scholarly journals Local-to-global Urysohn width estimates

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexey Balitskiy ◽  
Aleksandr Berdnikov
Keyword(s):  
Riemannian Manifold ◽  
Simplicial Complex ◽  
Metric Space ◽  
Low Dimensional ◽  
Dimensional Simplicial Complex

Abstract The notion of the Urysohn d-width measures to what extent a metric space can be approximated by a d-dimensional simplicial complex. We investigate how local Urysohn width bounds on a Riemannian manifold affect its global width. We bound the 1-width of a Riemannian manifold in terms of its first homology and the supremal width of its unit balls. Answering a question of Larry Guth, we give examples of n-manifolds of considerable ( n - 1 ) {(n-1)} -width in which all unit balls have arbitrarily small 1-width. We also give examples of topologically simple manifolds that are locally nearly low-dimensional.

scholarly journals Topological graph persistence

2020 ◽  
Vol 11 (1) ◽  
pp. 72-87
Author(s):  
Mattia G. Bergomi ◽  
Massimo Ferri ◽  
Lorenzo Zuffi
Keyword(s):  
Simplicial Complex ◽  
Independent Sets ◽  
Data Representation ◽  
Basic Tool ◽  
Topological Graph ◽  
Topological Information ◽  
One Dimensional ◽  
Low Dimensional ◽  
Dimensional Simplicial Complex

Abstract Graphs are a basic tool in modern data representation. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological constructions can be used to gain information otherwise concealed by the low-dimensional nature of graphs. We do this by extending previous work in homological persistence, and proposing novel graph-theoretical constructions. Beyond cliques, we use independent sets, neighborhoods, enclaveless sets and a Ramsey-inspired extended persistence.

scholarly journals Golodness and polyhedral products for two-dimensional simplicial complexes

2018 ◽  
Vol 30 (2) ◽  
pp. 527-532
Author(s):  
Kouyemon Iriye ◽  
Daisuke Kishimoto
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Two Dimensional ◽  
Dimensional Simplicial Complex

AbstractGolodness of two-dimensional simplicial complexes is studied through polyhedral products, and combinatorial and topological characterizations of Golodness of surface triangulations are given. An answer to the question of Berglund is also given so that there is a two-dimensional simplicial complex which is rationally Golod but not Golod over{\mathbb{Z}/p}.

Federer-Čech Couples

1969 ◽  
Vol 21 ◽  
pp. 842-864
Author(s):  
Micheal Dyer
Keyword(s):  
Simplicial Complex ◽  
Large Class ◽  
Metric Space ◽  
Topological Spaces ◽  
Compact Open Topology ◽  
Finite Dimensional ◽  
Cw Complexes

In (5),I considered two-term conditions in π-exact couples, of which the exact couple of Federer (7) is an example. Let M(X, Y)be the space of all maps from X to Y with the compact-open topology. Our aim in this paper is to construct a π-exact couple , where Xis a finite-dimensional (in the sense of Lebesgue) metric space and , a certain (rather large) class of spaces. Specifically, is the class of all topological spaces Xwhich possess the following property (P).(P) Let Y be a (possibly infinite) simplicial complex. There exists x0 ∈ X and y0 ∊ Y such that [X, x0]≃ [Y, y0].In § 5 it will be seen that contains all CW complexes and all metric absolute neighbourhood retracts (ANR)s.

k-Decomposable Monomial Ideals

2015 ◽  
Vol 22 (spec01) ◽  
pp. 745-756 ◽  
Author(s):  
Rahim Rahmati-Asghar ◽  
Siamak Yassemi
Keyword(s):  
Simplicial Complex ◽  
Monomial Ideals ◽  
Alexander Dual ◽  
Linear Quotients ◽  
Dimensional Simplicial Complex

In this paper we introduce a class of monomial ideals, called k-decomposable ideals. It is shown that the class of k-decomposable ideals is contained in the class of monomial ideals with linear quotients, and when k is large enough, the class of k-decomposable ideals is equal to the class of ideals with linear quotients. In addition, it is shown that a d-dimensional simplicial complex is k-decomposable if and only if the Stanley-Reisner ideal of its Alexander dual is a k-decomposable ideal, where k ≤ d. Moreover, it is shown that every k-decomposable ideal is componentwise k-decomposable.

A coarse Mayer–Vietoris principle

1993 ◽  
Vol 114 (1) ◽  
pp. 85-97 ◽  
Author(s):  
Nigel Higson ◽  
John Roe ◽  
Guoliang Yu
Keyword(s):  
Riemannian Manifold ◽  
Metric Space ◽  
Metric Spaces ◽  
Compact Operators ◽  
Cohomology Theory ◽  
Coarse Geometry ◽  
Locally Compact ◽  
Finite Propagation ◽  
Noncompact Riemannian Manifold

In [1], [4], and [6] the authors have studied index problems associated with the ‘coarse geometry’ of a metric space, which typically might be a complete noncompact Riemannian manifold or a group equipped with a word metric. The second author has introduced a cohomology theory, coarse cohomology, which is functorial on the category of metric spaces and coarse maps, and which can be computed in many examples. Associated to such a metric space there is also a C*-algebra generated by locally compact operators with finite propagation. In this note we will show that for suitable decompositions of a metric space there are Mayer–Vietoris sequences both in coarse cohomology and in the K-theory of the C*-algebra. As an application we shall calculate the K-theory of the C*-algebra associated to a metric cone. The result is consistent with the calculation of the coarse cohomology of the cone, and with a ‘coarse’ version of the Baum–Connes conjecture.

