Single loop space decompositions

Abstract

The method of single loop space decompositions, in which Ω X \Omega X is factored into minimal factors, is an important one for understanding the unstable homotopy of many simply-connected spaces X X . This paper begins with a survey of the major known theorems along these lines. We then give a necessary and sufficient condition for Ω X \Omega X to be decomposable as a product of spaces belonging to a certain list. We conclude with a nontrivial instance of an application of this condition.