Core Fundamental Groupoid Bundle, Its Sections and Relatedness of Homeomorphisms and Sections

Abstract

In this paper, we introduce bundle, fibre bundle and principal G-bundle structures on the Core fundamental groupoid keeping its standard projections and quotient topology intact. We give an explicit description of Core fundamental groupoids as such bundles including for the uniquely geodesic spaces and formulate some results on bundle maps. Further, we introduce sections on the Core fundamental groupoid bundle, and also, present some basic properties including composition and inverse with the help of the induced groupoid homomorphisms on the Core fundamental groupoids. With a group structure on the set of all continuous sections of the Core fundamental groupoid bundle, a group action has been built on the Core fundamental groupoid. A notion of relatedness of homeomorphism and section are defined and discussed the pushforward, pullback of sections and their properties. Finally, we investigate more about relatedness notions relΓ0(¯π1M)(f), relHomeo(M)(X) and same on the subsets based on both section related homeomorphisms and homeomorphism related sections. Further, some consequences based on an algebraic structure on the new class of relΓ0(¯π1M)(f), relHomeo(M)(X), etc. have been placed. We present an interrelationship between subsets of Homeo(M) and Γ 0 (¯π1M), which have nice applications in the leftinvariant sections and topological groups.