Analytical consideration of growth in population via homological invariant in algebraic topology (Q780490)

Summary: 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 space \(X\) and any point \(v^n\in X\), where \(v^n\) is the \(n\)-dimensional surface, the group \(\eta =(X, v^n)\) is called population of the space \(X\). The increasing sequence from \(v_i^n\in X\) to \(v_j {}^n\in X\) for \(i<j\) provides the bases for the population growth. A growth in population \(\eta =(X, v^n)\) occurs if \(v_i^n<v_j^n\) for all \(v_i {}^n\in X\) and \(v_j^n\in X\). This is described by the homological invariant \(H(\eta_k)=1\). The aim of this paper is to construct the homological invariant \(H(\eta_k)\) and use \(H(\eta_k)=1\) 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.