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

## Abstract

**:**

## 1. Introduction

- (Q1) We may ask if it is possible to propose the simple closed 6-surface $MS{S}_{6}$ in the picture $({\mathbb{Z}}^{3},6,18,MS{S}_{6})$ instead of $({\mathbb{Z}}^{3},6,26,MS{S}_{6})$.Hereafter, the operator “$\u266f$” means the digital connected sum (see Section 4 for the details).(Q2) How many types of $MS{S}_{6}\u266fMS{S}_{6}$ exist ?Let ${\mathcal{C}}_{6}^{n}:=\stackrel{n-\mathrm{times}}{\stackrel{\u23de}{MS{S}_{6}\u266f\cdots \u266fMS{S}_{6}}}$. Then we have the following queries:(Q3) How can we formulate ${\mathcal{C}}_{6}^{n},n\in \mathbb{N}\backslash \left\{1\right\}$ ?Given an $MS{S}_{18}$, we may raise the following query.(Q4) How many types of $MS{S}_{18}\u266fMS{S}_{18}$ exist ?Let ${\mathcal{C}}_{18}^{n}:=\stackrel{n-\mathrm{times}}{\stackrel{\u23de}{MS{S}_{18}\u266f\cdots \u266fMS{S}_{18}}}$. Then we have the following questions:(Q5) How can we formulate ${\mathcal{C}}_{18}^{n},n\in \mathbb{N}\backslash \left\{1\right\}$?(Q6) How about the almost fixed point property (AFPP for short) of ${\mathcal{C}}_{6}^{n},n\in \mathbb{N}$?(Q7) How about the AFPP of ${\mathcal{C}}_{18}^{n},n\in \mathbb{N}$?(Q8) What are some properties relating to the AFPP of a closed k-surface in ${\mathbb{Z}}^{3}$.

## 2. Basic Notions Involving Digital k-Surfaces and Connected Sums of Closed k-Surfaces

- It is natural to say that a digital image $(X,k)$ is k-disconnected if there are nonempty sets ${X}_{1},{X}_{2}\subset X$ such that $X={X}_{1}\cup {X}_{2}$, ${X}_{1}\cap {X}_{2}=\varnothing $ and further, there are no points ${x}_{1}\in {X}_{1}$ and ${x}_{2}\in {X}_{2}$ such that ${x}_{1}$ and ${x}_{2}$ are k-adjacent.
- We say that a digital image $(X,k)$ is k-connected (or k-path connected) if it is not k-disconnected. 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 ${\mathbb{Z}}^{n}$ is assumed to be a finite sequence ${\left({x}_{i}\right)}_{i\in {[0,l]}_{\mathbb{Z}}}\subset {\mathbb{Z}}^{n}$ such that ${x}_{i}$ and ${x}_{j}$ are k-adjacent if $|\phantom{\rule{0.166667em}{0ex}}i-j\phantom{\rule{0.166667em}{0ex}}|=1$ [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 ${\left({x}_{i}\right)}_{i\in {[0,l]}_{\mathbb{Z}}}\subset X$ such that $x={x}_{0}$ and $y={x}_{l}$.
- For a digital image $(X,k)$, the k-component of $x\in X$ 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 ${\left({x}_{i}\right)}_{i\in {[0,m]}_{\mathbb{Z}}}\subset {\mathbb{Z}}^{n}$ such that ${x}_{i}$ and ${x}_{j}$ are k-adjacent if and only if $|\phantom{\rule{0.166667em}{0ex}}i-j\phantom{\rule{0.166667em}{0ex}}|=1$ [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 ${\mathbb{Z}}^{n}$, denoted by $S{C}_{k}^{n,l}$ [17,19], $l\ge 4,l\in {\mathbb{N}}_{0}\backslash \left\{2\right\}$, ${\mathbb{N}}_{0}$ is the set of even natural numbers, means the finite set ${\left({x}_{i}\right)}_{i\in {[0,l-1]}_{\mathbb{Z}}}\subset {\mathbb{Z}}^{n}$ such that ${x}_{i}$ and ${x}_{j}$ are k-adjacent if and only if $|\phantom{\rule{0.166667em}{0ex}}i-j\phantom{\rule{0.166667em}{0ex}}|=\pm 1\left(mod\phantom{\rule{0.166667em}{0ex}}l\right)$.
- For a digital image $(X,k)$, a digital k-neighborhood of ${x}_{0}\in X$ with radius $\epsilon $ is defined in X as the following subset [17] of X$${N}_{k}({x}_{0},\epsilon ):=\{x\in X\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{l}_{k}({x}_{0},x)\le \epsilon \}\cup \left\{{x}_{0}\right\},\phantom{\rule{2.em}{0ex}}$$$${N}_{k}(x,1)={N}_{k}\left(x\right)\cap X.\phantom{\rule{2.em}{0ex}}$$

