# Analytical Consideration of Growth in Population via Homological Invariant in Algebraic Topology

Authors: Lewis Brew; William Obeng-Denteh; Fred Asante-Mensa
Journal: Journal of Mathematics
 (2020)

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## Abstract

This paper presents an abstract approach of analysing population growth in the field of algebraic topology using the tools of homology theory. For a topological spaceX and any point vn∈X, where vn is the n-dimensional surface, the group η=X,vn is called population of the space X. The increasing sequence from vin∈X to vjn∈X for i<j provides the bases for the population growth. A growth in population η=X,vn occurs if vin<vjn for all vin∈X and vjn∈X. This is described by the homological invariant Hηk=1. The aim of this paper is to construct the homological invariant Hηk and use Hηk=1 to analyse the growth of the population. This approach is based on topological properties such as connectivity and continuity. The paper made extensive use of homological invariant in presenting important information about the population growth. The most significant feature of this method is its simplicity in analysing population growth using only algebraic category and transformations.


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## Body

## 1. Introduction

Algebraic topology which provides computable property for identifying topological properties has gained importance and popularity in many diverse fields of mathematics and other branches of science. Though the subject algebraic topology is a relatively new field of mathematics, its contribution to the study of the dynamics in both metric and topological spaces cannot be overruled. For instance, in the study of the components of a space, it provides a lot of information about the number of the components and how they are linked together in the space. Not only does the information gained helps us to understudy both topological and geometric properties of the space, but also provides a method of reducing topological problems and continuous maps into algebraic problems, thereby making it appropriate to determine some homological invariants. The essence of the homological invariant in this study is that it does not change under the deformations for the same property. Rather, it assigns a particular number to a specific property under study. Population growth also remains to grow no matter how wide the population growth varies. This is assumed as a homological invariant under growth deformation. In this paper, a specific value is assigned to the homological invariant to represent the population growth. Though different methods such as statistical analysis and differential equation models have been developed and used in studying population growth in many research works, many are yet to be developed, which perhaps, may contribute meaningfully to the existing knowledge. Indeed, to study the growth of population of a country, one needs to be equipped with all the necessary information regarding the population of the country, and the approach to collect all this information is sometimes quite arduous in the field of statistics and differential equations. Research on population growth and its implication in Walker [1] revealed that nations with the fastest growing populations tend to rank high on global indices of hunger, poverty, and environmental degradation. Guney [2] also made it clear in his work that while population growth in developing countries affects sustainable development negatively, the population growth in developed countries affects sustainable growth positively. All these researches were carried out with enough information from the statistical data. Therefore, with a little information on population data, one finds it difficult to describe the process of population growth in statistics and other related fields. This therefore makes the application of the concept of algebraic topology which is abstract and requires fewer tasks which are very important in this study. Homological invariant in algebraic topology shortens this process and provides convenient techniques to the study of continuous process such as population growth. Indeed, homological invariants have proven to be a valuable tool for studying not only continuous process such as population growth but also for many continuous physical phenomena. For instance, Kinsey [3] used Euler characteristics as one of the topological invariants to study the surfaces of objects and concluded that the surface of an object is closed if the Euler characteristic is zero. This is one of the abstract techniques of describing a closed object. Hossenia et al. [4] also developed the concept of homotopy, one of the topological invariants via deformation and stabilization of the linear part of the homotopy equation. His method showed how the stability and convergence of a solution of equation was guaranteed. Many research works including others from [4, 5] which employed the concept of homological invariants were reviewed for this paper, but the major ideas of this paper were developed from the research works of Donald and Chang [5]. More often, the approach to the study of population growth is centered on statistics and differential equations [1, 6, 7]. In this paper, the novelty is the introduction of a homological invariant outside statistical and differential equations. The homological invariants we have introduced in this paper will describe the continuous paths between the 2-dimensional surfaces or groups and will be referred to as the growth of the population. In all cases, an increasing sequence of the paths of groups in the topological space is the population growth.


