Exactly 𝑘-to-1 maps and hereditarily indecomposable tree-like continua

Abstract

Abstract. In 1947, W.H. Gottschalk proved that no dendrite is the continuous, exactly k-to-1 image of any continuum if k ≥ 2. Since that time, no other class of continua has been shown to have this same property. It is shown that no hereditarily indecomposable tree-like continuum is the continuous, exactly k-to-1 image of any continuum if k ≥ 2.One of the earliest results concerning exactly k-to-1 maps between continua is W. H. Gottschalk's [2] result that no dendrite is the continuous exactly k-to-1 image of any continuum if k ≥ 2. Since Gottschalk's result, no other class of continua has been shown to repel exactly k-to-1 functions from continua in the manner that dendrites do. It is proved that no hereditarily indecomposable tree-like continuum is the continuous exactly k-to-1 image of any continuum if k ≥ 2. This result gives more information towards a resolution of a question posed by Nadler and Ward [10] i.e., which continua are k-to-1 images of continua, where k ≥ 2? The result also generalizes a result of J. Heath [5] who proved that no hereditarily indecomposable tree-like continuum is a two-to-one image of a continuum. It is known that for each k > 2 there exists a k-to-1 map between tree-like continua [4]. For more results concerning k-to-1 functions between continua, the reader is directed to a survey paper of J. Heath [7].A space is a compact metric space, a continuum is a nonempty, compact, connected metric space, and a map is a continuous function. If X and Y are spaces, then a map f from X into Y is said to be confluent if for any continuum L in the image, every component of f −1 (L) maps onto L. Lemma 1. Suppose that f is a confluent map onto a space