scholarly journals Homeomorphisms without the pseudo-orbit tracing property

1982 ◽  
Vol 88 ◽  
pp. 155-160 ◽  
Author(s):  
Nobuo Aoki
Keyword(s):  
Riemannian Manifold ◽  
Metric Space ◽  
Compact Riemannian Manifold ◽  
Compact Metric Space ◽  
Positive Dimension

Recently, A. Morimoto [5] proved that every isometry of a compact Riemannian manifold of positive dimension has not the pseudo-orbit tracing property, and that if a homeomorphism of a compact metric space has the pseudo-orbit tracing property then Eφ— 0φ(see § 1 for definition). The purpose of this paper is to show that every distal homeomorphism of a compact connected metric space has not the pseudo-orbit tracing property.

Towards optimal path computation in a simplicial complex

2019 ◽  
Vol 38 (8) ◽  
pp. 981-1009
Author(s):  
Subhrajit Bhattacharya
Keyword(s):  
Simplicial Complex ◽  
Metric Space ◽  
Search Algorithm ◽  
Optimal Path ◽  
Shortest Paths ◽  
Reconstruction Algorithm ◽  
Continuous Space ◽  
Rips Complex ◽  
Geodesic Paths ◽  
Configuration Manifold

Computing optimal path in a configuration space is fundamental to solving motion planning problems in robotics and autonomy. Graph-based search algorithms have been widely used to that end, but they suffer from drawbacks. We present an algorithm for computing the shortest path through a metric simplicial complex that can be used to construct a piece-wise linear discrete model of the configuration manifold. In particular, given an undirected metric graph, G, which is constructed as a discrete representation of an underlying configuration manifold (a larger “continuous” space typically of dimension greater than one), we consider the Rips complex, [Formula: see text], associated with it. Such a complex, and hence shortest paths in it, represent the underlying metric space more closely than what the graph does. Our algorithm requires only a local connectivity-based description of an abstract graph, [Formula: see text], and a cost/length function, [Formula: see text], as inputs. No global information such as an embedding or a global coordinate chart is required. The local nature of the proposed algorithm makes it suitable for configuration spaces of arbitrary topology, geometry, and dimension. We not only develop the search algorithm for computing shortest distances, but we also present a path reconstruction algorithm for explicitly computing the shortest paths through the simplicial complex. The complexity of the presented algorithm is comparable with that of Dijkstra’s search, but, as the results presented in this paper demonstrate, the shortest paths obtained using the proposed algorithm represent the geodesic paths in the original metric space significantly more closely.

scholarly journals Stratifications on the Ran Space

Order ◽  
2021 ◽  
Author(s):  
Jānis Lazovskis
Keyword(s):  
Simplicial Complex ◽  
Partial Order ◽  
Metric Space ◽  
Product Space ◽  
Simplicial Complexes ◽  
Real Numbers

AbstractWe describe a partial order on finite simplicial complexes. This partial order provides a poset stratification of the product of the Ran space of a metric space and the nonnegative real numbers, through the Čech simplicial complex. We show that paths in this product space respecting its stratification induce simplicial maps between the endpoints of the path.

scholarly journals Metric completions, the Heine-Borel property, and approachability

2020 ◽  
Vol 18 (1) ◽  
pp. 162-166
Author(s):  
Vladimir Kanovei ◽  
Mikhail G. Katz ◽  
Tahl Nowik
Keyword(s):  
Riemannian Manifold ◽  
Metric Space ◽  
Metric Spaces ◽  
Short Proof ◽  
Universal Cover

Abstract We show that the metric universal cover of a plane with a puncture yields an example of a nonstandard hull properly containing the metric completion of a metric space. As mentioned by Do Carmo, a nonextendible Riemannian manifold can be noncomplete, but in the broader category of metric spaces it becomes extendible. We give a short proof of a characterisation of the Heine-Borel property of the metric completion of a metric space M in terms of the absence of inapproachable finite points in ∗M.

Simplicial Complexes and Open Subsets of Non-Separable LF-Spaces

2011 ◽  
Vol 63 (2) ◽  
pp. 436-459 ◽  
Author(s):  
Kotaro Mine ◽  
Katsuro Sakai
Keyword(s):  
Simplicial Complex ◽  
Open Subset ◽  
Simplicial Complexes ◽  
Locally Finite ◽  
Direct Sum ◽  
Open Set ◽  
Finite Dimensional ◽  
Metric Topology ◽  
Dimensional Simplicial Complex

Abstract Let F be a non-separable LF-space homeomorphic to the direct sum , where . It is proved that every open subset U of F is homeomorphic to the product |K| × F for some locally finite-dimensional simplicial complex K such that every vertex v ∈ K(0) has the star St(v, K) with card St(v, K)(0) < 𝒯 = sup 𝒯n (and card K(0) ≤ 𝒯 ), and, conversely, if K is such a simplicial complex, then the product |K| × F can be embedded in F as an open set, where |K| is the polyhedron of K with the metric topology.

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