### 1.1. Background

In some previous works [6, 7], statistical and differential equation methods were used to study the growth of populations from a given data set. To some extent, these statistical and differential equation models have proven to be successful methods in the study of both linear and nonlinear growth of population. Though there are other equally important methods which also attempt to establish the growth of population, none of these methods adopted the concept of groups in the algebraic category. The algebraic topology used the concept of the homological invariant which preserves certain topological structures of the 2-dimensional surface (algebraic category). The objects of the group are limited to only the 2-dimensional surface. The paths (continuous maps) are helpful in producing the extension. The extension of the groups by the continuous maps in this study is referred to as the population growth.


## 1.1. Background

In some previous works [6, 7], statistical and differential equation methods were used to study the growth of populations from a given data set. To some extent, these statistical and differential equation models have proven to be successful methods in the study of both linear and nonlinear growth of population. Though there are other equally important methods which also attempt to establish the growth of population, none of these methods adopted the concept of groups in the algebraic category. The algebraic topology used the concept of the homological invariant which preserves certain topological structures of the 2-dimensional surface (algebraic category). The objects of the group are limited to only the 2-dimensional surface. The paths (continuous maps) are helpful in producing the extension. The extension of the groups by the continuous maps in this study is referred to as the population growth.


## 2. Main Concept

Population growth is developed from the objects of group and continuous maps. It occurs when one object is extended to another object by a path or line segment. By a path in population growth, we mean the continuous map. The populationη is a topological space X generated by the objects of vn.


### 2.1. Preliminary Propositions

Proposition 1.
The populationη in a particular year is the n-dimensional surface or n-simplex vin∈X for i=0,1,2,3,....Proposition 2.
For any populationη, the growth of population occurs if(1)Hηk=vk+1n−v0n=1,where n = , 2, 3, …, k = 1, 2, 3, 4, …, and v0n∈X is the initial n-dimensional surface.Proof of Proposition.
2. Let η denote the population value in a given year and the change in the population be a homomorphism or distance (d) from vin∈X to vjn∈X. In this case, vjn∈X is the new population size from vin∈X. Let k≥1 represent the years that growth occurs such that k=1 is the first year. At k=1, the population size from the initial population size v0n∈X is vk+1n∈X. Let v0n,vk+1n,… and d,2d,3d,… represent the objects of population and their corresponding homomorphisms or distances such that k=1=v2n=2v2=2,k=2=v3n=3v3=3,… and d=1,2d=2,3d=3,…. An increasing sequence of the individual objects of the space X increases the size of the population. This is given as(2)V=∑β=1βvn,(3)D=∑β=1β−1d.
Therefore, the total growth in the population is given as(4)Hηk=∑β=1βvn−∑β=1β−1d,which is the same as(5)Hηk=∑β=1vn−∑β=1d.
As the population grows from the first year to the second year, that is,n=2 and k=1, we have(6)Hη2=∑β=12vn−∑β=11d,which becomes(7)Hη2=vn+vn−d=1.
Similarly, atn=3 and k=3, we have(8)Hη3=∑β=13vn−∑β=12d,(9)Hη3=vn+vn+v3−d+d=1.
Therefore, at thek-th year, we have(10)Hηk=βvn−β−1d=1,which proves Proposition 2.Proposition 3.
Leta<b such that the closed interval a,b∈vin∈X defines the size of the i-th population. Let also c<d such that the closed interval c,d∈vjn∈X where c,d is larger than a,b. If f is the homomorphism from vin∈X to vjn∈X such that fa=c and fb=d, then the growth of population occurs. Thus, for a,b<c,d, c,d∈vjn∈X and a,b∈vin∈X, if f:a,b∈vin⟶c,d∈vjn. Such that fa=c and fb=d, and then the growth of population occurs. In terms of real values, population growth occurs when the final value of the previous population is the initial value of the current population.


