Regular topological group action yield sections of Core higher homotopy groupoid bundle

In this paper, we discuss sections of higher homotopy groupoid bundles those yield by group actions. We study some basic properties related to the left-invariant (G-invariant) of such sections and also present some conditions for such sections to be left-invariant (G-invariant). We concentrate a kind of group action on the Core higher homotopy groupoid bundle of certain topological spaces which undergo a regular topological group action. Interestingly, each element of orbit space yields continuous sections of the Core higher homotopy groupoid bundle having beautiful algebraic properties. Moreover, each of those continuous sections behaves nicely as a left-invariant section with respect to topological group action by G. Since higher homotopy groups are abelian, the set of such sections forms always an abelian group. Moreover, each element of the group is a left-invariant section, therefore, it is convenient to call them commutative group of left-invariant sections of the Core higher homotopy groupoid bundle. Further, we show this group is a topological invariant and study some basic properties of it. At the end of the paper, we give interrelations between the retraction map and this group.