RPT-Constructions

Our aim in this chapter is to represent the spaces of based and free loops Ω(X) and Λ(X) as well as other associated spaces of a simply connected, countable CW-complex X with a single 0-vertex, e, as CW-complexes of a certain kind. In the case of Ω(X), the complex will have a nondegenerate product corresponding to the usual path multiplication. The first example of such constructions is the reduced-product complex X_∞ of [63, James]. The complexes we construct generalize his construction and will be called constructions of the reduced-product type, or RPT-complexes for short.