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1
Content available remote

There Are No Essential Phantom Mappings from 1-dimensional CW-complexes

100%
Mardešić S.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2013
|
tom 61
|
nr 2
141-147
EN
A phantom mapping h from a space Z to a space Y is a mapping whose restrictions to compact subsets are homotopic to constant mappings. If the mapping h is not homotopic to a constant mapping, one speaks of an essential phantom mapping. The definition of (essential) phantom pairs of mappings is analogous. In the study of phantom mappings (phantom pairs of mappings), of primary interest is the case when Z and Y are CW-complexes. In a previous paper it was shown that there are no essential phantom mappings (pairs of phantom mappings) between CW-complexes if dim Y ≤ 1. In the present paper it is shown that there are no essential phantom mappings between CW-complexes if dim Z ≤ 1. In contrast, there exist essential phantom pairs of mappings between CW-complexes where dim Z = 1 and dim Y = 2. Moreover, there exist essential phantom mappings with dim Z = dim Y = 1 where Y is a CW-complex, but Z is not.
2
Content available remote

Coherent and strong expansions of spaces coincide

100%
Mardešić S.
Fundamenta Mathematicae
|
1998
|
tom 158
|
nr 1
69-80
EN
In the existing literature there are several constructions of the strong shape category of topological spaces. In the one due to Yu. T. Lisitsa and S. Mardešić [LM1-3] an essential role is played by coherent polyhedral (ANR) expansions of spaces. Such expansions always exist, because every space admits a polyhedral resolution, resolutions are strong expansions and strong expansions are always coherent. The purpose of this paper is to prove that conversely, every coherent polyhedral (ANR) expansion is a strong expansion. This result is obtained by showing that a mapping of a space into a system, which is coherently dominated by a strong expansion, is itself a strong expansion.
3
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There are no Phantom Pairs of Mappings to 1-Dimensional CW-Complexes

100%
Mardešić S.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2007
|
tom 55
|
nr 4
365-371
EN
Two mappings from a CW-complex to a 1-dimensional CW-complex are homotopic if and only if their restrictions to finite subcomplexes are homotopic.
4
Content available remote

On approximate inverse systems and resolutions

100%
Mardešić S.
Fundamenta Mathematicae
|
1993
|
tom 142
|
nr 3
241-255
EN
Recently, L. R. Rubin, T. Watanabe and the author have introduced approximate inverse systems and approximate resolutions, a new tool designed to study topological spaces. These systems differ from the usual inverse systems in that the bonding maps $p_{aa'}$ are not subject to the commutativity requirement $p_{aa' p_{a'a''} = p_{aa''}, a ≤ a' ≤ a''$. Instead, the mappings $p_{aa'}p_{a'a''}$ and $p_{aa''}$ are allowed to differ in a way controlled by coverings $U_a$, called meshes, which are associated with the members $X_a$ of the system. The purpose of this paper is to consider a more general and much simpler notion of approximate system and approximate resolution, which does not require meshes. The main result is a construction which associates with any approximate resolution in the new sense an approximate resolution in the previous sense in such a way that previously obtained results remain valid in the present more general setting.
5
Content available remote

Equivalence of the Borsuk and the ANR-system approach to shapes

45%
Mardešić S., Segal J.
Fundamenta Mathematicae
|
1971
|
tom 72
|
nr 1
61-68
6
Content available remote

Approximate polyhedra, resolutions of maps and shape fibrations

44%
Mardešić S.
Fundamenta Mathematicae
|
1981
|
tom 114
|
nr 1
53-78
7
Content available remote

Mapping approximate inverse systems of compacta

38%
Mardešić S., Segal J.
Fundamenta Mathematicae
|
1990
|
tom 134
|
nr 1
73-91
8
Content available remote

On the Whitehead theorem in shape theory II

38%
Mardešić S.
Fundamenta Mathematicae
|
1976
|
tom 91
|
nr 2
93-103
9
Content available remote

On the Whitehead theorem in shape theory I

38%
Mardešić S.
Fundamenta Mathematicae
|
1976
|
tom 91
|
nr 1
51-64
10
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Content available

Reducing inverse systems of monomorphisms

32%
Gordh Jr. G. R., Mardešić S.
Colloquium Mathematicum
|
1975
|
tom 33
|
nr 1
83-90
11
Content available remote

Shapes of compacta and ANR-systems

32%
Mardešić S., Segal J.
Fundamenta Mathematicae
|
1971
|
tom 72
|
nr 1
41-59
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