Finite-sheeted covering spaces and a Near Local Homeomorphism Property for pseudosolenoids

Boroński, Jan P.; Sturm, Frank

doi:10.1016/j.topol.2013.10.024

Cited by 4 publications

(9 citation statements)

References 26 publications

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“…Based on the results in [7] and [21] we state the following conjectures, which may also direct potential results for Fraïssé limits. Conjecture 1.12.…”

Section: Introductionmentioning

confidence: 60%

“…is a nonmetric analogue of the pseudo-arc; M is homogeneous and hereditarily equivalent; the natural projection π α : M → X α is a closed map for each α < ω 1 (cf. Claim 3.1.1 in [7]). In addition, M has covering dimension 1, due to the following result of Katuta.…”

Section: Preliminariesmentioning

confidence: 97%

“…As we shall show, the reason the pseudo-circle lacks approximate projective homogeneity is because it n-fold covers itself for every positive integer n. This is also crucial in the proof of Theorem 1.5. Since the set of all self-covering maps is a dense G δ in the set of all surjections of the pseudo-circle [7], in a sense there is a generic choice of maps that will prevent the pseudo-circle from having the projective homogeneity. In addition, Fearnley ([15], p.510) showed that any plane separating circle-like continuum K admits a degree 1 surjection from C to K. Composing with the self-coverings of the pseudo-circle we get the following two results 2 Theorem 1.8.…”

Section: Introductionmentioning

confidence: 99%

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On the Conjecture of Wood and projective homogeneity

Boroński

Smith

2018

Journal of Mathematical Analysis and Applications

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Abstract. In 2005 Kawamura and Rambla, independently, constructed a metric counterexample to Wood's Conjecture from 1982. We exhibit a new nonmetric counterexample of a spaceL, such that C 0 (L, C) is almost transitive, and show that it is distinct from a nonmetric space whose existence follows from the work of Greim and Rajagopalan in 1997. Up to our knowledge, this is only the third known counterexample to Wood's Conjecture. We also show that, contrary to what was expected, if a one-point compactification of a space X is R.H. Bing's pseudo-circle then C 0 (X, C) is not almost transitive, for a generic choice of points. Finally, we point out close relation of these results on Wood's conjecture to a work of Irwin and Solecki on projective Fraïssé limits and projective homogeneity of the pseudo-arc and, addressing their conjecture, we show that the pseudo-circle is not approximately projectively homogeneous.

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“…Based on the results in [7] and [21] we state the following conjectures, which may also direct potential results for Fraïssé limits. Conjecture 1.12.…”

Section: Introductionmentioning

confidence: 60%

Section: Preliminariesmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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On the Conjecture of Wood and projective homogeneity

Boroński

Smith

2018

Journal of Mathematical Analysis and Applications

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“…As any pseudo-suspension will have non-trivial cohomology, we know that the only continua other than the pseudo-circle that we will obtain in an inverse limit of Handel homeomorphisms of pseudo-circles are precisely these pseudo-solenoids. In a similar vein, while this inverse limit approach could be applied to finite sheeted covering spaces of pseudo-solenoids, this would not result in any new examples as any finite-sheeted covering space of P is again homeomorphic to P [9]. Theorem 9.2 For each P, the homeomorphism H P of the P-adic pseudo-solenoid P is uniformly rigid, minimal and weakly mixing.…”

Section: Then (1) H Is Weakly Mixing If and Only If Each H I Is Weakly Mixing (2) H Is Minimal If And Only If Each H I Is Minimal And (3)mentioning

confidence: 99%

New Exotic Minimal Sets from Pseudo-Suspensions of Cantor Systems

2021

Self Cite

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We develop a technique, pseudo-suspension, that applies to invariant sets of homeomorphisms of a class of annulus homeomorphisms we describe, Handel–Anosov–Katok (HAK) homeomorphisms, that generalize the homeomorphism first described by Handel. Given a HAK homeomorphism and a homeomorphism of the Cantor set, the pseudo-suspension yields a homeomorphism of a new space that combines features of both of the original homeomorphisms. This allows us to answer a well known open question by providing examples of hereditarily indecomposable continua that admit homeomorphisms with positive finite entropy. Additionally, we show that such examples occur as minimal sets of volume preserving smooth diffeomorphisms of 4-dimensional manifolds.We construct an example of a minimal, weakly mixing and uniformly rigid homeomorphism of the pseudo-circle, and by our method we are also able to extend it to other one-dimensional hereditarily indecomposable continua, thereby producing the first examples of minimal, uniformly rigid and weakly mixing homeomorphisms in dimension 1. We also show that the examples we construct can be realized as invariant sets of smooth diffeomorphisms of a 4-manifold. Until now the only known examples of connected spaces that admit minimal, uniformly rigid and weakly mixing homeomorphisms were modifications of those given by Glasner and Maon in dimension at least 2.

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“…Note that the continuum of the set of real numbers R is enriched with an algebraic linear-ordering relation forming a chain such that (R, •, +, * ) becomes an ordered field under the algebraic linear order * ∈ {<, >}. Recall that the concept of continuum and compact chain can be formulated in a metric space (X, d), where X is connected and compact [2,3]. It was noted by Lefschetz that any compact metric space can admit topological chains as well as chain complexes, where the concept of B − chains retains the topological notion of local connectedness [4].…”

Section: Introductionmentioning

confidence: 99%

Homotopy of Linearly Ordered Split–Join Chains in Covering Spaces of Foliated n-Manifold Charts

Bagchi

2023

Symmetry

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Topological spaces can be induced by various algebraic ordering relations such as, linear, partial and the inclusion-ordering of open sets forming chains and chain complexes. In general, the classifications of covering spaces are made by using fundamental groups and lifting. However, the Riesz ordered n-spaces and Urysohn interpretations of real-valued continuous functions as ordered chains provide new perspectives. This paper proposes the formulation of covering spaces of n-space charts of a foliated n-manifold containing linearly ordered chains, where the chains do not form topologically separated components within a covering section. The chained subspaces within covering spaces are subjected to algebraic split–join operations under a bijective function within chain-subspaces to form simply directed chains and twisted chains. The resulting sets of chains form simply directed chain-paths and oriented chain-paths under the homotopy path-products involving the bijective function. It is shown that the resulting embedding of any chain in a leaf of foliated n-manifold is homogeneous and unique. The finite measures of topological subspaces containing homotopies of chain-paths in covering spaces generate multiplicative and cyclic group varieties of different orders depending upon the types of measures. As a distinction, the proposed homotopies of chain-paths in covering spaces and the homogeneous chain embedding in a foliated n-manifold do not consider the formation of circular nerves and the Nachbin topological preordering, thereby avoiding symmetry/asymmetry conditions.

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Finite-sheeted covering spaces and a Near Local Homeomorphism Property for pseudosolenoids

Cited by 4 publications

References 26 publications

On the Conjecture of Wood and projective homogeneity

On the Conjecture of Wood and projective homogeneity

New Exotic Minimal Sets from Pseudo-Suspensions of Cantor Systems

Homotopy of Linearly Ordered Split–Join Chains in Covering Spaces of Foliated n-Manifold Charts

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