Local behavior and the Vietoris and Whitehead theorems in shape theory

Kozlowski, George; Segal, Jack

doi:10.4064/fm-99-3-213-225

Cited by 12 publications

(7 citation statements)

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“…For a proof of the first assertion of Theorem 4.1 in the compact metric case or in the locally simply-connected case see [18] or [17], respectively. The second assertion is well known, but it also follows from our proof below.…”

Section: Summary Of Resultsmentioning

confidence: 99%

Generalized universal covering spaces and the shape group

2007

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Abstract. If a paracompact Hausdorff space X admits a (classical) universal covering space, then the natural homomorphism ϕ : π 1 (X) →π 1 (X) from the fundamental group to its first shape homotopy group is an isomorphism. We present a partial converse to this result: a path-connected topological space X admits a generalized universal covering space if ϕ :This generalized notion of universal covering p : X → X enjoys most of the usual properties, with the possible exception of evenly covered neighborhoods: the space X is path-connected, locally path-connected and simply-connected and the continuous surjection p : X → X is universally characterized by the usual general lifting properties. (If X is first countable, then p : X → X is already characterized by the unique lifting of paths and their homotopies.) In particular, the group of covering transformations G = Aut( X p → X) is isomorphic to π 1 (X) and it acts freely and transitively on every fiber. If X is locally path-connected, then the quotient X/G is homeomorphic to X. If X is Hausdorff or metrizable, then so is X, and in the latter case G can be made to act by isometry. If X is path-connected, locally path-connected and semilocally simply-connected, then p : X → X agrees with the classical universal covering.A necessary condition for the standard construction to yield a generalized universal covering is that X be homotopically Hausdorff, which is also sufficient if π 1 (X) is countable. Spaces X for which ϕ : π 1 (X) →π 1 (X) is known to be injective include all subsets of closed surfaces, all 1-dimensional separable metric spaces (which we prove to be covered by topological R-trees), as well as so-called trees of manifolds which arise, for example, as boundaries of certain Coxeter groups.We also obtain generalized regular coverings, relative to some special normal subgroups of π 1 (X), and provide the appropriate relative version of being homotopically Hausdorff, along with its corresponding properties.General Assumption. Throughout this article, we consider a pathconnected topological space X with base point x 0 ∈ X. Recall that a continuous map p : X → X is called a covering of X, and X is called a covering space of X, if for every x ∈ X there is an open subset U of X with x ∈ U and such that U is evenly covered by p, that is, p −1 (U ) is the disjoint union of open subsets of X each of which is mapped homeomorphically onto U by p.In the classical theory, one assumes that X is, in addition, locally pathconnected and wishes to classify all path-connected covering spaces of X and to find among them a universal covering space, that is, a covering p : X → X with the property that for every covering q : X → X by a path-connected space X there is a covering q : X → X such that q•q = p. If X is locally pathconnected, we have the following well-known result, which can be found, for example, in [22] and [25]:

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Section: Summary Of Resultsmentioning

confidence: 99%

Generalized universal covering spaces and the shape group

2007

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“…These refinements allow one to recursively extend maps on simplicial complexes skeleton-wise. These extension methods, established in Section 4, are similar to methods found in [22,23].…”

Section: Introductionmentioning

confidence: 91%

“…We also put these extension methods to work in Section 6 where we identify conditions that imply Ψ n is an isomorphism. In [23], Kozlowski-Segal prove that if X is paracompact Hausdorff and U V n , then Ψ n is an isomorphism. In [18], Fischer and Zastrow generalize this result in dimension n " 1 by replacing "U V 1 " with "locally path connected and semilocally simply connected."…”

Section: Introductionmentioning

confidence: 99%

Elements of higher homotopy groups undetectable by polyhedral approximation

K.¹,

Brazas²

2022

Preprint

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When non-trivial local structures are present in a topological space X, a common approach to characterizing the isomorphism type of the n-th homotopy group πnpX, x0q is to consider the image of πnpX, x0q in the n-th Čech homotopy group πnpX, x0q under the canonical homomorphism Ψn : πnpX, x0q Ñ πnpX, x0q. The subgroup kerpΨnq is the obstruction to this tactic as it consists of precisely those elements of πnpX, x0q, which cannot be detected by polyhedral approximations to X. In this paper, we use higher dimensional analogues of Spanier groups to characterize kerpΨnq. In particular, we prove that if X is paracompact, Hausdorff, and U V n´1 , then kerpΨnq is equal to the n-th Spanier group of X. We also use the perspective of higher Spanier groups to generalize a theorem of Kozlowski-Segal, which gives conditions ensuring that Ψn is an isomorphism.

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“…Their approach requires some preparation, but Sect. 2.4 of Chapter 1 of [29] shows the equivalence with a simpler approach given in [23,24]. Nevertheless [29] is a very useful reference for material related to this section.…”

Section: Shape Of Spaces With Related Linear Orderingmentioning

confidence: 99%

“…An exposition of shape theory for the results in this section can be found in [29] by Mardešić and Segal. Their approach requires some preparation, but Section 2.4 of Chapter 1 of [29] shows the equivalence with a simpler approach given in [23] and [24]. Nevertheless [29] is a very useful reference for material related to this section.…”

Section: Shape Of Spaces With Related Linear Orderingmentioning

confidence: 99%

$$1/\kappa $$ 1 / κ -Homogeneous long solenoids

2016

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We study nonmetric analogues of Vietoris solenoids. Let be an ordered continuum, and let p = p 1 , p 2 , . . . be a sequence of positive integers. We define a natural inverse limit space S( , p), where the first factor space is the nonmetric "circle" obtained by identifying the endpoints of , and the nth factor space, n > 1, consists of p 1 p 2 . . . p n−1 copies of laid end to end in a circle. We prove that for every cardinal κ ≥ 1, there is an ordered continuum such that S( , p) is relation similar to one used to classify the additive subgroups of Q. Consequently, for each fixed , as p varies, there are exactly c-many different shapes, where c = 2 ℵ 0 , (and there are also exactly that many homeomorphism types) represented by S( , p).

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Local behavior and the Vietoris and Whitehead theorems in shape theory

Cited by 12 publications

References 0 publications

Generalized universal covering spaces and the shape group

Generalized universal covering spaces and the shape group

Elements of higher homotopy groups undetectable by polyhedral approximation

$$1/\kappa $$ 1 / κ -Homogeneous long solenoids

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