A Study of Functors Associated with Topological Groups

The aim of this paper is to construct functors associated with topological groups as well as to investigate these functors. More precisely, we prove that for a given topological groups G there always exists a contravariant functor F (G) from the homotopy category of pointed topological spaces and homotopy classes of base point preserving continuous maps to the category of groups and homomorphisms. We also prove that (i) the functor F (G) is natural in G in the sense that if the topological groups G and H have the same homotopy type then the groups F (G)(X) and F (H)(X) are isomorphic, for every pointed topological space X; and (ii) the functor F (G) is homotopy type invariant in the sense that if X and Y are two pointed spaces having the same homotopy type then the groups F (G)(X) are F (G)(Y ) are isomorphic. Moreover, given two topological groups G and H and a continuous homomorphism α : G → H, we show that there always exists a natural transformation between the functors F (G) and F (H) associated with topological groups G and H respectively.