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References (60)
-
Daniel Dugger, Daniel Isaksen (2004)
Topological hypercovers and 1-realizationsMathematische Zeitschrift, 246
-
Henry Adams, Johnathan Bush, F. Frick (2021)
The Topology of Projective Codes and the Distribution of Zeros of Odd MapsMichigan Mathematical Journal
-
Žiga Virk (2017)
1-Dimensional intrinsic persistence of geodesic spacesJournal of Topology and Analysis
-
F. Chazal, V. Silva, S. Oudot (2012)
Persistence stability for geometric complexesGeometriae Dedicata, 173
-
Nguyễn Như, Ta Cu (1989)
Probability measure functors preserving the ANR-property of metric spaces, 106
-
V. Bogachev (2018)
Weak convergence of measures on ℝ^{𝕕}Weak Convergence of Measures
-
(2018)
On spectral sequences arising from topological covers. Unpublished notes -
M Adamaszek, H Adams (2017)
The Vietoris?Rips complexes of a circlePac. J. Math., 290
-
Ž. Virk (2021)
A Counter-Example to Hausmann’s ConjectureFoundations of Computational Mathematics, 22
-
A. Nag'orko (2005)
Carrier and nerve theorems in the extension theory, 135
-
Matthew Zaremsky (2018)
Bestvina–Brady discrete Morse theory and Vietoris–Rips complexesAmerican Journal of Mathematics, 144
-
Günter Rote, Gert Vegter (2007)
Effective Computational Geometry for Curves and Surfaces Chapter 7 Computational Topology : An Introduction -
H. Edelsbrunner, D. Letscher, A. Zomorodian (2000)
Topological Persistence and SimplificationDiscrete & Computational Geometry, 28
-
C. Villani (2008)
Optimal Transport: Old and New -
P. Gillespie (2022)
A homological nerve theorem for open covers -
M. Katz (1989)
Diameter-extremal subsets of spheresDiscrete & Computational Geometry, 4
-
Henry Adams, Facundo M'emoli, Michael Moy, Qingsong Wang (2021)
The Persistent Topology of Optimal Transport Based Metric Thickenings -
Michal Adamaszek, Henry Adams, F. Frick, C. Peterson, Corrine Previte-Johnson (2014)
Nerve Complexes of Circular ArcsDiscrete & Computational Geometry, 56
-
Sunhyuk Lim, F. Mémoli, Osman Okutan (2020)
Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling RadiusArXiv, abs/2001.07588
-
A. Zomorodian, G. Carlsson (2004)
Computing Persistent HomologyDiscrete & Computational Geometry, 33
-
J. Latschev (2001)
Vietoris-Rips complexes of metric spaces near a closed Riemannian manifoldArchiv der Mathematik, 77
-
C. Villani (2003)
Topics in Optimal Transportation -
A. Björner (2003)
Nerves, fibers and homotopy groupsJ. Comb. Theory, Ser. A, 102
-
André Well (1952)
Sur les théorèmes de de RhamCommentarii Mathematici Helvetici, 26
-
M Adamaszek (2017)
1Pac. J. Math., 290
-
(2022)
Persistence stability for metric thickenings. Master’s thesis, Colorado State University -
E. Zeeman (1962)
Dihomology: I. Relations Between Homology TheoriesProceedings of The London Mathematical Society
-
K. Borsuk (1948)
On the imbedding of systems of compacta in simplicial complexesFundamenta Mathematicae, 35
-
(2016)
Micha(cid:32)l -
M Adamaszek (2016)
251Discrete Comput. Geom., 56
-
Henry Adams, Joshua Mirth (2017)
Metric thickenings of Euclidean submanifoldsTopology and its Applications
-
Email address : frick@cmu.edu (ˇZV) Faculty -
Michal Adamaszek (2011)
Clique complexes and graph powersIsrael Journal of Mathematics, 196
-
L. Evens (1991)
Cohomology of groups -
Michal Adamaszek, Henry Adams, F. Frick (2017)
Metric Reconstruction Via Optimal TransportSIAM J. Appl. Algebra Geom., 2
-
Marc Alexa, Michael Kazhdan, F. Chazal, D. Cohen-Steiner, L. Guibas, F. Mémoli, S. Oudot
Eurographics Symposium on Geometry Processing 2009 Gromov-hausdorff Stable Signatures for Shapes Using Persistence -
Stefan Friedl (2020)
Algebraic topologyGraduate Studies in Mathematics
-
P. Niyogi, S. Smale, S. Weinberger (2008)
Finding the Homology of Submanifolds with High Confidence from Random SamplesDiscrete & Computational Geometry, 39
-
D. Cohen-Steiner, H. Edelsbrunner, J. Harer (2005)
Stability of Persistence DiagramsDiscrete & Computational Geometry, 37
-
Michal Adamaszek, Henry Adams (2015)
The Vietoris-Rips complexes of a circleArXiv, abs/1503.03669
-
Matthew Wright (2016)
Introduction to Persistent Homology -
Karol Borsuk (1932)
Über eine Klasse von lokal zusammenhängenden RäumenFundamenta Mathematicae, 19
-
R. Fritsch, R. Piccinini (1990)
Cellular structures in topology -
Michael Moy (2022)