### 2.2. Homology

Generally, thenth homology group of the topological space X is defined as(11)Hn=KerdnImdn+1,where Kerdn, the kernel of the boundary operator, dn represents the number of enclosed paths (cycles) in X, and the image (Imdn+1) represents the edge (boundary) of the enclosed paths (cycles) in X. That is, Imdn+1⊆Kerdn. The topological space X consists of a finite set of points vi2∈X or 2-dimensional surfaces for the population η, and the increase in population is the increase from one vi2∈X to another. That is, vi2∈X increases to vj2∈X for i<j. Since vi2∈X is a 2-dimensional surface, zero, one, and two homology invariants can be estimated.


### 2.3. Computation ofH0 and H1 Homology Invariants

#### 2.3.1. Using 2-Dimensional Surface as the Simplicial Complex

In a situation wherevi2∈X is the 2-dimensional surface, the homology invariants are estimated as follows: If v02∈X is the initial population, and we assume that the 1st, 2nd, and 3rd years have the following population values v02, 2v02, and 3v02, then the zero-th homological invariant is estimated as(12)H0=v0n,2v0n,3v0n,…=Z,where Z is a constant and indicates the existence of an initial population.Ifv12∈X is an increment of v02∈X, then v02∈X⊂v12∈X. However, for all A∈v02∈X, KerA∈v02∈X and ImA∈v12∈X. Also, KerM∈v12∈X if M∈v12∈X. Therefore, the first homology invariant is H1=KerM/ImA since ImA⊂KerM. From Lagrange’s theorem, it can be deduced that the first homology invariant is H1=KerM/ImA=ϕ where ϕ∈Z is the extent of the increment.


#### 2.3.2. Lagrange’s Theorem

LetG be a group and H a subgroup of G, then in a finite group, the order of a subgroup divides the order of the group.Proof.
G is a group and H is a subgroup of G. ∀α,β∈G, let the class of α be α such that for k∈Z, we have kαi where i=1,2,… Consider kαi. If kα1, kα2, kα3,…, kαn are the n number of distinct classes, then G=∪i=1nkαi. Suppose the order of kαi is k denoted as k. If G=∪i=1nkαi and kαi are disjoint, then G=nk, hence k divides G.


#### 2.3.3. Using Differential Matrices from Smith Normal Form (SNF)

Consider the 2-dimensional rectangular surfaces as shown in Figures1 and 2, respectively.Figure 1
One 2-dimensional surface.Figure 2
Two 2-dimensional surfaces.Figure1 is the initial population and has only one triangular plane in the rectangle such that the vertices are not repeated. Figure 2 is the enlargement of Figure 1 and also has two triangular planes in the new rectangle where all the vertices are different. d1 and d2 are the differentials in the initial and final populations. From Figures 1 and 2, the differential matrices d1 and d2 are given as(13)(14)The Smith Normal Forms (SNF) from equations (13) and (14) are given in equations (15) and (16), respectively:(15)d1=543210100000001000000010000000100000001000000010,(16)d1=5,44,33,22,11,00,23,510010000000000.Using equation (15), H0=Z6−5−0=Z. Once again from equations (15) and (16), H1=Z7−2−5=Z0=1 which confirms equation (12). Thus, H0=Z shows the existence of the initial population which grows with respect to time as indicated by H1=1.


### 2.4. Topological Space from Data Set

X
=
ℜ
2 is the topological space. Thus, the set X together with the data points (years and the corresponding population values) as illustrated in Figure 1 is the topological space. A growth in population is an increasing sequence of the points, thus the 2-dimensional surface. With these points as the input, the homological invariants are identified.Supposevi2∈X where i=2,3,4,… are the data points of the space X. From Figure 3, vi2∈X are connected together by paths dvi2vj2 for all vi2,vj2∈X. The addition of paths indicates the addition of given distances di to vi2∈X and hence providing the increasing sequence of vi2∈X The growth of population is as a result of increasing sequence of vi2∈X which is also determined by the n-th homology invariant Hn=Kerdn/Imdn+1.Figure 3
Topological spaceX consisting of 2-dimensional surfaces.


