Topological Localization, Category and Cocategory

Abstract

It is easy to see that a localization (in the sense of [ 9 ]) of a simply connected co H -space (equivalently a simply connected space of Liusternik-Schnirelman category one) is again a co H -space. (All spaces in this paper will be pointed and have the based homotopy type of a connected CW complex; and all maps will preserve base-points.) We show that the category of a simply connected space does not increase on localizing. We give an example to show that the hypothesis simple-connectivity is crucial. In strong contrast, the dual result only requires connectivity.