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Papers quoted in Current Contents on project 1-01-253
Quoted papers: 22
Other papers: 0
Total: 22
Title: On universal compositions of maps
- Authors:
- Čerin, Zvonko (7994)
Journal: Publicacions Matematiques
Number: 2
ISSN: 0214-1493
Volume: 35
Year: 1991
Pages: from 363 to 374
Number of references: 10
Language: engleski
Summary: We introduce notions of F-universality and F-e-universality
formaps between compact Hausdorff spaces and explore the behavior of these
properties under the operation of composition of maps. We consider both the
quest for conditions on maps f and g which would imply that their
composition gf is either F-universal or F-e-universal and the quest for
consequences on f and g when the composition gf is either F-universal or
F-e-universal. In our approach F is an arbitrary class of maps. For a
special choice of F, the notion of F-universality reduces to Holsztynski's
notion of universality while F-e-universality reduces to Sanjurjo's notion
of proximate universality.
Keywords: universal map, proximately universal map, composition of maps, retraction, progression, liftable, approximate polyhedron, approximately right invertible, product of maps
Title: Proximately universal limits of approximate mappings
- Authors:
- Čerin, Zvonko (7994)
Journal: Mathematica Japonica
Number: 1
ISSN: 0025-5513
Volume: 37
Year: 1992
Pages: from 131 to 148
Number of references: 11
Language: engleski
Summary: We present sufficient conditions on an approximate mapping
F between approximate inverse systems in order that it's limit f is a
proximately alpha-universal map. The notion of proximate alpha-universality
is related to Jose M. R. Sanjurjo's concept of proximate universality for
maps between compact metric spaces and to the notion of weak universality
from fixed point theory. We also consider proximately beta-universal and
proximately gamma-universal maps and establish similar results for these
modifications. We get as corollaries some sufficient conditions on an
approximate inverse system implying that it's limit has the proximate fixed
point property.
Keywords: proximately universal map, proximately alpha-universal map, proximately beta-universal map, proximately gamma-universal map,approximate inverse system, approximate mapping, proximate ANR-system, proximate fixed point property
Title: Universal compositions
- Authors:
- Cammaroto, Filippo
- Gianella, Gian Mario
- Čerin, Zvonko (7994)
Journal: Mathematica Japonica
Number: 6
ISSN: 0025-5513
Volume: 37
Year: 1992
Pages: from 1061 to 1078
Number of references: 8
Language: engleski
Summary: We introduce four forms of universality for maps of compact
Hausdorff spaces and explore the behaviour of these properties under the
operation of composition of maps. We consider the quest for conditions on
maps f and g which would imply that their composition gf has one of these
four kinds of universality. These conditions are among the ones analogous
to various properties from shape theory. In order to keep track of many
possibilities we employ the matrix notation.
Keywords: universal map, proximately universal map, composition,retractive, right invertible, left invertible, liftable, progressive, movable, embedding
Title: Directors
- Authors:
- Cammaroto, Filippo
- Čerin, Zvonko (7994)
Journal: Atti Accademia Peloritana dei Pericolanti Classe I
di Scienze Fis. Mat. e Nat.
Number: 1
Volume: 68
Year: 1992
Pages: from 159 to 167
Number of references: 2
Language: engleski
Summary: We introduce a notion of directors and explore some of
their properties. There are sixteen different forms of directors. They
appear naturally in studies of relationships between various kinds of
universality for maps introduced recently by G. M.Gianella and the authors.
Keywords: director, universal map, progressive, liftable, direct sum, matrix
Title: Conductors
- Authors:
- Cammaroto, Filippo
- Gianella, Gian Mario
- Čerin, Zvonko (7994)
Journal: Supplemento ai Rendiconti del Circolo Matematico di
Palermo
Number: 1
Volume: 29
Year: 1992
Pages: from 41 to 50
Number of references: 2
Language: engleski
Summary: We introduce a notion of conductors and explore some of
their properties. There are sixteen different forms of conductors. They
appear naturally in studies of relationships between various kinds of
universality for maps introduced recently by F. Cammaroto, G. M. Gianella,
and the author.