## 2.1. Preliminary Propositions

Proposition 1.
The populationη in a particular year is the n-dimensional surface or n-simplex vin∈X for i=0,1,2,3,....Proposition 2.
For any populationη, the growth of population occurs if(1)Hηk=vk+1n−v0n=1,where n = , 2, 3, …, k = 1, 2, 3, 4, …, and v0n∈X is the initial n-dimensional surface.Proof of Proposition.
2. Let η denote the population value in a given year and the change in the population be a homomorphism or distance (d) from vin∈X to vjn∈X. In this case, vjn∈X is the new population size from vin∈X. Let k≥1 represent the years that growth occurs such that k=1 is the first year. At k=1, the population size from the initial population size v0n∈X is vk+1n∈X. Let v0n,vk+1n,… and d,2d,3d,… represent the objects of population and their corresponding homomorphisms or distances such that k=1=v2n=2v2=2,k=2=v3n=3v3=3,… and d=1,2d=2,3d=3,…. An increasing sequence of the individual objects of the space X increases the size of the population. This is given as(2)V=∑β=1βvn,(3)D=∑β=1β−1d.
Therefore, the total growth in the population is given as(4)Hηk=∑β=1βvn−∑β=1β−1d,which is the same as(5)Hηk=∑β=1vn−∑β=1d.
As the population grows from the first year to the second year, that is,n=2 and k=1, we have(6)Hη2=∑β=12vn−∑β=11d,which becomes(7)Hη2=vn+vn−d=1.
Similarly, atn=3 and k=3, we have(8)Hη3=∑β=13vn−∑β=12d,(9)Hη3=vn+vn+v3−d+d=1.
Therefore, at thek-th year, we have(10)Hηk=βvn−β−1d=1,which proves Proposition 2.Proposition 3.
Leta<b such that the closed interval a,b∈vin∈X defines the size of the i-th population. Let also c<d such that the closed interval c,d∈vjn∈X where c,d is larger than a,b. If f is the homomorphism from vin∈X to vjn∈X such that fa=c and fb=d, then the growth of population occurs. Thus, for a,b<c,d, c,d∈vjn∈X and a,b∈vin∈X, if f:a,b∈vin⟶c,d∈vjn. Such that fa=c and fb=d, and then the growth of population occurs. In terms of real values, population growth occurs when the final value of the previous population is the initial value of the current population.


## 2.2. Homology

Generally, thenth homology group of the topological space X is defined as(11)Hn=KerdnImdn+1,where Kerdn, the kernel of the boundary operator, dn represents the number of enclosed paths (cycles) in X, and the image (Imdn+1) represents the edge (boundary) of the enclosed paths (cycles) in X. That is, Imdn+1⊆Kerdn. The topological space X consists of a finite set of points vi2∈X or 2-dimensional surfaces for the population η, and the increase in population is the increase from one vi2∈X to another. That is, vi2∈X increases to vj2∈X for i<j. Since vi2∈X is a 2-dimensional surface, zero, one, and two homology invariants can be estimated.


## 2.3. Computation ofH0 and H1 Homology Invariants

### 2.3.1. Using 2-Dimensional Surface as the Simplicial Complex

In a situation wherevi2∈X is the 2-dimensional surface, the homology invariants are estimated as follows: If v02∈X is the initial population, and we assume that the 1st, 2nd, and 3rd years have the following population values v02, 2v02, and 3v02, then the zero-th homological invariant is estimated as(12)H0=v0n,2v0n,3v0n,…=Z,where Z is a constant and indicates the existence of an initial population.Ifv12∈X is an increment of v02∈X, then v02∈X⊂v12∈X. However, for all A∈v02∈X, KerA∈v02∈X and ImA∈v12∈X. Also, KerM∈v12∈X if M∈v12∈X. Therefore, the first homology invariant is H1=KerM/ImA since ImA⊂KerM. From Lagrange’s theorem, it can be deduced that the first homology invariant is H1=KerM/ImA=ϕ where ϕ∈Z is the extent of the increment.


