Embedding Solenoids

A generalized solenoid is an inverse limit space with bonding maps that are (regular) covering maps of closed compact manifolds. We study the embedding properties of solenoids in linear space and in foliations. A compact and connected topological space is called a continuum. A space X is homogeneous if for every x, y ∈ X there exists a homeomorphism h : X → X such that h(x) = y. The space is bihomogeneous if for every x, y ∈ X there exists a homeomorphism that switches x and y, i.e., h(x) = y and h(y) = x. Settling an old problem of Knaster, K. Kuperberg [11] showed that there exist continua that are homogeneous but not bihomogeneous. In the present paper we show that there exist homogeneous continua in R4 that are not bihomogeneous. It is unknown whether there exist such continua in Rk for some k < 4. The exmamples we provide are higher-dimensional analogues of the standard dyadic solenoid. We refer to these spaces as (generalized) solenoids. First we study embeddings of generalized solenoids in linear space. This turns out to be related to the study of embeddings of manifolds. Next we study embeddings of solenoids in foliated bundles. In the final part of our paper we generalize to higher dimensions a dynamical characterization of one-dimensional solenoids by Thomas. We also show that a conjecture by Oversteegen on a possible topological characterization of solenoids turns out to be related to an old conjecture of Williams on expanding attractors, that was settled by Farrell and Jones [6].