: Proposition 1. [17,18] Let (X, k 0 ) and (Y, k 1 ) be digital images in Z n 0 and Z n 1 , respectively. A function f : (X, k 0 ) → (Y, k 1 ) is (digitally) (k 0 , k 1 )-continuous if and only if for every x ∈ X f (N k 0 (x, 1)) ⊂ N k 1 ( f (x), 1).
In Proposition 1, in the case n 0 = n 1 and k := k 0 = k 1 , the map f is called a 'k-continuous' map. Since an n-dimensional digital image (X, k) is considered to be a set X in Z n with one of the k-adjacency relations of (3) (or a digital k-graph [27]), regarding a classification of n-dimensional digital images, we prefer the term a (k 0 , k 1 )-isomorphism (or k-isomorphism) as in [27] (see also [18]) to a (k 0 , k 1 )-homeomorphism (or k-homeomorphism) as in [28]. Definition 1. [27] (see also a (k 0 , k 1 )-homeomorphism in [28]) Consider two digital images (X, k 0 ) and (Y, k 1 ) in Z n 0 and Z n 1 , respectively. Then a map h : X → Y is called a (k 0 , k 1 )-isomorphism if h is a (k 0 , k 1 )-continuous bijection and further, h −1 : Y → X is (k 1 , k 0 )-continuous. Then we use the notation X ≈ (k 0 ,k 1 ) Y. Besides, in the case k := k 0 = k 1 , we use the notation X ≈ k Y.
The following notion of interior is often used in establishing a digital connected sum of digital closed k-surfaces.

Definition 2.
[5] Let c * := (x 0 , x 1 , · · · , x n ) be a closed k-curve in (Z 2 , k,k, c * ). A point x of c * , the complement of c * in Z 2 , is said to be interior to c * if it belongs to the boundedk-connected component of c * .
The following digital images MSC * 8 , MSC * 4 , and MSC * 8 in Z 2 [5,6,17] have essentially been used in establishing a connected sum and studying the digital fundamental group of a digital connected sum of closed k-surfaces. Thus we now recall them.
Based on the pointed digital homotopy in [29] (see also [28]), the following notion of k-homotopy relative to a subset A ⊂ X is often used in studying k-homotopic properties of digital images (X, k) in Z n . For a digital image (X, k) and A ⊂ X, we often call ((X, A), k) a digital image pair. Definition 3. [17,24,28] Let ((X, A), k 0 ) and (Y, k 1 ) be a digital image pair and a digital image in Z n 0 and Z n 1 , respectively. Let f , g : X → Y be (k 0 , k 1 )-continuous functions. Suppose there exist m ∈ N and a function H : Then we say that H is a (k 0 , k 1 )-homotopy between f and g [28]. • Furthermore, for all t ∈ [0, m] Z , assume that the induced map H t on A is a constant which follows the prescribed function from A to Y [17] (see also [5]). To be precise, Then we call H a (k 0 , k 1 )-homotopy relative to A between f and g, and we say that f and g are (k 0 , k 1 )-homotopic relative to A in Y, f (k 0 ,k 1 )rel.A g in symbols [17].
In Definition 3, if a k-continuous map f : X → X is k-homotopic to a certain constant map c {x 0 } , x 0 ∈ X, then we say that f is (pointed) k-null homotopic in (X, k) [28]. In Definition 3, if A = {x 0 } ⊂ X, then we say that F is a pointed (k 0 , k 1 )-homotopy at {x 0 } [28]. When f and g are pointed (k 0 , k 1 )-homotopic in Y, we use the notation f (k 0 ,k 1 ) g. In the case k := k 0 = k 1 and n 0 = n 1 , f and g are said to be pointed k-homotopic in Y and we use the notation f k g and f ∈ [g] which denotes the k-homotopy class of g. If, for some x 0 ∈ X, 1 X is k-homotopic to the constant map in the space X relative to {x 0 }, then we say that (X, x 0 ) is pointed k-contractible [28]. Indeed, motivated by this approach, the notion of strong k-deformation retract was developed in [30].
