Application of derivative maps to Homotopy

Abstract

The perspective of unification of mathematical concepts of Group Actions and Homotopy have been the bottom line of our study.  We exploit this to a higher degree by investigating the derivatives associated with these.  Heading in this direction, this paper is about linking the derivative of Homotopy and the derivative of Group Action.  Firstly, we verify if the derivative of a Group Action is itself a Group action and whether the derivative of a Homotopy is a Homotopy.  For this purpose, we take only special Group Actions and Homotopies restricted to the Euclidean space.  We then discuss when the derivative of Group Action is a Homotopy and vice-versa.  Thus our aim here is to find if the derivative of a Homotopy can lead to the existence of a related Group Action and the relevant criteria that must be satisfied for such a relation.  

This paper also investigates when the derivative of Homotopy between two functions is a Homotopy in addition to being a Group Action as well.  The derivative of Group Action and Homotopy are dealt with in an attempt to find if the derivative of Group Action is also a Homotopy.  Since the derivative of a special action is also an action, we verify if this action is a Homotopy.  Thus we interlink the derivative of the concepts of our study as theorems/propositions.  In particular, we obtain conditions for the derivative of a Homotopy to be a Group Action.Summarizing, this paper is about finding the derivative of Group Action and Homotopy if the presence of one leads to the existence of the other and if so when.