A STUDY OF THE GROUP OF COVERING TRANSFORMATION THROUGH FUNCTORS
The aim of this paper is to construct a functor \(Deck_x \) and we study the group of covering transformations through a functor \(Deck_x \). Let \(Deck_x (X)\) denote the set of all covering transformations of a covering \(p:(X,x)→(X,x_0)\), where \((X,x_0)\) is the arcwise connected, locally arcwise connected pointed topological space. In this paper we also study the group \(Deck_x (X)\) through the fundamental group functor \(π_1\). In this paper we show that: i) If \((X_1,x_1)\) and \((X_2,x_2)\) are equivalent covering of \((X,x_0)\) then the groups \(Deck_x (X_1)\) and \(Deck_x (X_2)\) are homomorphic; ii) \(Deck_x \) is a contravariant functor; iii) Let \((X_1,x_1)\) and \((X_2,x_2)\) are equivalent universal covering of \((X,x_0)\), where \(X\) is a connected manifold, then \[ Deck_x (X_1)/f_*π_1(X_1,x_1) Deck_x (X_2). \] , number=1, journal=Matematichki Bilten, author=Rana, Pravanjan Kr., year=2009, month=Jan., pages=21–24