Some mapping theorems for extensional dimension

Levin, Michael; Lewis, Wayne

doi:10.1007/bf02773061

Cited by 17 publications

(22 citation statements)

References 16 publications

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“…A space M is said to be a Krasinkiewicz space [9] if for any compactum X the function space C(X, M) contains a dense subset of Krasinkiewicz maps. Here, a map g : X → M, where X is compact, is said to be Krasinkiewicz [5] if every continuum in X is either contained in a fiber of g or contains a component of a fiber of g. The class of Krasinkiewicz spaces contains all Euclidean manifolds and manifolds modeled on Menger or Nöbeling spaces, all polyhedra (not necessarily compact), as well as all cones with compact bases (see [3], [5], [6], [8], [9]). Remark.…”

Section: Introductionmentioning

confidence: 99%

Parametric bing and Krasinkiewicz maps

Valov

2008

Topology and its Applications

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Using a factorization theorem due to Pasynkov [11] we provide a short proof of the existence and density of parametric Bing and Krasinkiewicz maps. In particular, the following corollary is established: Let f : X → Y be a surjective map between paracompact spaces such that all fibers f −1 (y), y ∈ Y , are compact and there exists a map g : X → I ℵ0 embedding each f −1 (y) into I ℵ0 . Then for every n ≥ 1 the space C * (X, R n ) of all bounded continuous functions with the uniform convergence topology contains a dense set of maps g such that any restriction g|f −1 (y), y ∈ Y , is a Bing and Krasinkiewicz map.2000 Mathematics Subject Classification. Primary 54E40; Secondary 54F15.

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Section: Introductionmentioning

confidence: 99%

Parametric bing and Krasinkiewicz maps

Valov

2008

Topology and its Applications

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“…The second inequality follows from Theorem 3.6. It turns out that the operation ⊕ nicely fits in the translation of some mapping theorems by Levin and Lewis [13] to the language of dimension types.…”

Section: Extension Theorymentioning

confidence: 69%

Dimension of the product and classical formulae of dimension theory

Dranishnikov

Levin

2013

Trans. Amer. Math. Soc.

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Let f : X −→ Y be a map of compact metric spaces. A classical theorem of Hurewicz asserts that dim X ≤ dim Y + dim f where dim f = sup{dim f −1 (y) : y ∈ Y }. The first author conjectured that dim Y + dim f in Hurewicz's theorem can be replaced by sup{dim(Y × f −1 (y)) : y ∈ Y }. We disprove this conjecture. As a byproduct of the machinery presented in the paper we answer in negative the following problem posed by the first author: Can for compact X the Menger-Urysohn formula dim X ≤ dim A + dim B + 1 be improved to dim X ≤ dim(A × B) + 1 ?On a positive side we show that both conjectures holds true for compacta X satisfying the equality dim(X × X) = 2 dim X.1 there are compacta X n and X m of dimensions n and m respectively with dim(X n ×X m ) = k [2]. We note that the inequality dim(X × Y ) ≤ dim X + dim Y always holds true.The first author conjectured that many classical formulas (inequalities) of dimension theory can be strengthen by replacing the sum of the dimensions by the dimension of the product. His believe was based on his results on the general position properties of compacta in euclidean spaces [5], [8]. Clearly, for two polyhedra K and L with transversal intersection in R n we have dim(K ∩ L) = n − (dim K + dim L). For compacta the corresponding formula is dim(X ∩ Y ) = n − dim(X × Y ). In particular, two compacta X and Y in general position in R n have empty intersection if and only if dim(X × Y ) < n.The next candidate for the improvement was the following classical theorem of Hurewicz.We note that the Hurewicz theorem applied to the projection X × Y → Y implies the inequality dim(X × Y ) ≤ dim X + dim Y . The first author proposed the following conjecture. Conjecture 1.2 ([8]) For a map of compactaNote that the Conjecture 1.2 holds true for nice maps like locally trivial bundles. It was known that the conjecture holds true when X is standard (compactum of type I in the sense of [12]). We call a compactum X standard if it has the property dim(X × X) = 2 dim X. It's not easy to come with an example of a compactum without this property. The Pontryagin surfaces satisfy it. First example of a non-standard compactum was constructed by Boltyanskii [1]. In this paper all non-standard compacta (compacta of type II in [12]) will be called Boltyanskii compacta. It is known that for all Boltyanskii compacta dim(X × X) = 2 dim X − 1.In this paper we disprove Conjecture 1.2. We will refer to the maps providing counterexamples to the conjecture as exotic maps.Positive results towards Conjecture 1.2 can be summarized in the following:Theorem 1.3 If a compactum X admits an exotic map f : X → Y then X is a Boltyanskii compactum. For every exotic map f : X → Y we have dim X = sup{dim(Y × f −1 (y)) | y ∈ Y } + 1.Another classical result in Dimension Theory where the first author hoped to replace the sum of the dimensions by the dimension of the product was the Menger-Urysohn Formula.

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“…In this section we apply Theorem 1.2 to extend some results from [14] which were established for maps between compact metric spaces. Extension theory, which was first introduced by Dranishnikov [5], is based on the following notion.…”

Section: Some Mapping Theorems For Extensional Dimensionmentioning

confidence: 95%

“…For a set A ⊂ X we write rdim X A ≤ n provided dim H ≤ n for every closed subset H of X which is contained in A. Next theorem is an analogue of Theorem 1.9 from [14]. (1) e-dimf ≤ K;…”

Section: Some Mapping Theorems For Extensional Dimensionmentioning

confidence: 99%

“…Theorem 1.2 was established in [21] in the case Y is strongly countable dimensional or M is a closed convex subset of a Banach space and Y a C-space. Theorem 1.2 is applied in the last Section 4 to show that some mapping theorems for extensional dimension established in [14] in the realm of compact metric spaces remain valid for general metric spaces.…”

Section: Introductionmentioning

confidence: 99%

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Parametric Bing and Krasinkiewicz maps: revisited

Valov

2011

Proc. Amer. Math. Soc.

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Abstract. Let M be a complete metric AN R-space such that for any metric compactum K the function space C(K, M ) contains a dense set of Bing (resp., Krasinkiewicz) maps. It is shown that M has the following property: If f : X → Y is a perfect surjection between metric spaces, then C(X, M ) with the source limitation topology contains a dense G δ -subset of maps g such that all restrictions g|f −1 (y), y ∈ Y , are Bing (resp., Krasinkiewicz) maps. We apply the above result to establish some mapping theorems for extensional dimension.

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Some mapping theorems for extensional dimension

Abstract: We present some results related to theorems of Pasynkov and Torunczyk on the geometry of maps of finite dimensional compacta.

Cited by 17 publications

References 16 publications

Parametric bing and Krasinkiewicz maps

Parametric bing and Krasinkiewicz maps

Dimension of the product and classical formulae of dimension theory

Parametric Bing and Krasinkiewicz maps: revisited

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