### 2.3.2. Lagrange’s Theorem

LetG be a group and H a subgroup of G, then in a finite group, the order of a subgroup divides the order of the group.Proof.
G is a group and H is a subgroup of G. ∀α,β∈G, let the class of α be α such that for k∈Z, we have kαi where i=1,2,… Consider kαi. If kα1, kα2, kα3,…, kαn are the n number of distinct classes, then G=∪i=1nkαi. Suppose the order of kαi is k denoted as k. If G=∪i=1nkαi and kαi are disjoint, then G=nk, hence k divides G.


### 2.3.3. Using Differential Matrices from Smith Normal Form (SNF)

Consider the 2-dimensional rectangular surfaces as shown in Figures1 and 2, respectively.Figure 1
One 2-dimensional surface.Figure 2
Two 2-dimensional surfaces.Figure1 is the initial population and has only one triangular plane in the rectangle such that the vertices are not repeated. Figure 2 is the enlargement of Figure 1 and also has two triangular planes in the new rectangle where all the vertices are different. d1 and d2 are the differentials in the initial and final populations. From Figures 1 and 2, the differential matrices d1 and d2 are given as(13)(14)The Smith Normal Forms (SNF) from equations (13) and (14) are given in equations (15) and (16), respectively:(15)d1=543210100000001000000010000000100000001000000010,(16)d1=5,44,33,22,11,00,23,510010000000000.Using equation (15), H0=Z6−5−0=Z. Once again from equations (15) and (16), H1=Z7−2−5=Z0=1 which confirms equation (12). Thus, H0=Z shows the existence of the initial population which grows with respect to time as indicated by H1=1.


## 2.3.1. Using 2-Dimensional Surface as the Simplicial Complex

In a situation wherevi2∈X is the 2-dimensional surface, the homology invariants are estimated as follows: If v02∈X is the initial population, and we assume that the 1st, 2nd, and 3rd years have the following population values v02, 2v02, and 3v02, then the zero-th homological invariant is estimated as(12)H0=v0n,2v0n,3v0n,…=Z,where Z is a constant and indicates the existence of an initial population.Ifv12∈X is an increment of v02∈X, then v02∈X⊂v12∈X. However, for all A∈v02∈X, KerA∈v02∈X and ImA∈v12∈X. Also, KerM∈v12∈X if M∈v12∈X. Therefore, the first homology invariant is H1=KerM/ImA since ImA⊂KerM. From Lagrange’s theorem, it can be deduced that the first homology invariant is H1=KerM/ImA=ϕ where ϕ∈Z is the extent of the increment.


## 2.3.2. Lagrange’s Theorem

LetG be a group and H a subgroup of G, then in a finite group, the order of a subgroup divides the order of the group.Proof.
G is a group and H is a subgroup of G. ∀α,β∈G, let the class of α be α such that for k∈Z, we have kαi where i=1,2,… Consider kαi. If kα1, kα2, kα3,…, kαn are the n number of distinct classes, then G=∪i=1nkαi. Suppose the order of kαi is k denoted as k. If G=∪i=1nkαi and kαi are disjoint, then G=nk, hence k divides G.