Vietoris–Rips metric thickenings of the circleJournal of Applied and Computational Topology
-
G. Carlsson (2009)
Topology and dataBulletin of the American Mathematical Society, 46
-
Henry Adams, Baris Coskunuzer (2021)
Geometric Approaches on Persistent HomologySIAM J. Appl. Algebra Geom., 6
-
Henry Adams, Johnathan Bush, F. Frick (2019)
METRIC THICKENINGS, BORSUK–ULAM THEOREMS, AND ORBITOPESMathematika
-
Žiga Virk (2019)
Rips Complexes as Nerves and a Functorial Dowker-Nerve DiagramMediterranean Journal of Mathematics, 18
-
Žiga Virk (2021)
Footprints of Geodesics in Persistent HomologyMediterranean Journal of Mathematics, 19
-
R. Bott, L. Tu (1982)
Differential forms in algebraic topology, 82
-
Žiga Virk (2017)
Approximations of 1-dimensional intrinsic persistence of geodesic spaces and their stabilityRevista Matemática Complutense, 32
-
J. Hausmann (1996)
On the Vietoris-Rips complexes and a Cohomology Theory for metric spaces -
M Adamaszek (2018)
597SIAM J. Appl. Algebra Geom., 2
-
M. Katz (1991)
On neighborhoods of the Kuratowski imbedding beyond the first extremum of the diameter functionalFundamenta Mathematicae, 137
-
M Adamaszek (2013)
295Isr. J. Math., 196
-
C. Dowker (1952)
TOPOLOGY OF METRIC COMPLEXES.American Journal of Mathematics, 74
-
Compositio Mathematica, T. Dieck (2019)
Partitions of unity in homotopy theory -
A. Vershik (2013)
Long History of the Monge-Kantorovich Transportation ProblemThe Mathematical Intelligencer, 35
-
Ellen Gasparovic, Maria Gommel, Emilie Purvine, R. Sazdanovic, Bei Wang, Yusu Wang, Lori Ziegelmeier (2017)
A Complete Characterization of the One-Dimensional Intrinsic Čech Persistence Diagrams for Metric GraphsarXiv: Algebraic Topology
-
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- 10.1007/s41468-022-00106-5
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Abstract
We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let U\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {U}}$$\end{document} be a cover of a separable metric space X by open sets with a uniform diameter bound. The Vietoris complex V(U)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {V}}({\mathcal {U}})$$\end{document} contains all simplices with vertex set contained in some U∈U\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$U \in {\mathcal {U}}$$\end{document}, and the Vietoris metric thickening Vm(U)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {V}}^\textrm{m}({\mathcal {U}})$$\end{document} is the space of probability measures with support in some U∈U\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$U \in {\mathcal {U}}$$\end{document}, equipped with an optimal transport metric. We show that Vm(U)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {V}}^\textrm{m}({\mathcal {U}})$$\end{document} and V(U)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {V}}({\mathcal {U}})$$\end{document} have isomorphic homotopy groups in all dimensions. In particular, by choosing the cover U\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {U}}$$\end{document} appropriately, we get isomorphisms between the homotopy groups of Vietoris–Rips metric thickenings and simplicial complexes πk(VRm(X;r))≅πk(VR(X;r))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\pi _k(\textrm{VR}^\textrm{m}(X;r))\cong \pi _k(\textrm{VR}(X;r))$$\end{document} for all integers k≥0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$k\ge 0$$\end{document}, where both spaces are defined using the convention “diameter <r\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$< r$$\end{document}” (instead of ≤r\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\le r$$\end{document}). Similarly, we get isomorphisms between the homotopy groups of Čech metric thickenings and simplicial complexes πk(Cˇm(X;r))≅πk(Cˇ(X;r))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\pi _k(\check{\mathrm {{C}}}^\textrm{m}(X;r))\cong \pi _k(\check{\mathrm {{C}}}(X;r))$$\end{document} for all integers k≥0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$k\ge 0$$\end{document}, where both spaces are defined using open balls (instead of closed balls).
Journal
Journal of Applied and Computational Topology – Springer Journals
Published: Jun 1, 2023
Keywords: Vietoris–Rips complexes; Čech complexes; Metric thickenings; Optimal transport; Nerve lemmas; Homotopy groups; 55N31 (Primary); 54E35; 55P10; 55U10 (Secondary)