Keywords: universal map, conductor, retractive, movable, matrix, product of spaces
Title: Lefschetz movable maps
- Authors:
- Čerin, Zvonko (7994)
Journal: Journal de Mathematiques Pures et Appliquees
Number: 1
ISSN: 0021-7824
Volume: 72
Year: 1993
Pages: from 81 to 103
Number of references: 25
Language: engleski
Summary: In an attempt to improve the Lefschetz fixed point theorem
we introduce the class of Lefschetz movable maps. These maps form a
subclass of Borsuk's nearly extendable maps. Its members satisfyan extra
condition concerning the behaviour of Lefschetz numbers of suitable
approximations. They all have fixed points even thought it might be
impossible to compute their Lefschets number.
We establish many nice properties of the new class. For example,
we study what type of domination preserves Lefschetz movability,
conditions under which compositions and products retain this property,
invariance with respect to shape, and characterizationin terms of
approximate resolutions. We also show that maps considered in Borsuk's
generalization of the Lefschetz fixed point theorem are Lefschetz movable.
The collections of spaces to which these techniques are applicable are
enlarged to classes of regularly movable and homology and homotopy calm
compacta.
Keywords: approximately movable map, Lefschetz movable map, nearly extendable map, Lefschetz number, generalized Lefschetz number, Granas movable map, domination, composition, product, homology stable map, approximate resolution, appropriate compactum
Title: Multi-valued functions and triviality
- Authors:
- Čerin, Zvonko (7994)
Journal: Topology Proceedings
Number: 1
Volume: 17
Year: 1992
Pages: from 1 to 27
Number of references: 12
Language: engleski
Summary: We give in this paper a description of a new extension of
the concept of contractible spaces to arbitrary topological spaces
analogous to the notion of spaces of trivial shape. We shall use
multi-valued functions with smaller and smaller images of points. Our
notion is modeled on the property that every map from a contractible space
is null-homotopic.
Keywords: contractible space, trivial shape, multi-valued function, sigma-small function, sigma-homotopy, CM-trivial, MD-trivial, O-space, like a class B, movable, tame
Title: Forced approximate resolutions
- Authors:
- Čerin, Zvonko (7994)
Journal: SUT Journal of Mathematics
Number: 2
Volume: 28
Year: 1992
Pages: from 105 to 119
Number of references: 4
Language: engleski
Summary: This paper is another contribution to the investigation of
recently defined notion of an approximate resolution of S. Mardesic and T.
Watanabe. The general idea is to improve
the classical theory of inverse systems and their limits where
we want to approximate a given space with spaces with good properties. We
introduce eight approximate type properties for classes of maps that we
call rectangular. For them we establish transition and domination results
and use these concepts in the problem of representing spaces as limits of
approximate resolutions with predefined bonding spaces and bonding maps.
Keywords: Approximate inverse system, approximate resolution, rectangular properties, net, fan, rim, fence, expansion, exposure
Title: Shape theory intrinsically
- Authors:
- Čerin, Zvonko (7994)
Journal: Publicacions Matematiques
Number: 2
Volume: 37
Year: 1993
Pages: from 317 to 334
Number of references: 6
Language: engleski
Summary: We prove in this paper that the category HM whose objects
are topological spaces and whose morphisms are homotopy classes of
multi-nets is naturally equivalent to the shape category Sh. The
description of the category HM was given earlier in the article "Shape
via multi-nets". We have shown there that the category
HM is equivalent to the shape category Sh only on a rather restricted class
of spaces. This class includes all comapct metric spaces where a similar
intrinsic description of the shape category using multi-valued functions
was given by Jose M. R. Sanjurjo. The present paper completes the task of
defining shape
theory of arbitrary topological spaces using multi-valued
functions that need not be continuous.
Keywords: homotopy theory, shape theory, multi-valued function, shape category, multi-net, category isomorphism, functor, normal cover, cofinite Cech system, canonical projection
Title: Recognizing approximate (A, B, C)-tameness
- Authors:
- Čerin, Zvonko (7994)
Journal: Acta Math. Univ. Comenianae
Number: 2
Volume: 62
Year: 1993
Pages: from 207 to 219
Number of references: 16
Language: engleski
Summary: In the paper "Forced approximate resolutions" the author
established some general conditions under which a topological space admits
an approximate resolution with predefined projections, bonding spaces,
and/or bonding maps. These conditions are expressed in terms of various
properties of maps of topological spaces described by approximately
commutative diagrams. Here we concentrate on the so called approximate
(A, B, C)-tameness, where A, B, and C are classes of maps. We
show that for suitable choices of classes A, B, and C these
results provide interesting consequences in dimension theory,
general topology, shape theory, and in the study of spaces which are like a
given class of spaces.
