Topological Coarse Shape Homotopy Groups

2016/04/04 by Ghanei, Fateme, Mirebrahimi, Hanieh, Mashayekhy, Behrooz +1
#Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1604.00969

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))=\checkπ_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 \checkπ_k^*^top preserves finite product of compact Hausdorff spaces. Also, we show that for a pointed topological space (X,x), \checkπ_k^top(X,x) can be embedded in \checkπ_k^*^top(X,x).

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