Topological Coarse Shape Homotopy Groups

TL;DR

This paper introduces a topology on coarse shape homotopy groups of topological spaces, making them Hausdorff topological groups, and explores their properties like second countability and product preservation.

Contribution

It extends the topology on shape morphisms to coarse shape homotopy groups, establishing their topological group structure and key properties.

Findings

The coarse shape homotopy groups form Hausdorff topological groups.

They preserve finite products of compact Hausdorff spaces.

Embedding of traditional into topological coarse shape homotopy groups is demonstrated.

Abstract

Uchillo-Ibanez et al. introduced a topology on the sets of shape morphisms between arbitrary topological spaces in 1999. In this paper, applying a similar idea, we introduce a topology on the set of coarse shape morphisms $Sh^*(X,Y)$, for arbitrary topological spaces $X$ and $Y$. In particular, we can consider a topology on the coarse shape homotopy group of a topological space $(X,x)$, $Sh^*((S^k,*),(X,x))=\check{\pi}_k^{*}(X,x)$, which makes it a Hausdorff topological group. Moreover, we study some properties of these topological coarse shape homotopoy groups such as second countability, movability and in particullar, we prove that $\check{\pi}_k^{*^{top}}$ preserves finite product of compact Hausdorff spaces. Also, we show that for a pointed topological space $(X,x)$, $\check{\pi}_k^{top}(X,x)$ can be embedded in $\check{\pi}_k^{*^{top}}(X,x)$.