A STUDY OF FUNCTOR ASSOCIATED WITH TRANSFORMATION GROUPS

Abstract

The aim of this paper is to construct a functor associated with transformation groups as well as investigate this functor.

In this paper we show that:-
i) for a given transformation group \((X,G)\), where \(X\) is a path connected pointed topological space with base point \(x_0\) and \(G\) is a group of homeomorphisms of \(X\), there always exists a covariant functor \(F\) from \(Tgh\) to \(Fgh\), where \(Tgh\) denotes the category of transformation groups and their continuous group homomorphisms and \(Fgh\) denotes the category of fundamental groups and their group homomorphisms;
ii) if the transformation groups \((X,G)\) and \((Y,H)\) have the same homotopy type, then the groups \(F(X,G)\) and \(F(Y,H)\) are isomorphic;
we also prove that
iii) The covariant functor \(F:Tgh\to Fgh\) is a homotopy type invariant.