Year: 1965
Type: article
Abstract: The purpose of this paper is two-fold.We define a class of spaces called solenoidal spaces, which generalize the solenoids of van Dantzig [13], and study their structure ( § §4,5).Then we use part of the structure developed to prove a theorem on the homogeneity of certain solenoidal spaces ( §6).A solenoidal space is the limit of an inverse limit sequence of "nice" spaces (the precise definition is below) where the bonding maps are regular covering maps (2).We will see that these spaces still have many of the properties of the classical solenoids.A space X is called homogeneous if for each pair x, y of points of X there is a homeomorphism of (X, x) onto (X, y).M. K. Fort, Jr.[8] asked the general question, "When is an inverse limit space homogeneous?"The pseudo-arc, which is known to be homogeneous (see Bing [l]), can be described roughly as an inverse limit of arcs where the bonding maps become sufficiently "crooked."Perhaps one can obtain some result on homogeneity in which "crookedness" of the bonding maps is one of the assumptions.(See Brown [4] for a precise definition of t-crooked map.)Probably one should restrict himself to the case where the factor spaces are 1-dimensional, for the following reason.Brown [4] has shown that an inverse limit of locally connected continua with sufficiently crooked bonding maps is hereditarily indecomposable; and Bing [2] has shown that if X is an ra-dimensional, hereditarily indecomposable continuum and n > 1, then X is not homogeneous.The word "probably" was used two sentences ago because of the fact that dimension may be lowered by taking an inverse limit of continua with "onto" bonding maps (although it can never be raised).The theorem in § 6 goes in the opposite direction from the preceding suggestion, by assuming that the bonding maps are "smooth." The assumption of local smoothness in the sense of differentiability will of course do no good.A kind of global smoothness is needed; this is why covering maps are appropriate.Case [5] has taken an inverse limit of universal curves where the bonding maps are regular covering maps to get a new example of a 1-dimensional homogeneous continuum containing arcs.Two of our theorems almost generalize two of his, but, as they stand, do not imply his.For notation and terminology on inverse limit sequences refer to [7].We
Author: M. C. McCord
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W1966464400