# Fibrewise Decomposition of Generalized Suspension Spaces and Loop Spaces

### Nobuyuki Oda

Fukuoka University, Japan

## Abstract

We work in the category $Top_{B}$ of fibrewise pointed topological spaces over $B$. Let $Γ$ be a co-Hopf space (which need not be co-associative) in $Top_{B}$. The $Γ_{B}$-suspension space $Γ_{B}X$ and the $Γ_{B}$-loop space $Γ_{B}X$ 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 $Γ_{B}X$ and $Γ_{B}$-loop space$Γ_{B}X$ 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.

## Cite this article

Nobuyuki Oda, Fibrewise Decomposition of Generalized Suspension Spaces and Loop Spaces. Publ. Res. Inst. Math. Sci. 30 (1994), no. 2, pp. 281–295

DOI 10.2977/PRIMS/1195166134