CE equivalence and shape equivalence of 1-dimensional compacta (Q1091629)

Finite dimensional compacta X and Y are CE equivalent if there exist a finite dimensional compactum Z and cell-like maps of Z onto both X and Y. While every cell-like map between finite dimensional compacta is a shape equivalence, \textit{S. Ferry} [Proc. Am. Math. Soc. 80, 154-156 (1980; Zbl 0542.57003)] has shown that there is a planar continuum X such that \(Sh(X)=Sh(S^ 1)\) but X is not CE equivalent to \(S^ 1\). The authors study the relationship between CE equivalence and shape equivalence for locally connected 1-dimensional compacta. They show that: (1) every path connected planar continuum is CE equivalent either to a finite bouquet of circles or to the Hawaiian earring; and (2) every locally connected 1- dimensional continuum is the domain f a cell-like map onto a planar continuum. It follows that CE equivalence and shape equivalence agree on the class of all locally connected 1-dimensional compacta. On the other hand, for every \(n\geq 1\) an example is given of an n-dimensional LC\({}^{n-2}\) continuum X in \(R^{n+1}\) such that \(Sh(X)=Sh(S^ 1)\) but X is not CE equivalent to \(S^ 1\).