On shape loop spaces

In this paper, considering the kth shape loop space (Omega) over cap (p)(k) (X, x), for an HPol(*)-expansion p : (X, x) -> ((X-lambda , x(lambda)), [p'], A) of a pointed topological space (X, x), first we show that (Omega) over cap (p)(k) (X, x) is an H-group for every topological space (X, x). Then we prove that Omega(c)(k) : Top(*) -> Top(*) is a functor, for all k is an element of N-0, where c is the tech HPol.-expansion of spaces. Moreover, with some assumptions, we show that if X is pi(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. (C) 2019 Elsevier B.V. All rights reserved.