Based on this k-homotopy, the notion of digital homotopy equivalence was firstly introduced in [31] (see also [32]), as follows: Definition 4. [31] (see also [32]) For two digital images (X, k) and (Y, k) in Z n , if there are k-continuous maps h : X → Y and l : Y → X such that the composite l • h is k-homotopic to 1 X and the composite h • l is k-homotopic to 1 Y , then the map h : X → Y is called a k-homotopy equivalence and is denoted by X k·h·e Y. Besides, we say that (X, k) is k-homotopy equivalent to (Y, k). In the case that the identity map 1 X is k-homotopy equivalent to a certain constant map c {x 0 } , x 0 ∈ X, we say that (X, k) is k-contractible.
In Definition 4, in the case X k·h·e Y, we say that (X, k) is the same k-homotopy type as (Y, k). In view of Definitions 3 and 4, we obviously see that the pointed k-contractibility implies the k-contractibility, the converse does not hold. Let (X, k) be k-contractible. Then it is obvious that any The digital k-fundamental group is induced from the pointed k-homotopy [28]. For a given digital image (X, k), by using several notions such as digital k-homotopy class [29], Khalimsky operation of two k-homotopy classes [29], trivial extension [28], the paper [28] defined the digital k-fundamental group, denoted by π k (X, x 0 ), x 0 ∈ X. Indeed, in digital topology there are several kinds of digital fundamental groups [33]. In addition, we have the following: If X is pointed k-contractible, then π k (X, x 0 ) is a trivial group [28]. Hereafter, we shall assume that each digital image (X, k) is k-connected.
Using the unique digital lifting theorem [17] and the homotopy lifting theorem [24] in digital covering theory [4,17,18,25,26], for a non-k-contractible space SC n,l k , we obtain the following: For a non-k-contractible SC n,l k , π k 1 (SC n,l k ) is an infinite cyclic group.
Namely, for an SC n,l k , l ≥ 6, it turns out that π k 1 (SC n,l k ) is an infinite cyclic group. Regarding Theorem 1, we see that SC n, 4 3 n −1 has the trivial group, n ≥ 2 [24,28] and further, SC 2,4 4 also has the trivial group because SC 2,4 4 is 4-contractible (see a certain idea from Example 1 below).
The following are proven in [5,7,17,18,28]. • Due to Theorem 1, it turns out that SC n,l k is not k-contractible if l ≥ 6.
In particular, both the non-8-contractibility of MSC 8 and the non-4-contractibility of MSC 4 play important roles in formulating a connected sum of two closed k-surfaces (see Section 4 for the details).
Hereafter, we denote the n-dimensional digital cube (or digital n-cube) with Based on the 6-contractibility of (I 3 , 6) (see [34]), using a similar method as the proof of it (see Remark 2 of [8]), it is obvious that (I n , k) is pointed k-contractible for any k-adjacency of Z n , where the k-adjacency is that of (3) according to the dimension "n".
Let us now examine if a k-isomorphism preserves a k-homotopy between two k-continuous maps.

Theorem 2.
A k-isomorphism preserves a k-homotopy.
Proof. Given two spaces X := (X, k), Y := (Y, k) in Z n , consider two k-continuous functions f , g : e., f k g. Besides, further assume two are also k-continuous maps from X to Y . Based on the given k-homotopy and the two k-isomorphisms h 1 and h 2 , we now define the new map Then, we obtain the following:
Proof. In Theorem 2, consider a k-contractible space (X, k) such that X k·h·e {x 0 } for some point x 0 ∈ X. Then, after replacing f (resp. g) by 1 X (resp. the constant map c {x 0 } ), we prove the assertion.
Proof. In Theorem 2 and Corollary 1, consider a pointed k-contractible space (X, k) such that 1 X is k-homotopic to the constant map in the space {x 0 } relative to {x 0 }. After replacing f (resp. g) with 1 X (resp. the constant map c {x 0 } ), we complete the proof.
Using a method similar to the proof of Theorem 2, we obtain the following: This section stresses some utilities of the minimal simple closed 6-, 18-, 26-surfaces, e.g., MSS 6 , MSS 18 , MSS 18 , MSS 26 [6] from the viewpoints of digital surface and digital homotopy theory. Indeed, these models for simple closed k-surfaces play important roles in digital homotopy theory, digital surface theory, and fixed point theory. Furthermore, these have been used in formulating connected sums of some simple closed k-surfaces, k ∈ {6, 18, 26} [5][6][7]. Besides, these were essentially used in proceeding with geometric realizations of digital k-surfaces [7,8].
