The orientability of matchbox manifolds

Aarts, Johannes; Hagopian, Charles L.; Oversteegen, Lex G.

doi:10.2140/pjm.1991.150.1

Cited by 10 publications

(24 citation statements)

References 15 publications

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“…This results inspired the subsequent works by McCord [29], Thomas [33], Hagopian [22], Mislove and Rogers [30], and Aarts, Hagopian and Oversteegen [3], all for 1-dimensional flow spaces.…”

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confidence: 62%

Manifold-like matchbox manifolds

Clark

Hurder

Lukina

2019

Proc. Amer. Math. Soc.

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A matchbox manifold is a generalized lamination, which is a continuum whose arccomponents define the leaves of a foliation of the space. The main result of this paper implies that a matchbox manifold which is manifold-like must be homeomorphic to a weak solenoid. DEFINITION 1.2. A continuum X is said to be manifold-like, if there exists n ≥ 1 such that for every ǫ > 0, there exists M ǫ ∈ M(n) and an ǫ-map f ǫ : X → M ǫ .The study of the properties of ǫ-maps and manifold-like continua has a long history in the study of the topology of spaces. Eilenberg showed in [19] that an ǫ-map, for ǫ > 0 sufficiently small, admits a left approximate inverse. Ganea studied the properties of compact, locally connected manifoldlike ANR's of dimension n in [20], and showed that such a space has the homotopy type of a closed n-manifold. Deleanu [12,13] showed that an n-dimensional connected polyhedron which is manifoldlike is a closed pseudo-manifold. Bob Edwards gave in his 1978 ICM address [18] an overview of the further applications of ǫ-approximations and homeomorphisms.Mardešić and Segal [26,27] studied the properties of manifold-like connected polyhedron, and gave conditions under which such spaces must be a topological manifold. These authors used a technique of approximation of the given continuum by an inverse limit of spaces, and noted that their results do not apply to a continuum which is not locally connected, such as the dyadic solenoid.The goal of this work is to characterize a class of manifold-like continua for which the Mardešić and Segal results do not apply. These are the matchbox manifolds as studied by the authors in [9,10,11], and discussed below. Our study of matchbox manifolds in these works was inspired by a result of Bing in [6]. Recall that a topological space X is homogeneous if for every x, y ∈ X, there exists a homeomorphism h : X → X such that h(x) = y. THEOREM 1.3. Let X be a homogeneous, circle-like continuum that contains an arc. Then either X is homeomorphic to the circle S 1 , or to an inverse limit of coverings of S 1 .

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“…This results inspired the subsequent works by McCord [29], Thomas [33], Hagopian [22], Mislove and Rogers [30], and Aarts, Hagopian and Oversteegen [3], all for 1-dimensional flow spaces.…”

supporting

confidence: 62%

Manifold-like matchbox manifolds

Clark

Hurder

Lukina

2019

Proc. Amer. Math. Soc.

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“…This result was conjectured by Bing [5]. Later, different proofs were given in [18] and [2]. Both these proofs concern the local product structure of a homogeneous one-dimensional space and, as a byproduct, yield another characterization.…”

Section: Theorem 1 (Hagopian [10]) a Homogeneous Metric Continuum Imentioning

confidence: 91%

Homogeneous weak solenoids

Fokkink

Oversteegen

2002

Trans. Amer. Math. Soc.

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118

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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 pathcomponents, but which is not homogeneous.

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“…A continuous onto map f : 1 and for every edge e 2 ∈ E 2 there is an edge e 1 ∈ E 1 such that f | e 2 \V 2 is a homeomorphism onto e 1 \ V 1 or a constant map. The map f : X 2 → X 1 is simplicial if it is simplicial relative to some vertex sets of X 1 and X 2 .…”

Section: Branched Matchbox Manifoldsmentioning

confidence: 99%

“…Since γ is an element of R(X), there is a continuous map g : X → R such that γ(x) = exp(2πig(x)). [1] in Br(X) as γ is homotopic to the identity element in Br(X). Hence the equivalence class of h is the identity element in lim − → G k , for ι : lim − → G k → Br(X) is an isomorphism.…”

Section: Therefore the Isomorphisms ιmentioning

confidence: 99%

“…In Section 2, using work of Aarts and Oversteegen ( [1]), Mardešić and Segal ([12]) and Rogers ([14]), we define compact connected orientable branched matchbox manifolds, and show that they all have presentations by orientation preserving maps of finite directed nondegenerate graphs. In Section 3, we show that the Bruschlinsky group with the winding order of a compact connected orientable branched matchbox manifold with the Simplicity Condition is order isomorphic to the direct limit of the graph groups with the standard order defined from the presentation (and therefore the winding and standard orders agree).…”

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confidence: 99%

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Ordered group invariants for one-dimensional spaces

2001

Fund. Math.

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Abstract. We show that the Bruschlinsky group with the winding order is a homeomorphism invariant for a class of one-dimensional inverse limit spaces. In particular we show that if a presentation of an inverse limit space satisfies the Simplicity Condition, then the Bruschlinsky group with the winding order of the inverse limit space is a dimension group and is a quotient of the dimension group with the standard order of the adjacency matrices associated with the presentation. Introduction.Ordered groups have been useful invariants for the classification of many different categories. A class of ordered groups, dimension groups, was used in the study of C * -algebras to classify AF-algebras ([6]), and Giordano, Herman, Putnam and Skau ([8,9]) defined (simple) dimension groups in terms of dynamical concepts to give complete information about the orbit structure of zero-dimensional minimal dynamical systems. Swanson and Volkmer ([15]) showed that the dimension group of a primitive matrix is a complete invariant for weak equivalence, which is called C * -equivalence by Bratteli, Jørgensen, Kim and Roush ([5]). And Barge, Jacklitch and Vago ([3]) showed that, for a certain class of one-dimensional inverse limit spaces, two spaces are homeomorphic if and only if their associated substitutions are weak equivalent, and that if two inverse limit spaces are homeomorphic and the squares of their connection maps are orientation preserving, then the dimension groups of the adjacency matrices associated with the substitutions are order isomorphic.A recent development ([2, 3, 4, 7, 8, 15]) is the refinement ofȞ 1 (X) as a topological invariant for certain one-dimensional spaces X, by making this group an ordered group. HereȞ 1 (X) is the direct limit of first cohomology groups on graphs approximating the space X. There is a natural order on the first cohomology of a graph (a coset is positive if it contains a nonnegative

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The orientability of matchbox manifolds

Cited by 10 publications

References 15 publications

Manifold-like matchbox manifolds

Manifold-like matchbox manifolds

Homogeneous weak solenoids

Ordered group invariants for one-dimensional spaces

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