Fibrewise Decomposition of Generalized Suspension Spaces and Loop Spaces

Abstract

We work in the category TopBB of fibrewise pointed topological spaces over B. Let Γ be a co-Hopf space (which need not be co-associative) in TopBB. The ΓB-suspension space ΓBX and the ΓB-loop space Γ*BX of a fibrewise pointed space X over B are defined as generalization of the usual suspension space ΣX and the loop space ΩX respectively, ΓB-suspension spaces and ΓB-loop spaces have some properties similar to those of the usual suspension spaces and loop spaces. This is an example of Eckmann-Hilton duality. In this paper, decomposition theorems of ΓB-suspension space ΓBX and ΓB-loop space Γ*BX are proved. Short exact sequences of homotopy sets involving ΓB-suspension spaces or ΓB-loop spaces are obtained in the category of algebraic loops.