Algebra of dimension theory

Dydak, Jerzy

doi:10.1090/s0002-9947-05-03690-1

Cited by 5 publications

(4 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…with the product of the signs ǫ 1 ⊗ ǫ 2 defined by ǫ ⊗ empty = ǫ, ǫ ⊗ ǫ = ǫ, ǫ = ±, and + ⊗− = −. It turns out that D 1 ⊞ D 2 is indeed a dimension type and its definition is justified by Theorem 2.6 (Bockstein Product Theorem [6], [13], [9]) For any two compacta X and Y…”

Section: Cohomological Dimensionmentioning

confidence: 99%

On dimensionally exotic maps

Dranishnikov

Levin

2014

Isr. J. Math.

View full text Add to dashboard Cite

We call a value y = f (x) of a map f : X → Y dimensionally regular if dim X ≤ dim(Y × f −1 (y)). It was shown in [5] that if a map f : X → Y between compact metric spaces does not have dimensionally regular values, then X is a Boltyanskii compactum, i.e. a compactum satisfying the equality dim(X ×X) = 2 dim X −1. In this paper we prove that every Boltyanskii compactum X of dimension dim X ≥ 6 admits a map f : X → Y without dimensionally regular values. Also we exhibit a 4-dimensional Boltyanskii compactum for which every map has a dimensionally regular value.

show abstract

Section: Cohomological Dimensionmentioning

confidence: 99%

On dimensionally exotic maps

Dranishnikov

Levin

2014

Isr. J. Math.

View full text Add to dashboard Cite

show abstract

“…The first detailed presentation of the theory was given in the survey [12]. Since then it was evolved in many papers and surveys [2], [7], [6], [5], [17], [10]. Our presentation here has features of both point of view on the subject, classical and modern.…”

Section: Bockstein Theorymentioning

confidence: 99%

Dimension of the product and classical formulae of dimension theory

Dranishnikov

Levin

2013

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

“…One of the main ideas of the paper is to treat ext-dim(X) ≤ SP (L) as the fundamental concept of cohomological dimension theory instead of dimG(X) ≤ n. In a subsequent paper [18] we show how properties of infinite symmetric products lead naturally to a calculus of graded groups which implies most of classical results of the cohomological dimension. The basic notion in [18] is that of homological dimension of a graded group which allows for simultanous treatment of cohomological dimension of compacta and extension properties of CW complexes.We introduce cohomology of X with respect to L (defined as homotopy groups of the function space SP (L) X ). As an application of our results we characterize all countable groups G so that the Moore space M (G, n) is of the same extension type as the Eilenberg-MacLane space K(G, n).…”

mentioning

confidence: 99%

“…In a subsequent paper [18] we will explain that Bockstein Theory plays the role of homological algebra in algebraic topology. In this paper we use Bockstein Theory to give necessary and sufficient conditions for SP (L) to have the same extension type as an Eilenberg-MacLane space K(G, n) (see Section 7).…”

mentioning

confidence: 99%

Extension theory of infinite symmetric products

Dydak

2004

Fund. Math.

Self Cite

View full text Add to dashboard Cite

Abstract. We present an approach to cohomological dimension theory based on infinite symmetric products and on the general theory of dimension called the extension dimension. The notion of the extension dimension ext-dim(X) was introduced by A. N. Dranishnikov [9] in the context of compact spaces and CW complexes. This paper investigates extension types of infinite symmetric products SP(L). One of the main ideas of the paper is to treat ext-dim(X) ≤ SP(L) as the fundamental concept of cohomological dimension theory instead of dim G (X) ≤ n. In a subsequent paper [18] we show how properties of infinite symmetric products lead naturally to a calculus of graded groups which implies most of the classical results on the cohomological dimension. The basic notion in [18] is that of homological dimension of a graded group which allows for simultaneous treatment of cohomological dimension of compacta and extension properties of CW complexes.We introduce cohomology of X with respect to L (defined as homotopy groups of the function space SP(L) X ). As an application of our results we characterize all countable groups G so that the Moore space M (G, n) is of the same extension type as the EilenbergMacLane space K(G, n). Another application is a characterization of infinite symmetric products of the same extension type as a compact (or finite-dimensional and countable) CW complex.

show abstract

Algebra of dimension theory

Cited by 5 publications

References 22 publications

On dimensionally exotic maps

On dimensionally exotic maps

Dimension of the product and classical formulae of dimension theory

Extension theory of infinite symmetric products

Contact Info

Product

Resources

About