Exactly two-to-one maps from continua onto arc-continua

Abstract

Continuing studies on 2-to-1 maps onto indecomposable continua having only arcs as proper non-degenerate subcontinua—called here arc-continua—we drop the hypothesis of tree-likeness, and we get some conditions on the arc-continuum image that force any 2-to-1 map to be a local homeomorphism. We show that any 2-to-1 map from a continuum onto a local Cantor bundleY is either a local homeomorphism or a retraction ifY is orientable, and that it is a local homeomorphism ifY is not orientable. Define X to be an arc-continuum if X is a (metric) continuum and each proper non-degenerate subcontinuum of X is an arc. In an earlier paper (5) we showed that there is no exactly 2-to-1 continuous map from any continuum onto a tree-like arc-continuum (to partially answer a question raised by Sam Nadler, Jr. and L. E. Ward, Jr. (14)) by first showing that any such map must be a local homeomorphism (i.e. a 2-fold covering map). In this paper we continue our study of exactly 2-to-1 maps from continua onto arc-continua, without the hypothesis of tree-likeness, and we have found some simple conditions on the arc-continuum image (different for orientable and non-orientable cases) that force any 2-to-1 map to be a local homeo- morphism. In the case of an indecomposable arc-continuum Y that is a local Cantor bundle, we show that any 2-to-1 map from a continuum onto Y is either a local homeomorphism or a retraction if Y is orientable (both situations can be realized), and any 2-to-1 map from a continuum onto Y is a local homeomorphism if Y is not orientable. Thus more is now known about what kinds of 2-to-1 maps are possible onto these types of spaces, including all solenoids. A decomposable arc-continuum is the union of two arcs, thus an arc or a simple closed curve. Harrold showed in 1940 (8) that the arc is not the