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Algebraic theory of fundamental dimension

100%
Nowak S.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Introduction......................................................................................................................................... 5 Chapter I Elementary topological characterizations of fundamental dimension........................... 6  1. Characterizations of fundamental dimension..................................................................... 6  2. The fundamental dimension of components of compacta.............................................. 9  3. The fundamental dimension of the union of two compacta............................................. 10 Chapter II Cohomology groups over local systems and generalized local systems................... 13  1. Local systems of groups......................................................................................................... 13  2. Cohomology with coefficients in local systems.................................................................. 16  3. The Künneth formula  4. Generalized local systems..................................................................................................... 20 Chapter III Homological characterizations of fundamental dimension........................................... 22  1. Deformability of maps and the number................................................................................ 23  2. Obstructions to deformability.................................................................................................. 24  3. Coefficients of cyclicity and ℱ-continua................................................................................. 25  4. Continua with fundamental dimension ≥ 3........................................................................ 28  5. Two algebraic lemmas............................................................................................................ 29  6. Continua with fundamental dimension equal to 1............................................................. 31  7. Continua with fundamental dimension equal to 2............................................................. 33  8. The main results....................................................................................................................... 34 Chapter IV Applications of the homological characterizations of fundamental dimension to the study of some special problems................................................................................................. 37  1. The fundamental dimension of the Cartesian product of a closed manifold and a continuum........................................................................................................................................ 37  2. The fundamental dimension of the Cartesian product of a curve and a continuum... 38  3. An example of a finite-dimensional continuum with an infinite family of shape factors and the fundamental dimension of the Cartesian product of polyhedra........................... 42  4. The fundamental dimension of the union of two compacta and of the quotient space............................................................................................................................................................ 43  5. The fundamental dimension of the suspension of a compactum.................................. 44  6. The fundamental dimension of the Cartesian product of approximative 1-connected compacta............................................................................................................................. 46  7. The fundamental dimension of a subset of manifold....................................................... 48 Final remarks and problems................................................................................................................... 50 References.................................................................................................................................................. 52 Index of symbols........................................................................................................................................ 54
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Stable cohomotopy groups of compact spaces

95%
Nowak S.
Fundamenta Mathematicae
|
2003
|
tom 180
|
nr 2
99-137
EN
We show that one can reduce the study of global (in particular cohomological) properties of a compact Hausdorff space X to the study of its stable cohomotopy groups $π^{k}_{s}(X)$. Any cohomology functor on the homotopy category of compact spaces factorizes via the stable shape category ShStab. This is the main reason why the language and technique of stable shape theory can be used to describe and analyze the global structure of compact spaces. For a given Hausdorff compact space X, there exists a metric compact space with the same stable shape iff the stable cohomotopy groups of X are countable. If $πⁿ_s(X) = 0$ for almost all n > 0 and the integral cohomology groups of X are countable (respectively finitely generated) for all n, then the k-fold suspension of X has the same stable shape as a finite-dimensional compact metric space (respectively a finite CW complex) for sufficiently large k. There is a duality between compact Hausdorff spaces and CW spectra under which stable cohomotopy groups of X correspond to homotopy groups of the CW spectrum $𝕎_{X}$ assigned to X and the class of all X with $ℭ^{s}(X) = max{k: π^{k}_s(X) ≠ 0} < ∞$ corresponds to the class of spectra bounded below. The notion of the cohomological dimension ℌ - dim X with respect to a generalized cohomology theory ℌ is studied. In particular we show that π - dim X ≥ ℌ - dim X for every ℌ and π - dim X = ∞ if $π - dim X > dim_{ℤ}X$, where π is the stable cohomotopy theory and $dim_{ℤ}X$ is the integral cohomological dimension. The following question remains open: does π - dim X coincide with dim X?
3
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Function spaces and shape theories

61%
Dydak J., Nowak S.
Fundamenta Mathematicae
|
2002
|
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.
4
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On the relationships between shape properties of subcompacta of $S^n$ and homotopy properties of their complements

60%
Nowak S.
Fundamenta Mathematicae
|
1987
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tom 128
|
nr 1
47-59
5
Content available remote

Remarks on the n-dimensional geometric measure of compacta

49%
Borsuk K., Nowak S., Spież S.
Fundamenta Mathematicae
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1984
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tom 121
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nr 1
59-71
6
Content available remote

Some remarks concerning the fundamental dimension of the cartesian product of two compacta

48%
Nowak S.
Fundamenta Mathematicae
|
1979
|
tom 103
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nr 1
31-41
7
Content available remote

Some extension and classification theorems for maps of movable spaces

42%
Nowak S.
Fundamenta Mathematicae
|
1985
|
tom 125
|
nr 2
105-113
8
Content available remote

On the fundamental dimension of approximatively 1-connected compacta

42%
Nowak S.
Fundamenta Mathematicae
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1975
|
tom 89
|
nr 1
61-79
9
Content available remote

Remarks on deformability

31%
Nowak S., Spież S.
Fundamenta Mathematicae
|
1985
|
tom 125
|
nr 2
95-103
10
Content available remote

Some properties of fundamental dimension

30%
Nowak S.
Fundamenta Mathematicae
|
1974
|
tom 85
|
nr 3
211-227
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