Fibrewise Decomposition of Generalized Suspension Spaces and Loop Spaces

Abstract

We work in the category \mathbf{Top}^B_B of fibrewise pointed topological spaces over B . Let Γ be a co-Hopf space (which need not be co-associative) in \mathbf{Top}^B_B . The Γ_B -suspension space Γ_BX and the Γ_B -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, Γ_B -suspension spaces and Γ_B -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 Γ_B -suspension space Γ_BX and Γ_B -loop space Γ^*_BX are proved. Short exact sequences of homotopy sets involving Γ_B -suspension spaces or Γ_B -loop spaces are obtained in the category of algebraic loops.