Topology and its Applications | 2019
On shape loop spaces
Abstract
Abstract In this paper, considering the kth shape loop space Ω ˇ k p ( X , x ) , for an HPol ⁎ -expansion p : ( X , x ) → ( ( X λ , x λ ) , [ p λ λ ′ ] , Λ ) of a pointed topological space ( X , x ) , first we show that Ω ˇ k p ( X , x ) is an H-group for every topological space ( X , x ) . Then we prove that Ω ˇ k c : T o p ⁎ ⟶ T o p ⁎ is a functor, for all k ∈ N 0 , where c is the Cech HPol ⁎ -expansion of spaces. Moreover, with some assumptions, we show that if X is π k -shape injective, then X is k-homotopically Hausdorff. Finally, we show that the long exact sequence of shape groups (coarse shape groups) of a pointed pair of certain spaces, is also exact as topological groups.