The coarse shape groups (Q2268843)

Coarse shape theory, \(\mathrm{Sh}^*\) [\textit{N. Koceić} and \textit{N. Uglešić}, Glas. Mat., III. Ser. 42, No. 1, 145--187 (2007; Zbl 1131.55005)] is a type of shape theory [\textit{S. Mardešić} and \textit{J. Segal}, Shape theory. The inverse system approach. North-Holland Mathematical Library, Vol. 26. Amsterdam etc.: North-Holland (1982; Zbl 0495.55001)] defined on the class of topological spaces. If we use \(\mathrm{Sh}\) to designate the classical shape theory of the preceding reference, then \(\mathrm{Sh}^*\) determines a coarser equivalence relation than that of \(\mathrm{Sh}\), although they agree if one restricts to the class of spaces having the homotopy type of polyhedra. There are also pointed coarse shape, \(\mathrm{Sh}_\star^*\) and pointed shape, \(\mathrm{Sh}_\star\). In this paper the author introduces functors \(\Check\pi_n^*:\mathrm{Sh}_\star^*\to\mathrm{Grp}\) (the category of groups) and compares them with the standard shape group functors \(\Check\pi_n:\mathrm{Sh}_\star\to\mathrm{Grp}\). The functor \(\Check\pi_n^*\) assigns to every pointed space \((X,\star)\) its \(n\)-th shape group \(\Check\pi_n^*(X,\star)\) which contains the \(n\)-th shape group \(\Check\pi_n(X,\star)\) as a subgroup. In general (e.g., for solenoids) \(\Check\pi_n(X,\star)=0\) does not imply that \(\Check\pi_n^*(X,\star)=0\), so coarse shape in such cases provides more information. It is proved that for pointed metrizable compacta \((X,\star)\), \(n\)-shape connectedness is equivalent to \(\Check\pi_k^*(X,\star)=0\) for every \(0\leq k\leq n\). For Part I, see the authors [Topology Appl. 156, No. 4, 710--720 (2009; Zbl 1166.54007)].