Exactly two-to-one maps from continua onto some tree-like continua

Abstract

Abstract. It is known that no dendrite (Gottschalk 1947) and no hereditarily indecomposable tree-like continuum (J. Heath 1991) can be the image of a continuum under an exactly 2-to-1 (continuous) map. This paper enlarges the class of tree-like continua satisfying this property, namely to include those tree-like continua whose nondegenerate proper subcontinua are arcs. This includes all Knaster continua and Ingram continua. The conjecture that all tree-like continua have this property, stated by S. Nadler Jr. and L. E. Ward Jr. (1983), is still neither confirmed nor rejected.

No arc-connected treelike continuum is the 2-to-1 image of a continuum

“…If f is a simple map from a continuum X onto a dendroid and f (x) = f (y), then either (1) there is an arc in X from x to y, or (2) there are disjoint twin arcs xx and yy in X such that for some indecomposable continuum I in X containing x and y , f (x ) is the center of f (I), or (3) there is an indecomposable continuum I in X containing x and y such that f (x) is the center of f (I).…”

“…Suppose |f We will extend the maximal triple T (m) to complete the contradiction. Using M as the continuum X in Lemma 6, and using x and y to denote the two inverse points of m, there are only three possibilities: (1) there is an arc in M from x to y, (2) there are disjoint twin arcs xx and yy in M and an indecomposable continuum J in M containing x and y such that f (x ) is the center of f (J), or (3) there is an indecomposable continuum J in M containing x and y such that f (x) is the center of f (J).…”

“…continuum is an arc, then again [2] it cannot be a 2-to-1 image; this class of continua includes all Knaster continua and the Ingram continua [7].…”

“…For the middle of the spectrum, it is known [2] that if a treelike continuum is the finite union of subcontinua each of which is not a 2-to-1 image, and no two of which intersect in more than one point, then the union also is not a 2-to-1 image. So, using this structure many hybrid treelike continua can be shown not to be 2-to-1 images.…”

Tree-like continua and 2-to-1 maps

“…That question is answered only for some cases. For instance, it is known that a tree-like continuum cannot be the image of a 2-to-1 map if it is hereditarily indecomposable [10], or if it is an indecomposable arc-continuum (every proper subcontinuum is an arc) [3]. In the decomposable case, Gottschalk [4] showed that no dendrite can be the image of a 2-to-1 map.…”