Homotopy Extension Property Research Articles

Two homotopy characterizations of G-ANR spaces for proper actions of Lie groups

For a Lie group G we consider the class G-M of all proper (in the sense of R. Palais) G-spaces that are metrizable by a G-invariant metric. It is proved that if each point x∈X of a proper G-space X admits a Gx-invariant neighbourhood U which is a Gx-ANE then X is a G-ANE, where Gx stands for the stabilizer of x. We give two equivariant homotopy characterizations of proper G-ANR spaces in the class G-M. One of them asserts that a G-space Y∈G-M is a G-ANR iff Y is locally G-contractible and every metrizable closed G-pair (X,A) with X∈G-M has the G-equivariant homotopy extension property with respect to Y (for short, property G-HEP). Another result establishes that a stronger homotopy property than G-HEP also characterizes G-ANR spaces in the class G-M.

Homotopy extension property for multi-valued functions

In this study, we introduce some well-known definitions and properties for multi-valued functions. We present new definitions such as, m-retraction, m-section, m-homeomorphism, m-HEP and reducible function. We give a new result on the relation between multi-homotopy groups and m-homeomorphism. We also deal with some properties of m-HEP.

Homotopy Theory in Digital Topology

Digital topology is part of the ongoing endeavor to understand and analyze digitized images. With a view to supporting this endeavor, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the most basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik–Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics.

A combinatorial characterization of Hurewicz cofibrations between finite topological spaces

We characterize the Hurewicz cofibrations between finite topological spaces, that is, the continuous functions between finite topological spaces that have the homotopy extension property with respect to all topological spaces. In particular, we show that cofibrations between connected non-empty finite topological spaces are homotopy equivalences.As a consequence of our characterization, we obtain a simple algorithm capable of determining whether a given continuous function between finite topological spaces is a cofibration.

Homotopy Extension Property in Homotopy Theory for Topological Semigroups

The purpose of this paper is to extend the concept of homotopy extension property in homotopy theory for topological spaces to its analogical structure in homotopy theory for topological semigroups. In this extension, we also give some results concerning on absolutely retract and its properties.

Designing and modeling cyberworlds using the incrementally modular abstraction hierarchy based on homotopy theory

For designing and modeling complicated and sophisticated systems such as cyberworlds, their mathematical foundation is critical. To realize it, two important properties called the homotopy lifting property (HLP) and homotopy extension property (HEP) are applied for designing and modeling a system in a bottom-up way and a top-down way, respectively. In this paper, an enterprise system and a real-time embedded system are considered as important socially emerging cases of cyberworlds, where the π-calculus processes for describing these behaviors formally, a Petri net for explaining process interactions, and XMOS XC programs are modeled and designed by our approach. The spaces in both properties are specified by the incrementally modular abstraction hierarchy by climbing down the abstraction hierarchy from the most abstract homotopy level to the most specific view level, while keeping invariants such as homotopy equivalence and topological equivalence.

A kaleidoscope as a cyberworld and its animation: linear architecture and modeling based on an incrementally modular abstraction hierarchy

AbstractAn incrementally modular abstraction hierarchy is known to effectively linearize cyberworlds and virtual worlds, which are combinatorially exploding and hardly managed. It climbs down from general level to specific model preserving the higher level modules as invariants. It not only prevents the combinatorial explosion but also benefits the reuse, development, testing and validation of cyberworld resources. By applying this incrementally modular abstraction hierarchy to a kaleidoscope animation, its architecture and modeling is also specified in this paper as a typical case of cyberworlds. In particular, a homotopy lifting property and a homotopy extension property, which satisfy a duality relation, are also described to show how a kaleidoscope world is systematically created top‐down from the whole system and bottom‐up from the components. Copyright © 2006 John Wiley & Sons, Ltd.

Absolute extensors and absolute neighborhood extensors in asymptotic categories

This paper deals with extending maps in asymptotic categories, i.e., in categories consisting of metric spaces and asymptotically Lipschitz coarsely proper maps. We demonstrate certain examples of absolute extensors and absolute neighborhood extensors. We give some conditions under which a version of Borsuk's homotopy extension theorem holds in these categories, and in answer to a problem posed by Dranishnikov in [Russian Math. Surveys 55 (2000) 1085] we show the failure of a general homotopy extension theorem. Finally, we show that a pair of an Hadamard space and its convex subspace has the homotopy extension property.

