Application of Homotopy to the Ageing Process of Human Body Within the Framework of Algebraic Topology

Abstract

This paper presents the ageing process of the human body from the homotopy concept in the algebraic topology. The compact connected human body with boundary is assumed to be topologically equivalent to a cylinder. The complex connected cylindrical shape of the body x=S^1*I, described by the functions f,h: x→x is a Cartesian product of the vertical interval I=[0,β] and a circle S^1. The initial state of the body x=S^1*I is the topological shape of the infant. The ageing process called the homotopy is the family of continuous functions f_t(x) on the interval I=[0,α] where α>β and t≥β. It is an increasing sequence of the function f_t(x) of the body x=S^1*I. The homotopy relates the topological shape of the infant to the topological shape of the adult. The homology theory assigns to the human body (x) a sequence of abelian groups H_n(x) for n=0,1,2…and to the continuous function a sequence of homomorphism. The homology group characterizes the number and continuity of the compact surface of human body. The study excludes genetic, hormonal, environmental and other factors in the ageing process and recognizes the fact that ageing continues throughout life, through childhood, and adolescence into adulthood. Topologically the infant is equal to the adult since the infant continuously grows into the adult. The study found an algebraic way of relating homotopy to the process of ageing of human body. It was also established that this could offer other topologists useful topological tips in the application of homotopy to other physical continuous processes.