Fibrewise Decomposition of Generalized Suspension Spaces and Loop Spaces

We work in the category Topf of fibrewise pointed topological spaces over B. Let r be a co-Hopf space (which need not be co-associativ e) in Topf. The /'^-suspension space rBX and the r^-loop space P\X of a fibrewise pointed space X over B are defined as generalizatio n of the usual suspension space Ji A" and the loop space QX respectively, /^-suspension spaces and /""s-loop spaces have some properties similar to those of the usual suspension spaces and loop spaces. This is an example of Eckmann-Hilto n duality. In this paper, decomposition theorems of /"^-suspension space rBX and Ts-loop space r%X are proved. Short exact sequences of homotopy sets involving /"^-suspension spaces or r#-loop spaces are obtained in the category of algebraic loops.