In order to study closed k-surfaces in Z n , let us recall some terminology from digital surface theory, as follows: A point x ∈ (X, k) is called a k-corner if x is k-adjacent to two and only two points y, z ∈ X such that y and z are k-adjacent to each other [2]. The k-corner x is called simple if y, z are not k-corners and if x is the only point k-adjacent to both y, z. (X, k) is called a generalized simple closed k-curve if what is obtained by removing all simple k-corners of X is a simple closed k-curve [2,9]. For a k-connected digital image (X, k) in X ⊂ Z 3 , we recall [1,2,6] In general, for a k-connected digital image (X, k) in Z n , n ≥ 3, we can state [7] |X| x := N 3 n −1 (x, 1) \ {x}.
We say that two subsets, (A, k) and (B, k) of (X, k), are k-adjacent if A ∩ B = ∅ and there are points a ∈ A and b ∈ B such that a and b are k-adjacent [19]. In particular, in the case that B is a singleton, say B = {x}, we say that A is k-adjacent to x.
Papers [5][6][7] introduced the notion of a closed k-surface in Z n , n ≥ 3 and various properties of it. However, in the present paper, we will stress the study of closed k-surfaces in Z 3 with the following approach in [3,9,10]. Definition 5. [3,10] Let (X, k) be a digital image in Z 3 , and X := Z 3 \ X. Then, X is called a closed k-surface if it satisfies the following.
Furthermore, if a closed k-surface X does not have a simple k-point, then X is called simple. (2) In the case (k,k) = (18, 6), (a) X is k-connected, (b) for each point x ∈ X, |X| x is a generalized simple closed k-curve. Furthermore, if the image |X| x is a simple closed k-curve, then the closed k-surface X is called simple.

Proposition 2. If given a digital image (X, k) is not k-connected, then it is not k-contractible.
Proof. Owing to the second property of Definition 3, the assertion is proved.
• (Correction) In the Figure 4c of [35], the given K-topological space (Z, κ 2 Z ) should be referred to as "non-K-retractible" instead of "K-retractible". From now on we denote a (simple) closed k-surface in Z 3 with S k , k ∈ {6, 18, 26}, which will be used in this paper. In particular, we will mainly consider an S k , k ∈ {6, 18, 26} in the picture as referred to in (1)
respectively. • Identify A k 0 \ A k 0 and f (A k 0 \ A k 0 ) by using the (k 0 , k 1 )-isomorphism f . Then, the quotient space Owing to Definition 6, S k 0 S k 1 is obtained in Z 3 . Besides, the digital topological type of S k 0 S k 1 absolutely depends on the choice of the subset A k 0 ⊂ S k 0 [7]. Furthermore, the k-adjacency of S k 0 S k 1 is required as follows: Remark 4. [5] In the quotient space S k 0 S k 1 := S k 0 ∪ S k 1 / ∼, the subsets A := S k 0 \ (A k 0 \ A k 0 ) and B := S k 1 \ f (A k 0 \ A k 0 ) in S k 0 S k 1 are assumed to be disjoint and there are no points x ∈ A and x ∈ B such that x and x are k-adjacent, where k := k 0 = k 1 . Then, the digital image (S k 0 S k 1 , k) is called a (digital) connected sum of S k 0 and S k 1 .
As mentioned in Remark 4, the requirement involving the k-adjacency of (S k 0 S k 1 , k) in Z 3 plays an important role in studying connected sums of closed k i -surfaces, i ∈ {0, 1}, k = k i . Indeed, it turns out that [8] (S k S k , k) is also a closed k-surface in the picture (Z 3 , k,k, S k S k ), where S k and S k are closed k-surfaces in the pictures (Z 3 , k,k, S k ) and (Z 3 , k,k, S k ), respectively.
This section explores several methods of formulating the digital connected sums MSS 6 MSS 6 , MSS 18 MSS 18 and an n-times iterated connected sum of MSS 6 and that of MSS 18 .
