Homogeneous weak solenoids
Robbert Fokkink,Lex Oversteegen +1 more
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A weak solenoid is an inverse limit space over manifolds with bonding maps that are covering maps as discussed by the authors, and strong solenoids are topologically equivalent to strong solens.Abstract:
A (generalized) weak solenoid is an inverse limit space over manifolds with bonding maps that are covering maps. If the covering maps are regular, then we call the inverse limit space a strong solenoid. By a theorem of M.C. McCord, strong solenoids are homogeneous. We show conversely that homogeneous weak solenoids are topologically equivalent to strong solenoids. We also give an example of a weak solenoid that has simply connected path-components, but which is not homogeneous.read more
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Homogeneous matchbox manifolds
Alex Clark,Steven Hurder +1 more
TL;DR: In this paper, it was shown that a homogeneous matchbox manifold of any finite dimension is homeomorphic to a McCord solenoid, thereby proving a strong version of a conjecture of Fokkink and Oversteegen.
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The discriminant invariant of Cantor group actions
TL;DR: In this article, the authors investigate the dynamical and geometric properties of weak solenoids, as part of the development of a "calculus of group chains" associated to Cantor minimal actions.
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Molino theory for matchbox manifolds
TL;DR: In this article, the Molino theory for equicontinuous matchbox manifolds is extended to the case of totally disconnected transversals, which is a special case of weak solenoids.
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Embedding solenoids in foliations
Alex Clark,Steven Hurder +1 more
TL;DR: In this paper, the authors show that if a smooth foliation F of a manifold M contains a compact leaf L with H 1 (L ; R ) not equal to 0, then there exists a foliation on M which is C 1 -close to F, and F has an uncountable set of solenoidal minimal sets contained in U that are pairwise non-homeomorphic.
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Classifying matchbox manifolds
TL;DR: In this article, it was shown that if the base manifolds satisfy a strong form of the Borel Conjecture, then return equivalence for the dynamics of their foliations implies the total spaces are homeomorphic.
References
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Book
The Topology of Fibre Bundles.
TL;DR: In this paper, a succint introduction to fiber bundles is provided, which includes such topics as differentiable manifolds and covering spaces, and a brief survey of advanced topics, such as homotopy theory and cohomology theory, before using them to study further properties of fibre bundles.
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A basic course in algebraic topology
TL;DR: In this article, the fundamental group of two-dimensional manifolds has been studied in the context of homology theory and its application in the theory of cocomology and product spaces.