**Proposition**

**1.**

**Definition**

**1.**

**Definition**

**2.**

- $(\star )$$MS{C}_{8}^{*}:=MS{C}_{8}\cup Int\left(MS{C}_{8}\right)$ [6], where $MS{C}_{8}$ is a digital image 8-isomorphic to the digital image, $MS{C}_{8}:=S{C}_{8}^{2,6}:=\{{c}_{0}=(0,0),{c}_{1}=(1,1),{c}_{2}=(1,2),{c}_{3}=(0,3),{c}_{4}=(-1,2),{c}_{5}=(-1,1)\}$.$(\star )$$MS{C}_{4}^{*}:=MS{C}_{4}\cup Int\left(MS{C}_{4}\right)$ [6], where $MS{C}_{4}$ is a digital image 4-isomorphic to the digital image, $MS{C}_{4}:=S{C}_{4}^{2,8}:=\{{v}_{0}=(0,0),{v}_{1}=(1,0),{v}_{2}=(2,0),{v}_{3}=(2,1),{v}_{4}=(2,2),{v}_{5}=(1,2),{v}_{6}=(0,2),{v}_{7}=(0,1)\}$.$(\star )$$MS{{C}_{8}^{\prime}}^{*}:=MS{C}_{8}^{\prime}\cup Int\left(MS{C}_{8}^{\prime}\right)$ [6], where $MS{C}_{8}^{\prime}$ is a digital image 8-isomorphic to the digital image, $MS{C}_{8}^{\prime}:=S{C}_{8}^{2,4}:=\{{w}_{0}=(0,0),{w}_{1}=(1,1),{w}_{2}=(0,2),{w}_{3}=(-1,1)\}$.

**Definition**

**3.**

- for all $x\in X,H(x,0)=f\left(x\right)$ and $H(x,m)=g\left(x\right)$;
- for all $x\in X$, the induced function ${H}_{x}:{[0,m]}_{\mathbb{Z}}\to Y$ given by${H}_{x}\left(t\right)=H(x,t)$ for all $t\in {[0,m]}_{\mathbb{Z}}$ is $(2,{k}_{1})$-continuous;
- for all $t\in {[0,m]}_{\mathbb{Z}}$, the induced function ${H}_{t}:X\to Y$ given by ${H}_{t}\left(x\right)=H(x,t)$ for all $x\in X$ is $({k}_{0},{k}_{1})$-continuous.Then we say that H is a $({k}_{0},{k}_{1})$-homotopy between f and g [28].
- Furthermore, for all $t\in {[0,m]}_{\mathbb{Z}}$, 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, ${H}_{t}\left(x\right)=f\left(x\right)=g\left(x\right)$ for all $x\in A$ and for all $t\in {[0,m]}_{\mathbb{Z}}$.

**Definition**

**4.**

**Theorem**

**1.**

- Due to Theorem 1, it turns out that $S{C}_{k}^{n,l}$ is not k-contractible if $l\ge 6$.

**Example**

**1.**

**Theorem**

**2.**

**Proof.**

- (1)
- for all ${x}^{\prime}\in {X}^{\prime}$, $G({x}^{\prime},a)={h}_{2}\circ f\circ {h}_{1}^{-1}\left({x}^{\prime}\right)$ and $G({x}^{\prime},b)={h}_{2}\circ g\circ {h}_{1}^{-1}\left({x}^{\prime}\right)$;
- (2)
- for all ${x}^{\prime}\in {X}^{\prime}$, the induced function ${G}_{{x}^{\prime}}:{[a,b]}_{\mathbb{Z}}\to {Y}^{\prime}$ defined by ${G}_{{x}^{\prime}}\left(t\right):=G({x}^{\prime},t)$ for all $t\in {[a,b]}_{\mathbb{Z}}$ is k-continuous;
- (3)
- for all $t\in {[a,b]}_{\mathbb{Z}}$, the induced function ${G}_{t}:{X}^{\prime}\to {Y}^{\prime}$ defined by ${G}_{t}\left({x}^{\prime}\right):=G({x}^{\prime},t)$ for all ${x}^{\prime}\in {X}^{\prime}$ is k-continuous.