At the moment, let us recall the previously-mentioned queries in Section 1: (Q1) After replacing (6,26) in Definition 5(1) with (6, 18), we may ask if it is possible to propose the simple closed 6-surface MSS 6 in the picture (Z 3 , 6, 18, MSS 6 ) instead of (Z 3 , 6, 26, MSS 6 ) . This query is a reminder of the importance of thek-adjacency of Z 3 \ S k of a simple closed k-surface S k in the picture (Z 3 , k,k, S k ). (Q2) Given the MSS 6 , how many models for MSS 6 MSS 6 exist ? Let C n 6 := n-times MSS 6 · · · MSS 6 . Then we also have the following question: (Q3) How can we formulate C n 6 , n ∈ N \ {1} ?
To address these queries, we now study some properties of MSS 6 and C n 6 . First of all, let us represent the question (Q1), as follows: Unlike the three cases of (1), we may ask if there are other binary relations (6,k) for MSS 6 , k ∈ {6, 18}. Similarly, using a method similar to the above approach, we cannot take the picture (Z 3 , 6, 6, MSS 6 ) for MSS 6 .
To address the above question (Q2), we have the following: Lemma 1. Given an MSS 6 , the only one type of MSS 6 MSS 6 exists up to 6-isomorphism.
Proof. In order to formulate MSS 6 MSS 6 , we should follow Definition 6 and Remark 4. In this situation, it is obvious that we obtain six cases of MSS 6 MSS 6 (see one of the cases in Figure 3a) which are 6-isomorphic to each other. Regarding the establishment of a connected sum MSS 6 MSS 6 , suppose some possibility of taking one of the points indicated by the numbers "8" or "7" in Figure 3a except the above-mentioned six points of MSS 6 , e.g., the point p of Figure 3b. Then we have a contradiction to Remark 4. Hence we have the only one type of MSS 6 MSS 6 as suggested in Figure 3a up to 6-isomorphism. Regarding the question (Q3), we obtain the following: Theorem 3. In the case of C n 6 , n ∈ N \ {1, 2}, many types of models for C n 6 exist.
Motivated by Theorem 1 of [8], we obtain the following:

Remark 6.
[7] Given a closed 6-surface S 6 in the picture (Z 3 , 6, 26, S 6 ), we obtain that S 6 MSS 6 is a simple closed 6-surface in the picture (Z 3 , 6, 26, S 6 MSS 6 ).  Based on the digital connected sums of MSS 6 , MSS 18 , MSS 18 , and MSS 26 introduced [5], in order to study them more systematically, we need to address the following query.

Existence of Only Two Types of C
(Q4) Given an MSS 18 , how many types of MSS 18 MSS 18 exist ?
Then the map of (16) is an 18-homotopy making C 2 18 18-contractible, i.e., 1 C 2 18 18 c {12} .  This section investigates if each of C 2 6 and C n 18 has the AFPP. In order to address the problems proposed with (Q6)-(Q8), let us now recall the category of digital topological spaces and further, the fixed point property and the almost fixed point property from the viewpoint of digital topology.
• We denote by DTC the category consisting of two data: The set of digital images (X, k) as Ob(DTC) and the set of (k 0 , k 1 )-continuous maps between every pair of digital images (X, k 0 ) and (Y, k 1 ) in Ob(DTC) as Mor(DTC) [18].
• We say that a digital image (X, k) in Z n has the fixed point property (for short FPP) [23] if for every k-continuous map f : (X, k) → (X, k) there is a point x ∈ X such that f (x) = x.
Due to the study of the non-FPP of a digital picture (or digital image) in [23](see Theorem 4.1 of [23]), it is clear that only the digital image (or a digital picture) (X, k) with |X| = 1 has the FPP because a singleton set obviously has the FPP in DTC. Thus we need to recall the following (see Theorem 4.1 of [23] and Remark 4.3 of [34]): Remark 10. [23,34] Only a digital image (X, k) with |X| = 1 has the FPP.
This property is obviously a certain implication of Theorems 3.3 and 4.1 of [23]. For the convenience of readers, we now confirm the assertion more precisely.
