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 $\Gamma$ be a co-Hopf space (which need not be co-associative) in ${\mathbf{Top}}_B^B$. The ${\Gamma }_B$-suspension space ${\Gamma }_{B}X$ and the ${\Gamma }_B$-loop space ${\Gamma }_B^{\ast }X$ of a fibrewise pointed space $X$ over $B$ are defined as generalization of the usual suspension space $\Sigma X$ and the loop space $\Omega X$ respectively, ${\Gamma }_B$-suspension spaces and ${\Gamma }_B$-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 ${\Gamma }_B$-suspension space ${\Gamma }_{B}X$ and ${\Gamma }_B$-loop space${\Gamma }_B^{\ast }X$ are proved. Short exact sequences of homotopy sets involving ${\Gamma }_B$-suspension spaces or ${\Gamma }_B$-loop spaces are obtained in the category of algebraic loops.