Since it is well known that every space has an approximate
resolution into polyhedra, an interesting question is to decide
what spaces have approximate resolutions with all bonding spaces
belonging to some subclass of polyhedra. We give necessary and
sufficient conditions on a space X in order that it has an approximate
resolution into polyhedra of dimension at most n, finite polyhedra,
countable polyhedra, locally finite polyhedra or locally countable
polyhedra, and contractible polyhedra. These
conditions are that X has covering dimension at most n, that X is
normally compact, that X is normally Lindelof, that X is strongly
paracompact, and that X has trivial shape.
Keywords: Approximately (A, B, C)-tame, approximates, approximate resolution, expansion, dimension, normally compact, normally Lindelof, strongly paracompact, trivial shape, C-like.
Title: Proper shape groups
- Authors:
- Čerin, Zvonko (7994)
Journal: Rendiconti dell'Istituto di Matematica dell'Universita di
Trieste
Number: I-II
Volume: 25
Year: 1993
Pages: from 89 to 125
Number of references: 11
Language: engleski
Summary: We give in this paper a description of proper shape groups
of arbitrary topological spaces. Our method is to use multi-valued
functions with smaller and smaller images of points. An analogous
intrinsic approach to shape groups of compact metric spaces was earlier
considered by Jose Sanjurjo. Our main result is a construction of a long
sequence of proper shape groups and the proof that it is weakly exact. On
compact spaces the proper shape
groups agree with the shape groups. We also study isomorphisms
induced on proper shape groups by trails in a space, where by a
trail we mean an analogue of the concept of a path in the proper
shape theory.
Keywords: Proper multi-valued function, proper homotopy group, proper shape group, proper pair, properly Z-pointed space, boundary operator, induced transformations, proper shape Z-groups sequence, weakly exact, trails induced isomorphisms
Title: Proper shape invariants: smoothness and calmness
- Authors:
- Čerin, Zvonko (7994)
Journal: Mathematica Pannonica
Number: 2
Volume: 5
Year: 1994
Pages: from 213 to 240
Number of references: 12
Language: engleski
Summary: We study properly (B, C)-smooth and properly C-calm spaces,
where
B and C denote classes of topological spaces. Both proper smoothness and
proper calmness are invariants of a recently invented author's proper shape
theory and are described by the use of proper multi-valued functions. The
main results deal
with preservation of these properties under various kinds of
proper maps. The dual notions are also examined.
Keywords: Proper multi-valued function, Mp-function, proper sigma-homotopy, proper multi-net, properly homotopic, properly smooth, properly calm, proper shape domination, proper surjection, proper injection
Title: Triviality via Multi-Valued Functions
- Authors:
- Čerin, Zvonko (7994)
Journal: Bollettino U. M. I.
Number: 7
Volume: 8-B
Year: 1994
Pages: from 477 to 507
Number of references: 17
Language: engleski
Summary: We describe a new extension to arbitrary topological spaces
of the notion of a contractible space analogous to the notion of
trivial shape. Our description uses multi-valued functions with
smaller and smaller images of points. The concept of a trivial
space is modelled on the fact that every map with values in a
contractible space is null-homotopic. In this fashion we get an
intrinsic description of spaces of trivial shape. One of the new
theorems that was not discovered earlier within inverse limit
approach to shape theory is the following. The union of trivial
spaces intersecting in a trivial space is also trivial.
