# Fibrewise Decomposition of Generalized Suspension Spaces and Loop Spaces

### Nobuyuki Oda

Fukuoka University, Japan

## Abstract

We work in the category **Top***BB* of fibrewise pointed topological spaces over *B*. Let *Γ* be a co-Hopf space (which need not be co-associative) in **Top***BB*. 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.