Towards the algebraic characterization of (coarse) shape path connectedness (Q1939214)

Coarse shape theory [\textit{N. Koceić Bilan} and \textit{N. Uglešić}, Glas. Mat., III. Ser. 42, No. 1, 145--187 (2007; Zbl 1131.55005)] generalizes shape theory and using the recent notion of coarse shape path connectedness introduced in [ibid. Ser. 46, No. 2, 489--503 (2011; Zbl 1236.55017)], the author describes actions that coarse shape paths induce on coarse shape groups and homotopy progroups. He presents interesting results for coarse shape path connected spaces: the independence of \(n\)-shape connectedness and independence of coarse shape groups from the choice of the base point. He also shows the relationship between (coarse) shape path connectedness and the triviality of some low dimensional (coarse) shape groups and homotopy pro-groups. In addition, the author presents a necessary (sufficient) condition for (coarse) shape path connectedness. Definitory results are: Theorem 4. Let \(X\) be a compactum. If there exists an \(x_0 \in X\) such that \(\check{\pi_0}(X,x_0)=0\) then, for every \(x_0'\in X\), there exists a coarse shape path from \(x_0\) to \(x_0'\). Theorem 6. Let \(X\) be a space. If there exists \(x_0 \in X\) such that \(\check{\pi_0}(X,x_0)=0\) and \(pro\)-\(\pi_1(X,x_0) \cong 0\) then, for every \(x_0' \in X\) there exists a shape path from \(x_0\) to \(x_0'\).