Keywords: normal cover, sigma-tau function, sigma-small function, sigma- homotopic functions, CM-trivial space, HM-domination, right placid function, left placid function, absolute M-retract, movable space
Title: Proper shape theory
- Authors:
- Čerin, Zvonko (7994)
Journal: Acta Sci. Math. (Szeged)
Number: 2
Volume: 59
Year: 1994
Pages: from 679 to 711
Number of references: 20
Language: engleski
Summary: We give in this paper a description of proper shape
theories of
arbitrary topological spaces. Our method is to use multi-valued
functions with smaller and smaller images of points. An analogous
intrinsic approach to shape theory of compact metric spaces was
earlier considered by Jose Sanjurjo. The author has extended it to
arbitrary topological spaces and the present paper shows how
this extension can be adapted to the proper case. The main result
is a construction of the proper shape category Shp whose objects
are topological spaces and whose morphisms are proper homotopy
classes of proper multi-nets. The category Shp relates to the proper
homotopy category Hp similarly as the shape category Sh relates to the
homotopy category H. On compact spaces the proper shape category agrees
with the shape category. For locally compact metrizable spaces, we show the
existence of a natural functor from our proper shape category into Ball's
proper shape
category which is similarly related to the original Ball and Sher proper
shape category.
Keywords: homotopic maps, proper map, multi-valued function, B-proper function, S-proper function, properly sigma-homotopic functions, proper multi-net, proper shape category, properly internally movable, properly internally calm
Title: Universal factors
- Authors:
- Cammaroto, Filippo
- Gianella, Gian Mario
- Čerin, Zvonko (7994)
Journal: Mathematica Japonica
Number: 1
ISSN: 0025-5513
Volume: 39
Year: 1994
Pages: from 185 to 197
Number of references: 12
Language: engleski
Summary: We study the following problem. Suppose that the
composition of
maps f and g is a universal map. Under what conditions will the
factors f and g also be universal? This question is here considered for
four different kinds of universality introduced
recently by the authors in the paper "Universal compositions".
The concept of a universal map was originally defined by
Holsztynski and is a map which has a coincidence point with
every other map. Our universalities are of the approximate
kind with shape theoretic flavor.
Keywords: universal map, liftable map, movable map, neighborhood extensor space, neighborhood restrictor space, extendable map, composition, factor, embedding, partial matrix
Title: Fiberwise shape theory
- Authors:
- Čerin, Zvonko (7994)
Journal: Collect. Math.
Number: 2
Volume: 45
Year: 1994
Pages: from 101 to 119
Number of references: 19
Language: engleski
Summary: We shall describe a modification of the fiberwise homotopy
theory
which we call the fiberwise shape theory. This is accomplished by
constructing the fiberwise shape category FsB. The category FsB is built
using multi-valued functions. Its objects are fiberwise topological spaces
while its morphisms are fiberwise homotopy classes of collections of
multi-valued functions which we call
fiberwise multi-nets. When the base space B is a single-element
space, the fiberwise shape category is isomorphic with the shape
category. Various authors have previously given other descriptions of
fiberwise shape categories under additional assumptions. Our description is
intrinsic in the sense that we do
not use any outside objects and not even absolute neighborhood retracts. It
is a fiberwise version of the author's extension to
arbitrary topological spaces of Sanjurjo's approach to shape theory via
small upper semi-continuous multi-valued functions.
Keywords: fiberwise topology, fiberwise homotopy theory, fiberwise normal cover, fiberwise sigma-homotopic functions, fiberwise multi-net, total space, base space, fiberwise internally movable, fiberwise internally calm, fiberwise absolute neighborhood retract
Title: Morphisms in proper shape theory
- Authors:
- Čerin, Zvonko (7994)
Journal: Mathematica Japonica
Number: 2
Volume: 41
Year: 1995
Pages: from 351 to 369
Number of references: 8
Language: engleski
Summary: This paper introduces four interesting classes of proper
maps
between topological spaces called proper MB-surjections, proper
NB-surjections, proper MB-injections, and proper NB-injections,
where B denotes a class of topological spaces. They are morphisms of a
recently invented author's proper shape theory and are described with the
use of multi-valued functions. This proper shape theory is applicable to
arbitrary topological spaces. Our
proper surjections and proper injections share many properties with
standard forms of these notions. There is a natural duality
between proper surjections and proper injections and between the M and the
N forms. These two possibilities reflect the fact that we can study global
properties of spaces by either mapping into them members of a class B or
mapping them into members of B.
Keywords: Proper multi-valued function, sigma-close functions, sigma-small function, proper sigma-homotopy, proper multi-net, properly homotopic, proper MB-surjection, proper MB-injection, proper NB-surjection, proper NB-injection
Title: Equivariant shape theory
- Authors:
- Čerin, Zvonko (7994)
Journal: Math. Proc. Camb. Phil. Soc.
