# Journal of the Ramanujan Mathematical Society

Volume 36, Issue 3, September 2021  pp. 231–241.

Core fundamental groupoid and covering projections

Authors:  Chidanand Badiger and T. Venkatesh
Author institution:Department of Mathematics, Rani Channamma University, Belagavi~591 156, Karnataka, India

Summary:  The notion of covering projection is an outstanding tool in computing the fundamental groups of some spaces. Here we have presented a few consequences based on covering projections and their induced groupoid homomorphisms on the Core fundamental groupoid. Developments about lifting correspondence and the existence of the lift in terms of the Core fundamental groupoids are discussed. Concerning the lift, we have introduced extended lifting correspondence and built a relation to it with the simply connected space. A~special kind of identification (quotient) on some space has been defined with the help of given any wide subgroupoid of the Core fundamental groupoid of the same space. Finally, a~special map is defined from identification space to base space, which becomes a covering projection, and moreover it is a topological homomorphism provided base space is a topological group. Further, we have given new characterizations to simply connected space employing the Core fundamental groupoid as in [6], in proposition 3.2, it is stated that Let M be a topological space then M is simply connected if and only if M is path-connected and the standard projection p: (π{1} M, I{p}) → (M, I{M}) is homeomorphism''. We~have discussed some results and related properties of induced groupoid homomorphisms on the Core fundamental groupoids alongside kernel and strong groupoid homomorphisms and their images as a subgroupoid.