On shape loop spaces (Q2417914)

The shape groups are well-known algebraic invariants in topology. These groups make a functorial relation between shape theory and algebraic topology. In coarse shape theory, which generalizes shape theory in many aspects, coarse shape groups have been introduced [\textit{N. Koceić Bilan}, Topology Appl. 157, No. 5, 894--901 (2010; Zbl 1192.55013)]. Since every shape group \(\check{\pi}_k(X,\cdot)\) naturally embeds in the coarse shape group \(\check{\pi}_k^\ast(X,\cdot)\), the study of coarse shape groups can be very useful, especially if \(\check{\pi}_k(X,\cdot)\) is trivial. There are several papers introducing a topology on the (coarse) shape groups in order to obtain a Hausdorff topological group [\textit{E. Cuchillo-Ibáñez} et al., Topology Appl. 94, No. 1--3, 51--60 (1999; Zbl 0939.54007); \textit{F. Ghanei} et al., ibid. 219, 17--28 (2017; Zbl 1365.55007)], or introducing a metric on the coarse shape groups [\textit{N. Koceić Bilan} and \textit{Z. Čuka}, Rad Hrvat. Akad. Znan. Umjet., Mat. Znan. 532, No. 21, 205--217 (2017; Zbl 1388.55009)]. In the present paper the authors show that the long exact sequence of (coarse) shape groups of a pointed (movable) pair of compacta is exact. Both results are known [\textit{S. Mardešić} and \textit{J. Segal}, Shape theory. The inverse system approach. North-Holland Mathematical Library, Vol. 26. Amsterdam - New York - Oxford: North-Holland Publishing Company. (1982; Zbl 0495.55001); \textit{N. Koceić Bilan}, Glas. Mat., III. Ser. 47, No. 1, 207--223 (2012; Zbl 1246.55012)] but here the authors prove the same property in the context of topological groups. They also consider the \(k\)-th shape loop space \(\check{\Omega}^{\boldsymbol{p}}_k(X,x)\) for the fixed expansion \(\boldsymbol{p}:(X,x)\to(\mathbf{X},\boldsymbol{x})\) of the pointed topological space \((X,x)\) which induces a functor \(\check{\Omega}^{\boldsymbol{p}}_k:\mathrm{Top}_\ast\to\mathrm{Top}_\ast\) keeping a homotopy for compact Hausdorff spaces.