Multifibrations. A class of shape fibrations with the path lifting property (original) (raw)
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A shape fibration with fibers of different shape
Pacific Journal of Mathematics, 1979
Let E and B be metric continua. Let p:E~>B be a shape fibration in the sense of Mardesic and Rushing. If B is arcwise connected, then all the fibers of p have the same shape. This is also true if B is connected by shape paths in the sense of Krasinkiewicz and Mine. It had been asked whether that this would be true without any assumptions other than B being a continuum. In this paper an example is given of a shape fibration p: E ~* B with E and B metric continua such that p has fibers of different shape.
Continuous functions induced by shape morphisms
Proceedings of the American Mathematical Society, 1973
Let C denote the category of compact Hausdorff spaces and continuous maps and H:C-*HC the homotopy functor to the homotopy category. Let S : C-<-SC denote the functor of shape in the sense of Holsztyñski for the projection functor //. Every continuous mapping / between spaces gives rise to a shape morphism S(f) in SC, but not every shape morphism is in the image of S. In this paper it is shown that if X is a continuum with xe X and A is a compact connected abelian topological group, then if F is a shape morphism from X to A, then there is a continuous map f:X-A such that/(x)=0 and S(f)=F. It is also shown that if f,g:X-*A are continuous whhf(x)=g(x)=0
Function spaces and shape theories
Fundamenta Mathematicae, 2002
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(HoT op) to Sh(pro − HoT op) by localizing pro − HoT op at shape equivalences. Similarly, we extend the strong shape category of Edwards-Hastings to sSh(pro − T op) by localizing pro − T op 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 ∈ AN R. A map f : X → Y of k-spaces is a strong shape equivalence if and only if the induced map f * : M ap(Y, P ) → M ap(X, P ) is a weak homotopy equivalence for all P ∈ AN R. One generalizes the concept of being a shape equivalence to morphisms of pro − HoT op without any problem and the only difficulty is to show that a localization of pro − HoT op at shape equivalences is a category (which amounts to showing that morphisms form a set). Due to peculiarities of function spaces, extending the concept of a strong shape equivalence to morphisms of pro − T op 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 * : M ap(Y, P ) → M ap(X, P ) is a homotopy equivalence for all P ∈ AN R, and one can extend it to morphisms of pro − T op. However, the authors do not know if the corresponding localization exists. Here are applications of our methods:
On the Shape Category of Compacta
Journal of the London Mathematical Society, 1986
The purpose of this paper is to prove several results concerning the shape category of a metric compactum in the sense of K. Borsuk. Among other things, we prove that if there exists a refinable map from a compactum X onto a compactum Y then the shape categories of A' and Y are coincident. We also characterize the shape category in terms of deformations of the diagonal. Finally, we introduce a new numerical shape invariant under the name of coefficient of movability and give a relation between the coefficient of movability of the total space, the coefficient of movability of the fibre and the shape category of the base space of a shape fibration.
Strong shape theory of continuous maps
TBILISI - MATHEMATICS, 2021
The work is motivated by the papers [Ba 1 ], [Ba 2 ], [Ba 7 ], [Ba 11 ], [Be] and [Be-Tu]. In particular, the strong homology groups of continuous maps were defined and studied in [Be] and [Be-Tu]. To show that the given groups are a homology type functor, it was required to construct a corresponding shape category. In this paper, we study this very problem. In particular, using the methods developed in [Ba 7 ], [Ma 3 ], the strong shape theory of continuous maps of compact metric spaces, the so-called strong fiber shape theory is constructed.
Fiber Strong Shape Theory for Topological Spaces
2017
In the paper we construct and develop a fiber strong shape theory for arbitrary spaces over fixed metrizable space 0 B . Our approach is based on the method of Mardešic'-Lisica and instead of resolutions, introduced by Mardešic', their fiber preserving analogues are used. The fiber strong shape theory yields the classification of spaces over 0 B which is coarser than the classification of spaces over 0 B induced by fiber homotopy theory, but is finer than the classification of spaces over 0 B given by usual fiber shape theory. Math. Sub. Class.:54C55, 54C56, 55P55.
Topology and its Applications, 1999
We prove that weak shape equivalences are monomorphisms in the shape category of uniformly pointed movable continua 5%~. We use an example of Draper and Keesling to show that weak shape equivalences need not be monomorphisms in the shape category. We deduce that Shhl is not balanced. We give a characterization of weak dominations in the shape category of pointed continua, in the sense of . We introduce the class of pointed movable triples (X, F, Y), for a shape morphism F :X -+ Y, and we establish an infinite-dimensional Whitehead theorem in shape theory from which we obtain, as a corollary, that for every pointed movable pair of continua (Y, X) the embedding j : X + Y is a shape equivalence iff it is a weak shape equivalence. 0 1999 Elsevier Science B.V. All rights reserved. Keywords: Weak shape equivalence; Shape category of uniformly pointed movable continua;
On maps with continuous path lifting
Fundamenta Mathematicae
We study a natural generalization of covering projections defined in terms of unique lifting properties. A map p : E → X has the "continuous path-covering property" if all paths in X lift uniquely and continuously (rel. basepoint) with respect to the compact-open topology. We show that maps with this property are closely related to fibrations with totally pathdisconnected fibers and to the natural quotient topology on the homotopy groups. In particular, the class of maps with the continuous path-covering property lies properly between Hurewicz fibrations and Serre fibrations with totally path-disconnected fibers. We extend the usual classification of covering projections to a classification of maps with the continuous path-covering property in terms of topological π 1 : for any path-connected Hausdorff space X, maps E → X with the continuous path-covering property are classified up to weak equivalence by subgroups H ≤ π 1 (X, x 0) with totally path-disconnected coset space π 1 (X, x 0)/H. Here, "weak equivalence" refers to an equivalence relation generated by formally inverting bijective weak homotopy equivalences.
A topology for the sets of shape morphisms
Topology and its Applications, 1999
We introduce a topology on the set of shape morphisms between arbitrary topological spaces X, Y , Sh(X, Y ). These spaces allow us to extend, in a natural way, some classical concepts to the realm of topological spaces. Several applications are given to obtain relations between shape theory and N-compactness and shape-theoretic properties of the spaces of quasicomponents.