## 2.3.3. Using Differential Matrices from Smith Normal Form (SNF)

Consider the 2-dimensional rectangular surfaces as shown in Figures1 and 2, respectively.Figure 1
One 2-dimensional surface.Figure 2
Two 2-dimensional surfaces.Figure1 is the initial population and has only one triangular plane in the rectangle such that the vertices are not repeated. Figure 2 is the enlargement of Figure 1 and also has two triangular planes in the new rectangle where all the vertices are different. d1 and d2 are the differentials in the initial and final populations. From Figures 1 and 2, the differential matrices d1 and d2 are given as(13)(14)The Smith Normal Forms (SNF) from equations (13) and (14) are given in equations (15) and (16), respectively:(15)d1=543210100000001000000010000000100000001000000010,(16)d1=5,44,33,22,11,00,23,510010000000000.Using equation (15), H0=Z6−5−0=Z. Once again from equations (15) and (16), H1=Z7−2−5=Z0=1 which confirms equation (12). Thus, H0=Z shows the existence of the initial population which grows with respect to time as indicated by H1=1.


## 2.4. Topological Space from Data Set

X
=
ℜ
2 is the topological space. Thus, the set X together with the data points (years and the corresponding population values) as illustrated in Figure 1 is the topological space. A growth in population is an increasing sequence of the points, thus the 2-dimensional surface. With these points as the input, the homological invariants are identified.Supposevi2∈X where i=2,3,4,… are the data points of the space X. From Figure 3, vi2∈X are connected together by paths dvi2vj2 for all vi2,vj2∈X. The addition of paths indicates the addition of given distances di to vi2∈X and hence providing the increasing sequence of vi2∈X The growth of population is as a result of increasing sequence of vi2∈X which is also determined by the n-th homology invariant Hn=Kerdn/Imdn+1.Figure 3
Topological spaceX consisting of 2-dimensional surfaces.


## 3. Homological Invariant and Population Growth

From equation (1), it was assumed that the condition Hηk=1 for k≥1 must be satisfied to indicate an increase from one of vi2∈X to another or to demonstrate growth in population η. However, the homological invariant Hηk which determines the growth of population is defined as(17)Hηk=βvn−β−1d,where βvi2 and β−1d denote the number of times the population increases and the extent to which population increases, respectively, for β = 2, 3, 4, …. From the first or initial data point to the second data point, there are two data points, that is, n = 2 and k=1, as shown in Figure 4.Figure 4
Change from the 1st point to the 2nd point.Therefore, we have twovi2∈X and only one distance (2–1)d between the two vi2∈X. Hence, the 1st homological invariant is(18)Hη1=2−1=1.The homological invariantHηk=1 indicates that there is a growth in the population from the first-year data point to the second-year data point. Similarly, from the first data point to the fourth data point, there are four vi2∈X with three distances between them, thus β=4 and k=3. The 3rd homological invariant from the first-year data point to the fourth-year data point is(19)Hη3=4−3=1.The homological invariantHη3=1 is an indication that there is a growth in population from the first to the fourth year.


## 4. Conclusion

The paper constructed and defined the homological invariant as an efficient tool to analyse the growth of the population. The topological information from a geometric object was applied to a data set from statistics in a 2-dimensional Euclidean plane as shown in Figure3 to obtain the information about the population growth. It was observed that the homological invariant offers another reliable technique to handle information regarding the population growth or the extension of the data set. An advantage of this method over the statistical and differential equation methods is that it simply explains the growth in the population using the time value and the homological invariant. In this work, it was assumed that anytime vi2∈X exceeds the path by one, extension occurs. By extension, we mean the changes in population values and this was determined using the homological invariant. Therefore, whenever the homological invariant condition Hηk=1 is satisfied as shown in equations (18) and (19), there is a growth in the population.The homological invariant method offered new important insights and information about the growth of population. The different methods of computing the homological invariantHη with known vi2∈X were presented. The methods were credible and simple. Its approach requires less information and was very simple when it was compared with statistics and differential equation approaches. The paper established a homological invariant to show that every population in a topological space grows. It is believed that the analysis of the population growth using a homological invariant within the algebraic topology concept is a contribution to knowledge. The most important result of this paper is the following proposition.Proposition 4.
For any populationη, the growth of population occurs if(20)Hηk=vnk+1−vn0=1.



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