Topology and its Applications
In this paper the relationship between CE equivalence and shape equivalence for locally connected, 1-dimensional compacta is investigated. Two theorems are proved. The first asserts that every path connected planar continuum is CE equivalent either to a bouquet of circles or to the Hawaiian earring. The second asserts that for every locally connected, 1-dimensional continuum X there is a cell-like map of X onto a planar continuum. It follows that CE equivalence and shape equivalence are the same for the class of all locally connected, 1-dimensional compacta. In addition, an example of Ferry is generalized to show that for every n≥1 there exists an n-dimensional, LCn-2 continuum Y such that Sh(Y)=Sh(S1) but Y is not CE equivalent to S1.
Daverman, R. J. and Venema, Gerard A., "CE equivalence and shape equivalence of 1-dimensional compacta" (1987). University Faculty Publications. 469.