Number: 2
Volume: 117
Year: 1995
Pages: from 303 to 320
Number of references: 16
Language: engleski
Summary: Equivariant shape theory is an improvement of equivariant
homotopy theory which could be regarded as an equivariant version
of Borsuk's shape theory. The main result in this paper is a description of
the equivariant shape category ShG whose objects are equivariant spaces or
G-spaces, i. e., topological spaces endowed with an action of a given
topological group G, and whose morphisms are equivariant homotopy or
G-homotopy classes of families of multi-valued functions which we call
equivariant multi-nets or G-multi-nets. Previously equivariant shape
theories
have been described only under the assumptions that the group G is either
finite or compact. We also study classes of G-spaces on which equivariant
shape and equivariant homotopy coincide, look for conditions under which a
G-map f between G-spaces X and Y is an equivariant shape equivalence, and
give some characterizations of G-spaces with trivial equivariant shape.
Keywords: Equivariant shape theory, equivariant homotopy theory, G-space, G-map, equivariant multi-net, internally G-movable, internally G-calm, equivariant shape equivalence, equivariant shape category, approximately G-contractible space
Title: Proximate topology and shape theory
- Authors:
- Čerin, Zvonko (7994)
Journal: Proceedings of the Royal Society of Edinburgh
Number: 4
Volume: 125A
Year: 1995
Pages: from 595 to 615
Number of references: 15
Language: engleski
Summary: Most of the development of shape theory was in the
so-called outer shape theory, where the shape of spaces is described with
the help of some outside objects.
This paper belongs to the so-called inner shape theory, in which the
shape of spaces is described intrinsically without the use of any outside
gadgets. We give a description of shape theory that does not need absolute
neighborhood retracts. We prove that the category HN whose objects are
topological spaces and whose morphisms are proximate homotopy classes of
proximate nets is naturally equivalent to the shape category Sh. The
description of the category HN for compact metric spaces was given earlier
by Jose M. R. Sanjurjo. We also give three applications of this new
approach to shape theory.
Keywords: Homotopy theory, shape theory, proximate topology, normal cover, alpha-beta-continuous functions, sigma-close functions, sigma-homotopic functions, proximate net of functions, proximately homotopic proximate nets, cofinite Cech system
Title: Upper Homotopy and Shape Morphisms
- Authors:
- Čerin, Zvonko (7994)
Journal: Atti Sem. Mat. Fis. Univ. Modena
Number: 1
Volume: 43
Year: 1995
Pages: from 41 to 52
Number of references: 15
Language: engleski
Summary: We consider families of multi-valued functions between
topological spaces X and Y which more and more resemble single-valued
functions. We introduce a notion of upper homotopy for these families and
prove that there is a one-to-one correspondence between the set of all
upper homotopy classes and the set Sh(X, Y) of all shape morphisms from X
into a space Y that has an ANR-resolution with onto projection maps. In the
case when X and Y are compact metric spaces, Jose M. R. Sanjurjo gave
analogous descriptions of shape morphisms using multi-valued functions.
Keywords: Multi-valued function, sigma-small function, sigma-close functions, upper sigma-homotopic functions, normal cover, upper C-map, upper homotopic, O-space, approaching map, category isomorphism
Title: Homotopy groups for C*-algebras
- Authors:
- Čerin, Zvonko (7994)
Journal: Bolyai Society Mathematical Studies
Volume: 4
Year: 1993
Pages: from 29 to 45
Number of references: 6
Language: engleski
Summary: This paper belongs to the part of mathematics which could
be described as the algebraic topology of C*-algebras. The idea is to study
analogues of notions from algebraic topology. Some authors have been using
the term "non-commutative" algebraic topology whenever referring to results
which represent extensions to arbitrary C*-algebras from commutative
C*-algebras (which by Gelfand duality are identified with locally compact
Hausdorff spaces). Our point of view is to consider notions for C*-algebras
resembling corresponding notions from homotopy theory and algebraic
topology without insisting that in the case of commutative C*-algebras they
reduce (via Gelfand duality) to these notions. This change of approach will
be illustrated here with homotopy groups.
Keywords: Homotopy theory, algebraic topology, homotopy groups, C*-algebra, Gelfand duality, relative homotopy groups, *-homomorphism, long exact homotopy groups sequence, C*-fibering, induced transformations
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