A STUDY OF THE GROUP OF COVERING TRANSFORMATION THROUGH FUNCTORS

Abstract

The aim of this paper is to construct a functor \(Deck_{\tilde{x}}\) and we study the group of covering transformations through a functor \(Deck_{\tilde{x}}\). Let \(Deck_{\tilde{x}}(\widetilde{X})\) denote the set of all covering transformations of a covering \(p:(\widetilde{X},\tilde{x})\to(X,x_0)\), where \((X,x_0)\) is the arcwise connected, locally arcwise connected pointed topological space.

In this paper we also study the group \(Deck_{\tilde{x}}(\widetilde{X})\) through the fundamental group functor \(\pi_1\).

In this paper we show that:

i) If \((\widetilde{X}_1,\tilde{x}_1)\) and \((\widetilde{X}_2,\tilde{x}_2)\) are equivalent covering of \((X,x_0)\) then the groups \(Deck_{\tilde{x}}(\widetilde{X}_1)\) and \(Deck_{\tilde{x}}(\widetilde{X}_2)\) are homomorphic;

ii) \(Deck_{\tilde{x}}\) is a contravariant functor;

iii) Let \((\widetilde{X}_1,\tilde{x}_1)\) and \((\widetilde{X}_2,\tilde{x}_2)\) are equivalent universal covering of \((X,x_0)\), where \(X\) is a connected manifold, then
\[
Deck_{\tilde{x}}(\widetilde{X}_1)/f_*\pi_1(\widetilde{X}_1,\tilde{x}_1)
\cong
Deck_{\tilde{x}}(\widetilde{X}_2).
\]