Homotopy Extension 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.

Topological finiteness properties of monoids, I: Foundations

We initiate the study of higher dimensional topological finiteness properties of monoids. This is done by developing the theory of monoids acting on CW complexes. For this we establish the foundations of $M$-equivariant homotopy theory where $M$ is a discrete monoid. For projective $M$-CW complexes we prove several fundamental results such as the homotopy extension and lifting property, which we use to prove the $M$-equivariant Whitehead theorems. We define a left equivariant classifying space as a contractible projective $M$-CW complex. We prove that such a space is unique up to $M$-homotopy equivalence and give a canonical model for such a space via the nerve of the right Cayley graph category of the monoid. The topological finiteness conditions left-$\mathrm{F}_n$ and left geometric dimension are then defined for monoids in terms of existence of a left equivariant classifying space satisfying appropriate finiteness properties. We also introduce the bilateral notion of $M$-equivariant classifying space, proving uniqueness and giving a canonical model via the nerve of the two-sided Cayley graph category, and we define the associated finiteness properties bi-$\mathrm{F}_n$ and geometric dimension. We explore the connections between all of the these topological finiteness properties and several well-studied homological finiteness properties of monoids which are important in the theory of string rewriting systems, including $\mathrm{FP}_n$, cohomological dimension, and Hochschild cohomological dimension. We also develop the corresponding theory of $M$-equivariant collapsing schemes (that is, $M$-equivariant discrete Morse theory), and among other things apply it to give topological proofs of results of Anick, Squier and Kobayashi that monoids which admit presentations by complete rewriting systems are left-, right- and bi-$\mathrm{FP}_\infty$.

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.

Quantum Field Theories on Categories Fibered in Groupoids

We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as bundles, connections and spin structures. Using right Kan extensions, we can assign to any such theory an ordinary quantum field theory defined on the category of spacetimes and we shall clarify under which conditions it satisfies the axioms of locally covariant quantum field theory. The same constructions can be performed in a homotopy theoretic framework by using homotopy right Kan extensions, which allows us to obtain first toy-models of homotopical quantum field theories resembling some aspects of gauge theories.

The Homotopy Lifting Theorem for Semiprojective $C^*$-Algebras

We prove a complete analog of the Borsuk Homotopy Extension Theorem for arbitrary semiprojective $C^*$-algebras. We also obtain some other results about semiprojective $C^*$-algebras: a partial lifting theorem with specified quotient, a lifting result for homomorphisms close to a liftable homomorphism, and that sufficiently close homomorphisms from a semiprojective $C^*$-algebra are homotopic.

The covering homotopy extension problem for compact transformation groups

It is shown that the orbit space of universal (in the sense of Palais) G-spaces classifies G-spaces. Theorems on the extension of covering homotopy for G-spaces and on a homotopy representation of the isovariant category ISOV are proved.

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.

Cofibrations in the category of Frölicher spaces: Part I

Cofibrations are defined in the category of Fr\olicher spaces by weakening the analog of the classical definition to enable smooth homotopy extensions to be more easily constructed, using flattened unit intervals. We later relate smooth cofibrations to smooth neighborhood deformation retracts. The notion of smooth neighborhood deformation retract gives rise to an analogous result that a closed Fr\olicher subspace $A$ of the Fr\olicher space $X$ is a smooth neighborhood deformation retract of $X$ if and only if the inclusion $i: A\hookrightarrow X$ comes from a certain subclass of cofibrations. As an application we construct the right Puppe sequence.

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.

Perturbation theorems for positive eigenvalues of countably condensing maps

Using a homotopy extension theorem, we give a perturbation theorem for countably condensing maps in a more general setting. Moreover, we prove perturbation theorems for positive eigenvalues of countably condensing maps in locally convex topological vector spaces which include the case of condensing maps due to Jerofsky [1].

Fixed point theorems for condensing admissible maps in topological vector spaces

Introducing the notion of c-measure of noncompactness due to S. Hahn, we give a homotopy extension theorem and the fixed point property for pseudo-condensing admissible maps in general topological vector spaces. Moreover, these results are applied to obtain a Leray-Schauder type fixed point theorem for pseudo-condensing admissible maps in topological vector spaces which includes many known results.

P-embeddings, AR and ANR spaces

Generalizations of Borsuk’s characterization theorems for AR and ANR spaces (for metrizable spaces) are provided. Moreover, a simple proof of Morita’s theorem is given, in generalization of Borsuk’s homotopy extension theorem.

Strong cofibrations and fibrations in enriched categories

We define strong cofibrations and fibrations in suitably enriched categories using the relative homotopy extension resp. lifting property. We prove a general pairing result, which for topological spaces specializes to the well-known pushout-product theorem for cofibrations. Strong cofibrations and fibrations give rise to cofibration and fibration categories in the sense of homotopical algebra. We discuss various examples; in particular, we deduce that the category of chain complexes with chain equivalences and the category of categories with equivalences are symmetric monoidal proper closed model categories.

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.

Fixed points of compact approachable multimaps in topological vector spaces

Applying the homotopy extension theorem for compact approachable vector fields with star-shaped values, we prove the existence of fixed points and eigenvalues of compact approachable multimaps in topological vector spaces. In particular, we give the Birkhoff-Kellogg Theorem for compact approachable multimaps with star-shaped values in normal spaces.

Generalized Leray-Schauder principles for compact adminisible multifunctions

We establish the Leray–Schauder type theorems for very general classes of multifunctions, which are called admissible. Our admissible classes contain compositions of important multifunctions in nonlinear analysis and algebraic topology. Moreover, our arguments are elementary, without using the concept of degree of maps or theory of homotopy extensions. The Leray–Schauder principle [LS], one of the most important theorems in nonlinear analysis, was first proved for a Banach space in the context of degree theory. In [N], Nagumo extended the degree theory to locally convex topological vector spaces and his results can be used to generalize the Leray–Schauder principle. Variations of the principle were due to Browder [B] for Banach spaces and to Schaefer [Sc2] for locally convex topological vector spaces without using degree theory. Schaefer’s version has important applications to integral equations. Later Potter [Po] generalized the results of Browder and Schaefer. However, those authors considered single-valued maps and adopted boundary conditions particular to the so-called Leray–Schauder condition (LS). There are many other authors who obtained generalized versions of the Leray–Schauder type theorems (see the references).