COMPACTA LYING IN THE $ n$-DIMENSIONAL UNIVERSAL MENGER COMPACTUM AND HAVING HOMEOMORPHIC COMPLEMENTS IN IT

Chigogidze, A.

doi:10.1070/sm1988v061n02abeh003219

Cited by 17 publications

(14 citation statements)

References 9 publications

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“…It is closely related to previous papers [1,8,[3][4][5] and can be considered as their natural continuation. We gave a detailed discussion of certain phenomena in [5] that are inherent in this theory (in comparison with Hubert cube manifold theory) and that are clearly manifested even in the formulations of such important components of the theory as stability and triangulation theorems [8,5] as well as the open embedding theorem [5].…”

Section: Introductionmentioning

confidence: 99%

Classification theorem for Menger manifolds

Chigogidze¹

1992

Proc. Amer. Math. Soc.

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Abstract.We introduce the notion of the n-homotopy kernel of a Menger manifold and prove the following theorem: Menger manifolds are w-homotopy equivalent if and only if the «-homotopy kernels are homeomorphic This paper is devoted to some problems induced by the internal development of Menger manifold theory. It is closely related to previous papers [1,8,[3][4][5] and can be considered as their natural continuation. We gave a detailed discussion of certain phenomena in [5] that are inherent in this theory (in comparison with Hubert cube manifold theory) and that are clearly manifested even in the formulations of such important components of the theory as stability and triangulation theorems [8,5] as well as the open embedding theorem [5]. Let us consider the last proposition more formally (see §1 for definitions). It states that each Menger (ai + l)-manifold N contains a Z-set Z satisfying the following conditions: (1) Z is homeomorphic to N; (2) the complement N-Z admits an open embedding into Mn+l (throughout the paper Mk denotes the Ac-dimensional universal Menger compactum). It should be observed (it was remarked in [5], too) that there is a certain arbitrariness in the above formulation connected with the choice of homeomorphism from condition (1). It will be shown that under minimal restrictions the mentioned arbitrariness can be eliminated. At the same time this specifies the second condition, because in this way the complement N-Z will acquire completely correct sense and will depend only on the Menger (ai + l)-manifold N. Moreover, this complement (which will be called an Ai-homotopy kernel of N and will be denoted by Ker(/V)) becomes a very important object associated with N. Indeed, if we bear in mind the strong parallel between Hubert cube manifold theory and Menger (ai + 1)-manifold theory then we shall be able to conclude confidently that Ker(/V) plays the role of the product N x [0, 1 ) (compare our modified open embedding theorem below and Chapman's open embedding theorem for Hubert cube manifolds [2]). Moreover, the consideration of Ai-homotopy kernels gives us the possibility to solve the problem of classification of Menger (ai + l)-manifolds by «-homotopy type. Our main result states in this connection that Menger (ai + l)-manifolds are Ai-homotopy equivalent if and only if their Ai-homotopy kernels are homeomorphic (compare with Chapman's theorem that states that

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Section: Introductionmentioning

confidence: 99%

Classification theorem for Menger manifolds

Chigogidze¹

1992

Proc. Amer. Math. Soc.

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“…In this paper we generalize a result of Chigogidze [2] by establishing the following result about closed subsets of the universal Menger compactum pk+x .…”

Section: Introductionmentioning

confidence: 99%

“…We begin with the proof that (1) implies (2). Let U be a neighborhood of X, and let K be a Z-set in the interior of U that is a spine of U relative to X.…”

Section: Weak Z-sets In Pk+x-manifoldsmentioning

confidence: 99%

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A complement theorem in the universal Menger compactum

Sher¹

1994

Proc. Amer. Math. Soc.

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Abstract.A. Chigogidze has shown that two Z-sets in the universal Menger compactum of dimension k + 1 have the same fc-shape if and only if their complements are homeomorphic. We show that this result holds for weak Zsets. The class of weak Z-sets, defined herein and analogous to the weak Z-sets in Q , contains but is larger than the class of Z-sets. We give some examples of weak Z-sets in the universal Menger compactum and in Q that are not Z-sets.

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“…Algebraic models for n-types (the so called cat n -groups) were found in [24]. Approximately at the same time n-homotopies (and subsequently even n-shapes [7], [6]) begun to play a substantial role in the revitalized theory of Menger manifolds [2], [5] (see [8] for a discussion of categorical connections between n-homotopies and homotopies via the theories of manifolds modeled on Menger and Hilbert cubes respectively). The following theorem has been proved in [16].…”

Section: Theorem Bmentioning

confidence: 99%

Topological Model Categories Generated by Finite Complexes

Chigogidze

Karasev

2003

Monatshefte f�r Mathematik

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Abstract. Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov's notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them -describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups -is obtained by letting L = {point}. The other -describing n-homotopy equivalences between at most (n + 1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ≤ n -by letting L = S n+1 , n ≥ 0.

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COMPACTA LYING IN THE $ n$-DIMENSIONAL UNIVERSAL MENGER COMPACTUM AND HAVING HOMEOMORPHIC COMPLEMENTS IN IT

Cited by 17 publications

References 9 publications

Classification theorem for Menger manifolds

Classification theorem for Menger manifolds

A complement theorem in the universal Menger compactum

Topological Model Categories Generated by Finite Complexes

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