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

## 3. Utilities of the Minimal Simple Closed 6-, 18- and 26-Surfaces; $MS{S}_{6}$, $MS{S}_{18}$, $MS{S}_{18}^{\prime},MS{S}_{26}^{\prime}$

**Definition**

**5.**

- (1)
- In the case $(k,\overline{k})\in \{(26,6),(6,26)\}$, for each point $x\in X$,
- (a)
- ${\left|X\right|}^{x}$ has exactly one k-component k-adjacent to x;
- (b)
- $|\overline{X}{|}^{x}$ has exactly two $\overline{k}$-components which are $\overline{k}$-adjacent to x; we denote by ${C}^{x\phantom{\rule{0.166667em}{0ex}}x}$ and ${D}^{x\phantom{\rule{0.166667em}{0ex}}x}$ these two components; and
- (c)
- for any point $y\in {N}_{k}\left(x\right)\cap X$ (or ${N}_{k}(x,1)$ in $(X,k)$), ${N}_{\overline{k}}\left(y\right)\cap {C}^{x\phantom{\rule{0.166667em}{0ex}}x}\ne \varphi $ and ${N}_{\overline{k}}\left(y\right)\cap {D}^{x\phantom{\rule{0.166667em}{0ex}}x}\ne \varphi $.

Furthermore, if a closed k-surface X does not have a simple k-point, then X is called simple. - (2)
- In the case $(k,\overline{k})=(18,6)$,
- (a)
- X is k-connected,
- (b)
- for each point $x\in X$, ${\left|X\right|}^{x}$ is a generalized simple closed k-curve.

**Remark**

**2.**

**Proposition**

**2.**

**Proof.**

- (Correction) In the Figure 4c of [35], the given K-topological space $(Z,{\kappa}_{Z}^{2})$ should be referred to as “non-K-retractible” instead of “$K$-retractible”.

## 4. Several Types of Models for ${\mathcal{C}}_{6}^{n}:=\stackrel{n-\mathrm{times}}{\stackrel{\u23de}{MS{S}_{6}\u266f\cdots \u266fMS{S}_{6}}}$

**Definition**

**6.**

- Consider ${A}_{{k}_{0}}^{\prime}\subset {A}_{{k}_{0}}\subset {S}_{{k}_{0}}$ and take ${A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime}\subset {S}_{{k}_{0}}$, where ${A}_{{k}_{0}}{\approx}_{({k}_{0},4)}MS{C}_{4}^{*}$ or ${A}_{{k}_{0}}{\approx}_{({k}_{0},8)}MS{C}_{8}^{*}$, or ${A}_{{k}_{0}}{\approx}_{({k}_{0},8)}MS{{C}_{8}^{\prime}}^{*}$, and further, ${A}_{{k}_{0}}^{\prime}{\approx}_{({k}_{0},4)}$$Int\left(MS{C}_{4}\right)$ or ${A}_{{k}_{0}}^{\prime}{\approx}_{({k}_{0},8)}$$Int\left(MS{C}_{8}\right)$, or ${A}_{{k}_{0}}^{\prime}{\approx}_{({k}_{0},8)}$$Int\left(MS{C}_{8}^{\prime}\right)$, respectively.
- Let $f:{A}_{{k}_{0}}\to f\left({A}_{{k}_{0}}\right)\subset {S}_{{k}_{1}}^{\prime}$ be a $({k}_{0},{k}_{1})$-isomorphism. Remove ${A}_{{k}_{0}}^{\prime}$ and $f\left({A}_{{k}_{0}}^{\prime}\right)$ from ${S}_{{k}_{0}}$ and ${S}_{{k}_{1}}$, respectively.
- Identify ${A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime}$ and $f({A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime})$ by using the $({k}_{0},{k}_{1})$-isomorphism f. Then, the quotient space ${S}_{{k}_{0}}^{\prime}\cup {S}_{{k}_{1}}^{\prime}/\sim $ is obtained by $i\left(x\right)\sim f\left(x\right)\in {S}_{{k}_{1}}^{\prime}$ for $x\in {A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime}$ and is denoted by ${S}_{{k}_{0}}\u266f{S}_{{k}_{1}}$, where ${S}_{{k}_{0}}^{\prime}={S}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime}$, ${S}_{{k}_{1}}^{\prime}={S}_{{k}_{1}}\backslash f\left({A}_{{k}_{0}}^{\prime}\right)$, and the map $i:{A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime}\to {S}_{{k}_{0}}^{\prime}$ is the inclusion map.

