Published Paper Details:

Several mathematical concepts share a common feature that acts as a link between them. One such connection between the mathematical concepts of Group, Group Actions, and Homotopy is that they all involve projective maps. This underlying commonality allows us to establish a relationship between them, prompting a focused study of Group Actions and Homotopy to uncover this connection. While these concepts have each been the subject of extensive individual research, there has been little effort to determine if one can lead to the other and the conditions for such an interlinking function. Additionally, identifying Group Actions can be more challenging than finding Homotopic maps. Therefore, we anticipate that discovering an interlink could be beneficial. By relating these concepts on a common platform, we aim to determine if they can coexist through a single function. As a step in this direction, this paper is a part of the work that is about the condition for functions to be Homotopic under a Group Action and conversely. Here we derive the condition for a Homotopy to induce a group action. Our aim is to find a single map which can give rise to a Homotopy and a Group Action simultaneously. The extension of Homotopy to sets, vector-fields etc., serves to meet this purpose. Similarly, we extend Group Actions to general sets (Set-Action). We obtain conditions for a Homotopy to be a group, Action. This study serves as an approach to find Group Actions using existing Homotopy. Thus, we show here that a Homotopy leads to group action between two functions with special properties and theorems on these.