Metrizable shape and strong shape equivalences

In this paper we construct a functor ˆ : proTop ! proANR which extends Mardeysic correspondence that assigns to every metrizable space its canonical ANR-resolution. Such a functor allows one to deÞne the strong shape category of prospaces and, moreover, to deÞne a class of spaces, called strongly Þbered, that play for strong shape equivalences the role that ANR- spaces play for ordinary shape equivalences. In the last sec- tion we characterize SSDR-promaps, as deÞned by Dydak and Nowak, in terms of the strong homotopy extension property considered by the author.

Comparison Theorems in Homotopy Theory Via Operations on Homotopy Sets

In this paper we present comparison theorems in semicubical homotopy theory. Topological versions are more or less well known and have been obtained amongst others by A. DOLD ([6], [7]), I. M. JAMES ([11]), P. R. HEATH ([9]), S. P. LAM ([15]), and the first author ([2], [3]). A partially abstract approach has been given by the first author in [3]. We use operations on homotopy sets and the notion of CHEP (covering homotopy extension property).

On extending continuous functions into a metrizable AE

We say that a subset S of a topological space X is M-embedded ( M N 0 -embedded) in X if every map from S to a (separable) metrizable AE can be extended over X. Characterizations of M-and M N O -embedding are given and we prove that S is M-embedded ( M N O -embedded) in X iff( X,S) has the Homotopy Extension Property with respect to every (seperable) ANR space.

On the very weak and rather weak homotopy-extension property

On the very weak and rather weak homotopy-extension property

C-embedding and the homotopy extension property

The purpose of this paper is to give a condition under which the union of two C-embedded subsets of a space X is C-embedded in X, and apply it to obtain a homotopy extension theorem stating that a pair ( X, A) of spaces has the homotopy extension property with respect to a separable Čech complete ANR for metric spaces iff A is C-embedded in X.

On composite loop functors

P

�ber einige formale Eigenschaften von Faserungen und h-Faserungen

The aim of this paper is to study some formal properties of fibrations (=continuous maps which have the covering homotopy property), cofibrations (=maps which have the homotopy extension property), h-fibrations ([2],6.4) and h-cofibrations([2],2.2). We introduce the notion of a generalized homotopy system in a category (2). This notion will be selfdual. To show that fibrations and cofibrations, h-fibrations and h-cofibrations are dual notions, no use will be made of adjoint functors. Our approach admits the transition from a categoryC to other categories, e.g. to the categoryCB of objects over a given object B ofC.

Induced fibrations and cofibrations

Introduction. It is well known that any map is homotopically equivalent to a fiber map, i.e., to the projection of the total space on the base in a fibration. Simple examples, however, reveal that there are maps which fail to be homotopically equivalent to any inclusion of a fiber in the total space, and the problem of characterizing the maps which are equivalent to such inclusions was first raised in [14]. By the first remark, the map under consideration may be assumed from the beginning to be a fiber map p: E -B, and the problem is thus converted into that of characterizing the fibrations p which are equivalent to induced fibrations. There are two immediate necessary conditions: the fiber F of p must have the homotopy type of some loop space fl Y, and the inclusion i: F -* E must map the generalized homotopy group 7r(X, fQF) into the center of n(X, QE) for any space X; the latter is a mild generalization of the well known fact that the boundary operator in the homotopy sequence of a fibration E -* B -* Y maps XT2( Y) into the center of 7r1(E). The first result in this direction is due to Serre and asserts that p is an induced fibration if B is 1-connected and F has a single nonvanishing (Abelian) homotopy group. This result was generalized [6], [12], [17], [15] to allow F to have at most m -1 nonvanishing homotopy groups in consecutive dimensions provided it has the homotopy type of a loop space and B is (m -)-connected. In the first two sections of this paper we give results which allow F to have 2m -1 nonvanishing homotopy groups in consecutive dimensions provided B is (m 1)-connected and F has the homotopy type of a loop space under a homotopy equivalence fulfilling a certain condition which involves the operation of QB on F. In case F has m nonvanishing homotopy groups in consecutive dimensions, the sufficient condition in order that p be induced is expressed by means of the vanishing of a certain Whitehead product; this result answers a question raised in [15]. Dually, any map is homotopically equivalent to an inclusion A -> X having the homotopy extension property, but few maps are equivalent to identification maps resulting by shrinking to a point such a subset A of X. This leads to the problem of characterizing the cofibrations A -> X which are induced by maps of some space Y into A. Our results extend the range of applicability of some previous results [8], [12] by imposing an additional condition which involves the cooperation of the suspension EA on the cofiber C obtained from X by shrinking A to a point; the condition is conveniently expressed in terms of the Hopf invariant of a cofibration