Homogeneous matchbox manifolds

Alex Clark, Steven Hurder
2012 Transactions of the American Mathematical Society  
We prove that a homogeneous matchbox manifold is homeomorphic to a McCord solenoid, thereby proving a strong version of a conjecture of Fokkink and Oversteegen, which is a general form of a conjecture of Bing. A key step in the proof shows that if the foliation of a matchbox manifold has equicontinuous dynamics, then it is minimal. Moreover, we then show that a matchbox manifold with equicontinuous dynamics is homeomorphic to a weak solenoid. A result of Effros is used to conclude that a
more » ... clude that a homogeneous matchbox manifold has equicontinuous dynamics, and the main theorem is a consequence. The proofs of these results combine techniques from the theory of foliations and pseudogroups, along with methods from topological dynamics and coding theory for pseudogroup actions. These techniques and results provide a framework for the study of matchbox manifolds in general, and exceptional minimal sets of smooth foliations. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3152 ALEX CLARK AND STEVEN HURDER where for ≥ 0, M is a compact, connected, n-dimensional manifold without boundary, and the maps p +1 : M +1 → M are proper covering maps. A Vietoris solenoid is a 1-dimensional solenoid, where each M is a circle. If all of the defined compositions of the covering maps p are normal coverings, then S is said to be a McCord solenoid. McCord solenoids are homogeneous [31], and conversely, Fokkink and Oversteegen showed in [22] that any homogeneous n-dimensional solenoid is homeomorphic to a McCord solenoid. An n-dimensional foliated space M is a continuum which has a local product structure [12, 36] ; that is, every point of M has an open neighborhood homeomorphic to an open subset of R n times a compact metric space (the local transverse model). The leaves of the foliation F of M are the maximal connected components with respect to the fine topology on M induced by the plaques of the local product structure. Precise definitions are given in Section 2. A matchbox manifold is a foliated space M such that the local transverse models are totally disconnected. Intuitively, a 1-dimensional matchbox manifold M has local coordinate charts U which are homeomorphic to a "box of matches". Manifolds and n-dimensional solenoids provide examples of matchbox manifolds. As remarked above, every homogeneous 1-dimensional matchbox manifold is homeomorphic to a circle or a solenoid [1]. Our primary result is the generalization of this 1-dimensional result to n-dimensions, thereby proving a strong version of a conjecture of Fokkink and Oversteegen [22, Conjecture 4] under a smoothness assumption, as clarified in Section 2. Theorem 1.2. Let M be a homogeneous smooth matchbox manifold. Then M is homeomorphic to a McCord solenoid. In particular, M is minimal. As a consequence of Theorem 1.2 and the impossibility of codimension-one embeddings of solenoids as shown in [14] , we obtain the following corollary, which is a generalization of the result of Prajs [39] that any homogeneous continuum in R n+1 which contains an n-cube is an n-manifold. 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. The work [16] by the authors studies the problem of finding smooth embeddings of solenoids into foliated manifolds with codimension q ≥ 2. It is an open problem, in general, to determine the lowest codimension q > 1 in which a given solenoid can be embedded, either into a compact manifold, or as a minimal set for a C r -foliation of a compact manifold. The proof of the main theorem involves drawing an important connection between homogeneity and equicontinuity, based on the fundamental result of Effros that transitive continuous actions of Polish groups are micro-transitive [5, 19, 51, 52] . As a step in the proof of Theorem 1.2, we show in Theorem 5.2 that Effros' Theorem implies that a homogeneous matchbox manifold is equicontinuous. Combining the results of Theorem 8.9 and Proposition 10.1, we obtain: Theorem 1.4. A smooth matchbox manifold M is homeomorphic to an n-dimensional solenoid if and only if M is equicontinuous.
doi:10.1090/s0002-9947-2012-05753-9 fatcat:ja4wfdmczffb5hp4whkznq4bu4