On snake cones, alternating cones and related constructions (Q2843835)

This paper is a continuation of papers by \textit{K. Eda, U. H. Karimov} and \textit{D. Repovš} [Fundam. Math. 195, No. 3, 193--203 (2007; Zbl 1148.54016) and Mediterr. J. Math. 10, No. 1, 519--528 (2013; Zbl 1263.54043)]. In these papers, the snake cone functor \(SC(-)\) and the alternating cone functor \(AC(-)\) were defined. In the current paper, the authors introduce variants of these constructions, the collapsed snake cone \(CSC(-)\) and the collapsed alternating cone \(CAC(-)\), which are respectively obtained from \(SC(-)\) and \(AC(-)\) by further collapsing, and investigate homotopy equivalences among the four spaces \(SC(X, x_0)\), \(AC(X, x_0)\), \(CSC(X, x_0)\), and \(CAC(X, x_0)\). The main results under review are:NEWLINENEWLINE\noindent i) If a path-connected compact Hausdorff space \(X\) is semi-locally strongly contractible at \(x_0\), then \(SC(X, x_0)\) (reps., \(AC(X, x_0)\)) is homotopy equivalent to \(CSC(X, x_0)\) (reps., \(CAC(X, x_0)\)).NEWLINENEWLINE\noindent iii) \(SC(S^1, x_0)\) is homotopy equivalent to \(AC(S^1, x_0)\) for every \(x_0\in S^1\).NEWLINENEWLINE\noindent iii) Let \(\mathbb H\) be the Hawaiian earring with the base point \(o\). Then \(SC({\mathbb H}, o)\) is not homotopy equivalent to \(AC({\mathbb H}, o)\) or \(CSC({\mathbb H}, o)\), \(AC({\mathbb H}, o)\) is not homotopy equivalent to \(CAC({\mathbb H}, o)\), but \(CSC({\mathbb H}, o)\) is homotopy equivalent to \(CAC({\mathbb H}, o)\). \smallskipNEWLINENEWLINE\noindent iv) Let \(T\) be the \(2\)-dimensional torus with the base point \(z_0\). Then \(SC(T, z_0)\) is not homotopy equivalent to \(AC(T, z_0)\) (and hence \(CSC(T, z_0)\) is not homotopy equivalent to \(CAC(T, z_0)\)).NEWLINENEWLINE\noindent v) Let \({\mathbb H}_T\) be the Hawaiian tori wedge, which is defined as the Hawaiian earring with each circle replaced by a torus. Then \(\pi_1({\mathbb H}_T)\) is isomorphic to the free \(\sigma\)-product of countably many copies of the free abelian group of rank two; \(\pi_2({\mathbb H}_T)\) is trivial; and \(H_2({\mathbb H}_T)\) is isomorphic to the free abelian group on countably many generators, where the generators are associated with the fundamental cycles of the tori.