Homogeneous matchbox manifolds (Q2838119)

In 1960 R. H. Bing proved that if \(X\) is a homogeneous circle-like continuum that contains an arc, then either \(X\) is homeomorphic to a circle or to a Vietoris solenoid. He also raised the following question. If \(X\) is a homogeneous continuum, and if every proper subcontinuum of \(X\) is an arc, must the \(X\) be a circle or a solenoid? Affirmative answers to this question were given by Hagopian (1977), Mislove, Rogers (1989), and Aarts, Hagopian and Oversteegen (1991). In this paper the generalization of this result to \(n\)-dimensional matchbox manifolds, for \(n\geq 1\), is proven.NEWLINENEWLINEAn \(n\)-dimensional solenoid is an inverse limit \(S=\lim_{\leftarrow}\{p_{\ell+1}: M_{\ell +1}\rightarrow M_{\ell}\}\), where for \(\ell \geq 0\), \(M_{\ell}\) is a compact, connected, \(n\)-dimensional manifold without boundary. The maps \(p_{\ell+1}: M_{\ell +1}\rightarrow M_{\ell}\) are proper covering maps. A Vietoris solenoid is a \(1\)-dimensional solenoid, where each \(M_{\ell}\) is a circle. If all of the defined compositions of the covering maps \(p_{\ell}\) are normal coverings, then \(S\) is said to be a McCord solenoid.NEWLINENEWLINEAn \(n\)-dimensional foliated space \(M\) is a continuum which has a local product structure: every point of \(M\) has an open neighborhood homeomorphic to an open subset of \(\mathbb{R}^n\) times a compact metric space. More precisely, a continuum \(M\) is a foliated space of dimension \(n\) if there exists a compact separable metric space \(X\), and for each \(x\in M\) there is a compact subset \(T_x\subset X\), an open subset \(U_x\subset M\), and a homeomorphism defined on its closure \(\varphi_{x} : \bar{U_x}\rightarrow [-1, 1]^n\times T_x\) such that \(\varphi_x(x) = (0, w_x)\), where \(w_x\in {}^{\text{Int}}(T_x)\). The subspace \(T_x\) of \(X\) is called the local transverse model at \(x\). A matchbox manifold is a foliated space \(M\) such that the local transverse models are totally disconnected.NEWLINENEWLINEThe following are the main results of the paper.NEWLINENEWLINE{ Theorem 1.2.} Let \(M\) be a homogeneous smooth matchbox manifold. Then \(M\) is homeomorphic to a McCord solenoid. In particular, \(M\) is minimal.NEWLINENEWLINE{ Corollary 1.3.} Let \(M\) be a homogeneous, smooth \(n\)-dimensional matchbox manifold which embeds in a closed orientable \((n +1)\)-dimensional manifold. Then \(M\) is a manifold.NEWLINENEWLINE{ Theorem 1.4.} A smooth matchbox manifold \(M\) is homeomorphic to an \(n\)-dimensional solenoid if and only if \(M\) is equicontinuous.