Solenoids of pseudo-arcs

The search for homogeneous, circle-like continua progresses. There are three known homogeneous plane continua: the circle, the pseudo-arc [1], and the circle of pseudo-arcs [3]. C. E. Burgess [4] has shown that a circle-like homogeneous plane continuum must be one of these three. Burgess ' main tool, a theorem in [10], states that the pseudo-arc is the only homogeneous, hereditarily indecomposable, circle-like continuum. Solenoids are examples of homogeneous, nonplanar, circle-like continua. R. H. Bing [2] has shown that a homogeneous circle-like continuum that contains an arc must be a solenoid. C. L. Hagopian [8] has announced that a homogeneous circle-like continuum that does not contain an arc must contain a pseudo-arc. The purpose of this paper is to construct uncountably many homogeneous circle-like continua that do not contain arcs. These continua have the same relationship to the solenoids as the circle of pseudo-arcs has to a circle; hence they are called solenoids of pseudo-arcs. In the last section, we use the circle of pseudo-arcs to show that homogeneity is not a Whitney property. The appropriate terms are defined there. We adopt the notation of [3] except that we enlarge the class of circle-like continua to include circularly chainable, arc4ike continua. In particular, a snake-like continuum G is called a continuous snake-like arc of pseudo-arcs {gxl0 • x < 1} if there xists an open map f of G onto [0,1] such t at f-l(x) is the pseudo-arc gx ' Then {gx} is a continuous collection of pseudo-arcs filling G such that the corresponding decomposition space is an arc. Each pseudo-arc gx is called a pseudo-arc of G. The continua go and gl are called the end elements of {gx}. We use [ga,g b] to denote the sum of all continua in the collection {gxla < x < b}. The map f is called an indexing map, and [0,1] an indexing arc. (Note that a local homeomorphism betwee