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The Whitehead and the Smale theorems in shape theory

100%
Dydak J.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS §1. Introduction................................................................................................................................... 5 §2. Some classes of objects and morphisms in pro-categories..................................................... 5 §3. Shape category.................................................................................................................................... 14 §4. Deformation dimension..................................................................................................................... 16 §5. Some properties of n-equivalences of pro-$H_0$ ...................................................................... 18 §6. The Whitehead theorems in shape and pro-homotopy.............................................................. 26 §7. Criteria for stability in shape and pro-homotopy........................................................................... 29 §8. The Smale theorem in shape theory............................................................................................... 37 References.................................................................................................................................................. 49
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Extension theory of infinite symmetric products

95%
Dydak J.
Fundamenta Mathematicae
|
2004
|
tom 182
|
nr 1
53-77
EN
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 Eilenberg-MacLane 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.
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Function spaces and shape theories

61%
Dydak J., Nowak S.
Fundamenta Mathematicae
|
2002
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tom 171
|
nr 2
117-154
EN
The purpose of this paper is to provide a geometric explanation of strong shape theory and to give a fairly simple way of introducing the strong shape category formally. Generally speaking, it is useful to introduce a shape theory as a localization at some class of "equivalences". We follow this principle and we extend the standard shape category Sh(HoTop) to Sh(pro-HoTop) by localizing pro-HoTop at shape equivalences. Similarly, we extend the strong shape category of Edwards-Hastings to sSh(pro-Top) by localizing pro-Top at strong shape equivalences. A map f:X → Y is a shape equivalence if and only if the induced function f*:[Y,P] → [X,P] is a bijection for all P ∈ ANR. A map f:X → Y of k-spaces is a strong shape equivalence if and only if the induced map f*: Map(Y,P) → Map(X,P) is a weak homotopy equivalence for all P ∈ ANR. One generalizes the concept of being a shape equivalence to morphisms of pro-HoTop without any problem and the only difficulty is to show that a localization of pro-HoTop at shape equivalences is a category (which amounts to showing that the morphisms form a set). Due to peculiarities of function spaces, extending the concept of strong shape equivalence to morphisms of pro-Top is more involved. However, it can be done and we show that the corresponding localization exists. One can introduce the concept of a super shape equivalence f:X → Y of topological spaces as a map such that the induced map f*: Map(Y,P) → Map(X,P) is a homotopy equivalence for all P ∈ ANR, and one can extend it to morphisms of pro-Top. However, the authors do not know if the corresponding localization exists. Here are applications of our methods: Theorem. A map f:X → Y of k-spaces is a strong shape equivalence if and only if $f × id_Q: X ×_k Q → Y ×_k Q$ is a shape equivalence for each CW complex Q. Theorem. Suppose f: X → Y is a map of topological spaces. (a) f is a shape equivalence if and only if the induced function f*: [Y,M] → [X,M] is a bijection for all M = Map(Q,P), where P ∈ ANR and Q is a finite CW complex.(b) If f is a strong shape equivalence, then the induced function f*: [Y,M] → [X,M] is a bijection for all M = Map(Q,P), where P ∈ ANR and Q is an arbitrary CW complex. (c) If X, Y are k-spaces and the induced function f*: [Y,M] → [X,M] is a bijection for all M = Map(Q,P), where P ∈ ANR and Q is an arbitrary CW complex, then f is a strong shape equivalence.
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Strong Cohomological Dimension

61%
Dydak J., Koyama A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2008
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tom 56
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nr 2
183-189
EN
We characterize strong cohomological dimension of separable metric spaces in terms of extension of mappings. Using this characterization, we discuss the relation between strong cohomological dimension and (ordinal) cohomological dimension and give examples to clarify their gaps. We also show that $Ind_{G} X = dim_{G} X$ if X is a separable metric ANR and G is a countable Abelian group. Hence $dim_{ℤ} X = dim X$ for any separable metric ANR X.
5
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Bimorphisms in pro-homotopy and proper homotopy

61%
Dydak J., del Portal F. R. R.
Fundamenta Mathematicae
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1999
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tom 160
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nr 3
269-286
EN
A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. The category is balanced if every bimorphism is an isomorphism. In the paper properties of bimorphisms of several categories are discussed (pro-homotopy, shape, proper homotopy) and the question of those categories being balanced is raised. Our most interesting result is that a bimorphism f:X → Y of $tow(H_0)$ is an isomorphism if Y is movable. Recall that $\tow(H_0)$ is the full subcategory of $pro-H_0$ consisting of inverse sequences in $H_0$, the homotopy category of pointed connected CW complexes.
6
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Borsuk-Sieklucki theorem in cohomological dimension theory

43%
Boege M., Dydak J., Jiménez R., Koyama A., Shchepin E.
Fundamenta Mathematicae
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2002
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tom 171
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nr 3
213-222
EN
The Borsuk-Sieklucki theorem says that for every uncountable family ${X_{α}}_{α∈A}$ of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that $dim (X_{α} ∩ X_{β}) = n$. In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is $clc^{n+1}_{ℤ}$, where n ≥ 1, and G is an Abelian group. Let ${X_{α}}_{α∈J}$ be an uncountable family of closed subsets of X. If $dim_{G}X = dim_{G}X_{α} = n$ for all α ∈ J, then $dim_{G}(X_{α}∩ X_{β}) = n$ for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski [C-K]. Independently, Dydak and Koyama [D-K] proved it for G being an arbitrary principal ideal domain and posed the question of validity of the Theorem for quasicyclic groups (see Problem 1 in [D-K]). As applications of the Theorem we investigate equality of cohomological dimension and strong cohomological dimension, and give a characterization of cohomological dimension in terms of a special base.
7
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A generalization of cohomotopy groups

30%
Dydak J.
Fundamenta Mathematicae
|
1975-1976
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tom 90
|
nr 1
77-98
8
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Pointed and unpointed shape and pro-homotopy

30%
Dydak J.
Fundamenta Mathematicae
|
1980
|
tom 107
|
nr 1
57-69
9
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On elementary maps

25%
Dydak J.
Colloquium Mathematicum
|
1974
|
tom 31
|
nr 1
67-69
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