**Remark**

**4.**

- (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 $MS{S}_{6}$ in the picture $({\mathbb{Z}}^{3},6,18,MS{S}_{6})$ instead of $({\mathbb{Z}}^{3},6,26,MS{S}_{6})$.This query is a reminder of the importance of the $\overline{k}$-adjacency of ${\mathbb{Z}}^{3}\backslash {S}_{k}$ of a simple closed k-surface ${S}_{k}$ in the picture $({\mathbb{Z}}^{3},k,\overline{k},{S}_{k})$.(Q2) Given the $MS{S}_{6}$, how many models for $MS{S}_{6}\u266fMS{S}_{6}$ exist ?Let ${\mathcal{C}}_{6}^{n}:=\stackrel{n-\mathrm{times}}{\stackrel{\u23de}{MS{S}_{6}\u266f\cdots \u266fMS{S}_{6}}}$. Then we also have the following question:(Q3) How can we formulate ${\mathcal{C}}_{6}^{n},n\in \mathbb{N}\backslash \left\{1\right\}$ ?

**Remark**

**5.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Remark**

**6.**

## 5. Existence of Only Two Types of ${\mathcal{C}}_{18}^{n}:=\stackrel{n-\mathrm{times}}{\stackrel{\u23de}{MS{S}_{18}\u266f\cdots \u266fMS{S}_{18}}},n\ge 2$

**Remark**

**7.**

- (1)
- The set $MS{S}_{18}$ cannot be a simple closed 18-surface in the picture $({\mathbb{Z}}^{3},18,18,MS{S}_{18})$.
- (2)
- The set $MS{S}_{26}^{\prime}$ cannot be a simple closed 26-surface in the picture $({\mathbb{Z}}^{3},26,18,MS{S}_{26}^{\prime})$.

**Theorem**

**4.**

- (1)
- Only two types of $MS{S}_{18}\u266fMS{S}_{18}$ exist up to 18-isomorphism.
- (2)
- In the case of ${\mathcal{C}}_{18}^{n},n\in \mathbb{N}\backslash \{1,2\}$, only two methods are admissible in establishing ${\mathcal{C}}_{18}^{n}$ up to 18-isomorphism.

**Proof.**

- (Case 1) Based on the cases of (12) (1)–(2), in the case that we follow the method suggested in Figure 4a, we obtain $MS{S}_{18}\u266fMS{S}_{18}=MS{S}_{18}$ [5]. Eventually, if we take this process for obtaining ${\mathcal{C}}_{18}^{n}$, then we have ${\mathcal{C}}_{18}^{n}=MS{S}_{18}$.(Case 2) Based on the cases of (12) (3), according to the method suggested in Figure 4b, i.e., in the case $MS{S}_{18}\u266fMS{S}_{18}\ne MS{S}_{18}$, we now prove that there is only one type of $MS{S}_{18}\u266fMS{S}_{18}$ up to 18-isomorphism. To be precise, after identifying two sets denoted by the set $\{1,2,3,4\}$ of $MS{S}_{18}$ (see Figure 4b), we obtain $MS{S}_{18}\u266fMS{S}_{18}$. Hence, we have only one way to proceed to $MS{S}_{18}\u266fMS{S}_{18}$ as proposed in Figure 4b up to 18-isomorphism. Eventually, we uniquely obtain ${\mathcal{C}}_{18}^{n}$ in terms of ${\mathcal{C}}_{18}^{n}:={\mathcal{C}}_{18}^{n-1}\u266fMS{S}_{18}$. □

**Remark**

**8.**

**Corollary**

**4.**

## 6. Digital 18-Contractibility of ${\mathcal{C}}_{18}^{n}$ and Simply k-Connectedness of ${\mathcal{C}}_{k}^{n}$, $k\in \{6,18,26\}$

