Local Contractibility Research Articles

Smooth and proper maps with respect to a fibration

Abstract This paper explain how the geometric notions of local contractibility and properness are related to the $\Sigma$ -types and $\Pi$ -types constructors of dependent type theory. We shall see how every Grothendieck fibration comes canonically with such a pair of notions—called smooth and proper maps—and how this recovers the previous examples and many more. This paper uses category theory to reveal a common structure between geometry and logic, with the hope that the parallel will be beneficial to both fields. The style is mostly expository, and the main results are proved in external references.

Hyperinvariant multiscale entanglement renormalization ansatz: Approximate holographic error correction codes with power-law correlations

We consider a class of holographic tensor networks that are efficiently contractible variational ansatze, manifestly (approximate) quantum error correction codes, and can support power-law correlation functions. In the case when the network consists of a single type of tensor that also acts as an erasure correction code, we show that it cannot be both locally contractible and sustain power-law correlation functions. Motivated by this no-go theorem, and the desirability of local contractibility for an efficient variational ansatz, we provide guidelines for constructing networks consisting of multiple types of tensors that can support power-law correlation. We also provide an explicit construction of one such network, which approximates the holographic HaPPY pentagon code in the limit where variational parameters are taken to be small.

Lower bounds for codimension-1 measure in metric manifolds

We establish Euclidean-type lower bounds for the codimen\-sion-1 Hausdorff measure of sets that separate points in doubling and linearly locally contractible metric manifolds. This gives a quantitative topological isoperimetric inequality in the setting of metric manifolds, in the sense that lower bounds for the codimension-1 measure of a set depend not on some notion of filling or volume but rather on in-radii of complementary components. As a consequence, we show that balls in a closed, connected, doubling, and linearly locally contractible metric n -manifold (M,d) with radius 0 < r \leq \mathrm {diam}(M) have n -dimensional Hausdorff measure at least c \cdot r^n , where c>0 depends only on n and on the doubling and linear local contractibility constants.

Topological properties in Whitney blocks

Let C(X) be the hyperspace of subcontinua of a continuum X. A Whitney block is a set of the form μ−1([s,t]), where μ:C(X)→[0,1] is a Whitney map and 0≤s<t≤1. In this paper, we study the following implication: if X has property P, then each Whitney block in C(X) has property P. We consider the following properties: connectedness im kleinen, being absolute neighborhood retract, local contractibility, and m-mutual aposyndesis.

Injective objects in the category of stratified spaces

The paper deals with the category of stratified spaces. For this category we establish the existence of the absolute extensors and study their connection with the absolute extensors in the category of filtered spaces. To this end, we introduce and investigate the absolute extensors reflecting the properties of both the filtered category and the stratified category. We determine the contractibility property as well as the local contractibility property for the stratified absolute extensors. It is shown that the family of strata for a stratified absolute neighbourhood extensor is equicontinuously locally extendable.

Contractibility and fixed point property: the case of Khalimsky topological spaces

AbstractBased on the notions of both contractibility and local contractibility, many works were done in fixed point theory. The present paper concerns a relation between digital contractibility and the existence of fixed points of digitally continuous maps. In this paper, establishing a new digital homotopy named by a K-homotopy in the category of Khalimsky topological spaces, we prove that in digital topology, whereas contractibility implies local contractibility, the converse does not hold. Furthermore, we address the following problem, which remains open. Let X be a Khalimsky (K- for short) topological space with K-contractibility. Then we may pose the following question: does the space X have the fixed point property (FPP)? In this paper, we prove that not every K-topological space with K-contractibility has the FPP.

Groups of uniform homeomorphisms of covering spaces

In this paper we deduce a local deformation lemma for uniform embeddings in a metric covering space over a compact manifold from the deformation lemma for embeddings of a compact subspace in a manifold. This implies the local contractibility of the group of uniform homeomorphisms of such a metric covering space under the uniform topology. Furthermore, combining with similarity transformations, this enables us to induce a global deformation property of groups of uniform homeo- morphisms of metric spaces with Euclidean ends. In particular, we show that the identity component of the group of uniform homeomorphisms of the standard Euclidean n-space is contractible.

