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.
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