**Lemma**

**2.**

**Definition**

**7.**

**Lemma**

**3.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Corollary**

**5.**

**Theorem**

**5.**

**Proof.**

**Remark**

**9.**

**Theorem**

**6.**

**Proof.**

**Corollary**

**6.**

**Proof.**

**Corollary**

**7.**

**Proof.**

## 7. Non-almost Fixed Point Property of ${\mathcal{C}}_{k}^{n},k\in \{6,18\}$

- We denote by $DTC$ the category consisting of two data: The set of digital images $(X,k)$ as $Ob\left(DTC\right)$ and the set of $({k}_{0},{k}_{1})$-continuous maps between every pair of digital images $(X,{k}_{0})$ and $(Y,{k}_{1})$ in $Ob\left(DTC\right)$ as $Mor\left(DTC\right)$ [18].
- We say that a digital image $(X,k)$ in ${\mathbb{Z}}^{n}$ has the fixed point property (for short FPP) [23] if for every k-continuous map $f:(X,k)\to (X,k)$ there is a point $x\in X$ such that $f\left(x\right)=x$.

**Proof.**

- We say that a digital image $(X,k)$ in ${\mathbb{Z}}^{n}$ 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 $x\in X$ such that $f\left(x\right)=x$ or $f\left(x\right)$ is k-adjacent to x.

**Theorem**

**7.**

**Lemma**

**4.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Corollary**

**8.**

**Theorem**

**9.**

**Proof.**

**Definition**

**8.**

- (1)
- ${S}_{k}(\subset {\mathbb{Z}}^{3})$ is k-isomorphic to $(X,k)$ and
- (2)
- $({\mathbb{Z}}^{3}\backslash {S}_{k},\overline{k})$ is $\overline{k}$-isomorphic to $({\mathbb{Z}}^{3}\backslash X,\overline{k})$.

**Remark**

**11.**

**Proposition**

**4.**

**Proof.**

**Remark**

**12.**

## 8. Conclusions and Further Work

## Funding

## Conflicts of Interest

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**Figure 3.**(

**a**) Process of constructing $MS{S}_{6}\u266fMS{S}_{6}$ [5]; (

**b**) Configuration of ${\mathcal{C}}_{6}^{3}:=MS{S}_{6}\u266fMS{S}_{6}\u266fMS{S}_{6}$.

**Figure 4.**Explanation of the only two types of $MS{S}_{18}\u266fMS{S}_{18}$ in terms of the processes via (

**a**) or (

**b**) [5].

**Figure 5.**(

**a**) Explanation of the process of establishing ${\mathcal{C}}_{18}^{3}:={\mathcal{C}}_{18}^{2}\u266fMS{S}_{18}$. (

**b**) Configuration of an 18-homotopy $H:{\mathcal{C}}_{18}^{n+1}\times {[0,m+{m}^{\prime}]}_{\mathbb{Z}}\to {\mathcal{C}}_{18}^{n+1},{m}^{\prime}\ge 1$.

**Figure 6.**Configuration of the 18-homotopy of (16) involving the 18-contractibility of ${\mathcal{C}}_{18}^{2}$ (see the proof of Corollary 6).

**Figure 7.**(

**a**) Configuration of the AFPP of $MS{S}_{6}$. (

**b**) Configuration of the non-AFPP of ${\mathcal{C}}_{6}^{2}:=MS{S}_{6}\u266fMS{S}_{6}$. (

**c**) In case $MS{S}_{18}\u266fMS{S}_{18}\ne MS{S}_{18}$, configuration of the non-AFPP of ${\mathcal{C}}_{18}^{2}$.

**Figure 9.**Digital topological properties of the non-AFPP of the minimal simple closed k-surfaces $MS{S}_{6}$, $MS{S}_{18}$, $MS{S}_{18}^{\prime}$, $MS{S}_{26}^{\prime}$, ${\mathcal{C}}_{6}^{2}$, and ${\mathcal{C}}_{18}^{n}$.

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**MDPI and ACS Style**

Han, S.-E.
Digital *k*-Contractibility of an *n*-Times Iterated Connected Sum of Simple Closed *k*-Surfaces and Almost Fixed Point Property. *Mathematics* **2020**, *8*, 345.
https://doi.org/10.3390/math8030345

**AMA Style**

Han S-E.
Digital *k*-Contractibility of an *n*-Times Iterated Connected Sum of Simple Closed *k*-Surfaces and Almost Fixed Point Property. *Mathematics*. 2020; 8(3):345.
https://doi.org/10.3390/math8030345

**Chicago/Turabian Style**

Han, Sang-Eon.
2020. "Digital *k*-Contractibility of an *n*-Times Iterated Connected Sum of Simple Closed *k*-Surfaces and Almost Fixed Point Property" *Mathematics* 8, no. 3: 345.
https://doi.org/10.3390/math8030345