Localization, Algebraic Loops and H-Spaces I

If (Y, µ) is an H-Space (here all our spaces are assumed to be finitely generated) with homotopy associative multiplication µ. and X is a finite CW complex then [X, Y] has the structure of a nilpotent group. Using this and the relationship between the localizations of nilpotent groups and topological spaces one can demonstrate various properties of [X,Y] (see [1], [2], [6] for example). If µ is not homotopy associative then [X, Y] has the structure of a nilpotent loop [7], [9]. However this algebraic structure is not rich enough to reflect certain significant properties of [X, Y]. Indeed, we will show that there is no theory of localization for nilpotent loops which will correspond to topological localization or will restrict to the localization of nilpotent groups.