Proof. To wit the assertion, when establishing the notion of AFPP in [23] (see the bottom of the page 179 of [23]), we obviously find that Rosenfeld [23] stated two theorems such as Theorems 3.3 and 4.1 of [23] relating to the above assertion. More precisely, as mentioned in the above part (see the part just below Section 4 of [23]), a paper [23] finally mentioned the AFPP of an n-dimensional digital picture (I n , 3 n − 1) or a general picture (X, 3 n − 1) in Z n . For instance, for the case of ([a, b] Z , 2), a = b, Rosenfeld [23] proved the AFPP of it (see Theorem 3.3 of [23]). To be precise, for any 2-continuous self-map f of ([a, b] Z , 2), it turns out that ([a, b] Z , 2) has the AFPP instead of the FPP. Then, Theorem 3.3 implies that not every 2-continuous self-map f of ([a, b] Z , 2) support the FPP of it. However, the assertion supports the AFPP of ([a, b] Z , 2) instead of the FPP. Obviously, take a point x ∈ [a, b] Z and N 2 (x, 1) ⊂ [a, b] Z . Then consider any point x ( = x) ∈ N 2 (x, 1) and further, according to Theorem 3.3 of [23], consider a self-map f of ([a, b] Z , 2) defined by Then, the map f is obviously 2-continuous and f implies that ([a, b] Z , 2) does not have the FPP. As a good example, consider a simple digital interval ([0, 1] Z , 2) and consider the self-map f of it, say f (0) = 1 and f (1) = 0 which supports Theorem 3.3 of [23], which implies the AFPP of it instead of the FPP. Similarly, as mentioned in the beginning part of Section 4 of [23], the paper [23] proved that the n-dimensional case (I n , 3 n − 1) or a general picture (X, 3 n − 1) in Z n (see Theorem 4.1 of [23]) has the AFPP instead of the FPP. Eventually, with the same method as above, for any general digital image (X, k) in Z n , we confirm the assertion of Remark 10.
Owing to Remark 10, it turns out that the study of the FPP in DTC is very trivial. Henceforth, Rosenfeld [23] firstly studied the almost fixed point property for digital images. Hence we need to stress the AFPP in DTC.
• We say that a digital image (X, k) in Z n has the almost fixed point property (for short AFPP) [23] if for every k-continuous self-map f of (X, k), there is a point Furthermore, a paper [8] proved that each of MSS 18 and MSS 18 does not have the AFPP (see Theorem 7 below). Thus the study of the AFPP of C n k , n ∈ N \ {1}, k ∈ {6, 18} remains. Let us now address this issue. To address these two queries, we first prove the non-AFPP of MSS 6 , as follows: Lemma 4. MSS 6 does not have the AFPP.
Proof. Consider the set MSS 6 in Figure 7a(1). Then, let f be a self-map of MSS 6 which is the composite of the three times reflections of MSS 6 according to the three xy-, yz-, and xz-planes in R 3 (see the image of the map f on the set MSS 6 of Figure 7a(2)). Whereas the map f of Figure 7a is obviously a 6-continuous self-bijection of MSS 6 , it does not support the AFPP of MSS 6 .
Before proving the assertion, due to Lemma 1, we recall that C 2 6 uniquely exists up to 6-isomorphism.
Proof. Consider the set C 2 6 in Figure 7a(2). Then assume a self-map g of C 2 6 which is the composite of the three times reflections of MSS 6 according to the three xy-, yz-, and xz-planes in R 3 (see the image of the map g of C 2 6 in Figure 7a(2)). Whereas the map g is obviously a 6-continuous bijection, it does not support the AFPP of C 2 6 .
Corollary 8. Let C n 6 be assumed as the set formulated via the method suggested in Figure 3b(1). The image C n 6 in the binary picture (Z 3 , 6, 26, C n 6 ) does not have the AFPP.
As a generalization of the non-AFPP of MSS 18 referred to in Theorem 7, we obtain the following: (Case 2) In case MSS 18 MSS 18 = MSS 18 , let us now prove the non-AFPP of C n 18 . With the hypothesis, by Theorem 4, we see that C n 18 has the shape suggested in Figure 7c (just an example for C 2 18 in Figure 7c). Then, let h be a self-map of C n 18 which is the composite of the three times reflections of C n 18 according to the xy-, yz-, and xz-planes in R 3 . Whereas the map h is obviously an 18-continuous map, it does not support the AFPP of C n 18 . In order to generalize Theorem 9, we need the following notion which is stronger than the isomorphism of Definition 1.

Remark 11.
Comparing the isomorphism of Definition 1 and that of Definition 8, we observe that they are different.