(Non)connectedness and (non)homogeneity

We discuss an approach to a problem posed by A.V. Arhangel'skii and E.K. van Douwen on a possibility to present a compact space as a continuous image of a homogeneous compact space. Then we suggest some ways of proving nonhomogeneity of τ-powers of a space X using points of local connectedness (or local contractibility) and components of path connectedness of X.

Weak convergence of currents and cancellation

In this article, we study the relationship between the weak limit of a sequence of integral currents in a metric space and the possible Hausdorff limit of the sequence of supports. Due to cancellation, the weak limit is in general supported in a strict subset of the Hausdorff limit. We exhibit sufficient conditions in terms of topology of the supports which ensure that no cancellation occurs and that the support of the weak limit agrees with the Hausdorff limit of the supports. We use our results to prove countable \({{\fancyscript H}^m}\)-rectifiability of the Gromov-Hausdorff limit of sequences of Lipschitz manifolds Mn all of which are λ-linearly locally contractible up to some scale r0. In the Appendix, we show that the Gromov-Hausdorff limit need not be countably \({{\fancyscript H}^m}\) -rectifiable if the Mn have a common local geometric contractibility function which is only concave (and not linear). We also relate our results to work of Cheeger-Colding on the limits of noncollapsing sequences of manifolds with nonnegative Ricci curvature.

Local contractibility of the homeomorphism group of ℝ n

The goal of this paper is to give a modified exposition of the main part of the proof of the local contractibility theorem. The derivation of general theorems from the special case of Euclidean space remains intact. The exposition is rather detailed and is aimed, in particular, at correcting an inaccuracy in the original proof.

On the number of homotopy types of fibres of a definable map

In this paper we prove a single exponential upper bound on the number of possible homotopy types of the fibres of a Pfaffian map in terms of the format of its graph. In particular, we show that if a semi-algebraic set S⊂Rm+n, where R is a real closed field, is defined by a Boolean formula with s polynomials of degree less than d, and π: Rm+n→Rn is the projection on a subspace, then the number of different homotopy types of fibres of π does not exceed s2(m+1)n(2m nd)O(nm). As applications of our main results we prove single exponential bounds on the number of homotopy types of semi-algebraic sets defined by fewnomials, and by polynomials with bounded additive complexity. We also prove single exponential upper bounds on the radii of balls guaranteeing local contractibility for semi-algebraic sets defined by polynomials with integer coefficients.

On the structure of the group of Lipschitz homeomorphisms and its subgroups, II

We consider the group of Lipschitz homeomorphisms of a Lipschitz manifold and its subgroups. First we study properties of Lipschitz homeomorphisms and show the local contractibility and the perfectness of the group of Lipschitz homeomorphisms. Next using this result we can prove that the identity component of the group of equivariant Lipschitz homeomorphisms of a principal G-bundle over a closed Lipschitz manifold is perfect when G is a compact Lie group.

Effectiveness for Embedded Spheres and Balls

We consider arbitrary dimensional spheres and closed balls embedded in Rn as ⫫01 classes. Such a strong restriction on the topology of a ⫫01 class has computability theoretic repercussions. Algebraic topology plays a crucial role in our exploration of these consequences; the use of homology chains as computational objects allows us to take algorithmic advantage of the topological structure of our ⫫01 classes. We show that a sphere embedded as a ⫫01 class is necessarily located, i.e., the distance to the class is a computable function, or equivalently, the class contains a computably enumerable dense set of computable points. Similarly, a ball embedded as a ⫫01 class has a dense set of computable points, though not necessarily c.e. To prove location for balls, it is sufficient to assume that both it and its boundary sphere are ⫫01. However, the converse fails, even for arcs; using a priority argument, we prove that there is a located arc in R2 without computable endpoints. Finally, the requirement that the embedding map itself be computable is shown to be stronger than the other effectiveness criteria considered. A characterization in terms of computable local contractibility is stated the proof will be the subject of a sequel.