As a generalization of Theorems 8 and 9, and Corollary 8, we obtain the following: Proposition 4. Consider a (simple) closed k-surface S k in (Z 3 , k,k, S k ), k ∈ {6, 18, 26} with the binary relations of (11). If it is (k,k)-isomorphic to (X, k) in the picture (Z 3 , k,k, X) and the set X is symmetric according to each of xy-, yz-, and xz-planes of R 3 , then S k does not have the AFPP.
Proof. With the hypothesis, we proceed with the following several steps for proving the assertion. For convenience we may assume S k := {s i | i ∈ [1, m] Z } for some m ∈ Z. (Step 1) Take a (k,k)-isomorphism h from S k to (X, k) in the given digital pictures (see Figure 8), where X := {x i | i ∈ [1, m] Z , x i := h(s i )}. Namely, we may assume a (k,k)-isomorphism h : ( Step 2) Given the set (X, k), proceed to the composite of the three times of different reflections of (X, k) according to the certain xy-, yz-, and xz-planes in R 3 which is a k-continuous bijection (or a k-isomorphism). Then we denote the composite with the self-map f of (X, k). For convenience, put f (x i ) = x j , i, j ∈ [1, m] Z and we see i = j. ( Step 3) We denote the digital image being proceeded with (Step 2) with (X , k), i.e., f (X) := X := {x j | x j = f (x i ) | j ∈ [1, m] Z }. Then we see that the k-isomorphism f supports the non-AFPP (see the proof of Theorem 8). Indeed, although the set X is equal to the set X, the subscript of each of all elements is completely changed from x i to x j , i = j.
(Step 4) After assigning each element s i ∈ S k with s j ∈ S k such that we obtain the set S k := {s j | j ∈ [1, m] Z }. Indeed, although S k = S k as a set, we see that each element s i ∈ S k is changed into another element s j ∈ S k . Consider the map h : (X(= X ), k) → S k (= S k ) defined by h (x j ) = s j ∈ S k = S k , j ∈ [1, m] Z . ( Step 5) We finally obtain the composite of h, f , and h (see Figure 8), i.e., such that Finally, we see that the composite h • f • h is a certain k-continuous bijection (or a k-isomorphism) of S k which does not support the AFPP of S k .

Conclusions and Further Work
After formulating C n k , k ∈ {6, 18, 26}, the present paper proved that there are only two types of connected sums MSS 18 MSS 18 up to 18-isomorphism, only one type of MSS 6 MSS 6 up to 6-isomorphism and further, several types of connected sums C 3 6 := MSS 6 MSS 6 MSS 6 . Furthermore, it turns out that there are several types of connected sums for C 3 6 := MSS 6 MSS 6 MSS 6 . Besides, in case MSS 18 MSS 18 = MSS 18 up to 18-isomorphism, we proved that C n 18 := n-times MSS 18 · · · MSS 18 uniquely exists up to 18-isomorphism. In addition, we proved the digital k-contractibility of C n k := n-times MSS k · · · MSS k , k ∈ {18, 26} and further, the simply k-connectedness of C n k , k ∈ {6, 18, 26}, n ∈ N. Finally, we explored the non-AFPP of each of C 2 6 , C n 18 and C n 26 . In view of several homotopic properties of MSS 6 , MSS 18 , MSS 18 , and MSS 26 and further, the non-AFPP of them and their connected sums, we obtain the following: As a further work, based on Proposition 4, we need to further study the AFPP of C n 6 , n ∈ N \ {1, 2} according to the processes associated with Figure 3b(2), (3), and (4). As mentioned above, some homotopic features of the models MSS 6 , MSS 18 , MSS 18 , MSS 26 play important roles in digital topology and digital geometry because each of them can be considered to be the typical sphere-like model in Euclidean topology. Hence, the features referred to in Figure 9 facilitate studying many objects involving AFPP for digital images. Furthermore, the notion of digital connected sum also plays a crucial role in digital geometry because it can contribute to formulating another surface from two given surfaces. Besides, using the new topological structures in [36], we can study the FPP and AFPP of S k as subspaces of the newly-established topological structures. Finally, considering the geometric realization of a digital k-surface with an SST-structure in [37], we can deal with them from the viewpoint of computational geometry. In addition, after establishing a certain cone metric on a digital image [38][39][40][41][42], we need to further compare the current digital metric spaces using a length of simple k-path with cone metric spaces.