Strongly Normal Sets of Tiles in N Dimensions

The first and third authors and others [2,8,9,10,11,12] have studied sets of “tiles” (a generalization of pixels or voxels) in two and three dimensions that have a property called strong normality (SN): For any tile P, only finitely many tiles intersect P, and any nonempty intersection of these tiles must also intersect P. This paper presents extensions of the basic results about SN sets of tiles to n dimensions. One of our results is that if SN holds for every n + 1 or fewer tiles in a locally finite set of tiles in Rn, then the entire set of tiles is SN. Other results are that SN is equivalent to hereditary local contractibility, that simpleness of a tile in an SN set of tiles is equivalent to contractibility of its shared subset, and that deletion of a simple tile in an SN set of tiles preserves the homotopy type of the union of all the tiles.

On the structure of the group of Lipschitz homeomorphisms and its subgroups

We consider the group of Lipschitz homeomorphisms of a Lipschitz manifold and its subgroups. First we study properties of Lipschitz homeomorphisms and show the local contractibility and the perfectness of the group of Lipschitz homeomorphisms. Next using this result we can prove that the identity component of the group of equivariant Lipschitz homeomorphisms of a principal G-bundle over a closed Lipschitz manifold is perfect when G is a compact Lie group.

On the structure of the group of Lipschitz homeomorphisms and its subgroups

We consider the group of Lipschitz homeomorphisms of a Lipschitz manifold and its subgroups. First we study properties of Lipschitz homeomorphisms and show the local contractibility and the perfectness of the group of Lipschitz homeomorphisms. Next using this result we can prove that the identity component of the group of equivariant Lipschitz homeomorphisms of a principal G-bundle over a closed Lipschitz manifold is perfect when G is a compact Lie group.

Smooth p-adic analytic spaces are locally contractible

0 Introduction 293 1 Piecewise RS -linear spaces 298 2 R-colored polysimplicial sets 308 3 R-colored polysimplicial sets of length l 313 4 The skeleton of a nondegenerate pluri-stable formal scheme 327 5 A colored polysimplicial set associated with a nondegenerate poly-stable fibration 336 6 p-Adic analytic and piecewise linear spaces 346 7 Strong local contractibility of smooth analytic spaces 355 8 Cohomology with coefficients in the sheaf of constant functions 362

On Continuity Properties of the Modulus of Local Contractibility

LetMXbe the set of all metrics compatible with a given topology on a locally contractible spaceXand let for each triplez=(ρ,x,ε)∈MX×X×(0,∞), Δ(z) be the set of all positive δ such that the open δ-neighborhood ofxis contractible in the open ε-neighborhood ofxin metric ρ. We prove several continuity properties of the map Δ:MX×X×(0,∞)→(0,∞) and then, using a selection theorem for non-lower semicontinuous mappings, show that Δ admits a continuous singlevalued selection. Similar, but somewhat different properties are also demonstrated for the modulus Δnof localn-connectedness.

心臓腔内血流性状の定量的評価

For the purpose of non-invasive quantitative evaluation of the intracardiac blood flow dynamics, a new method for obtaining the stream lines and velocity vector distribution of the blood flow has been developed in our laboratory. In this study, characteristics of intracardiac flow were investigated in normals and several cases with myocardial infarction by using this method. Flow velocity data were obtained from ca 150 sampling points on the apical long axis cross section plane of the heart by the use of pulsed Doppler method of 3.5 MHZ ultrasound. The stream lines and velocity vector distribution were calculated from the 2-dimensional distribution data of the velocity by applying the stream function theory. From the 2-dimensional distribution of the stream line and velocity vectors, the flow speed and direction are clearly evaluated.And spreading or gathering flow, rotating flow are eddies in the ventricle occurred during one cardiac cycle are also estimated. From the results obtained from the pathological cases, it is considered that the flow state and dynamics reflect the changes in local myocardial contractibility and extensibility and that this new method will serve as a useful and reliable technique for evaluating the intraventricular flow dynamics and myocardial functions.

Local Contractibility, Cell-Like Maps, and Dimension

Local Contractibility, Cell-Like Maps, and Dimension