Algebraic & Geometric Topology Volume 24, issue 2 (2024)

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

Author index

To appear

Other MSP journals

Vietoris–Rips persistent homology, injective metric spaces, and the filling radius

Sunhyuk Lim, Facundo Mémoli and Osman Berat Okutan

Algebraic & Geometric Topology 24 (2024) 1019–1100

DOI: 10.2140/agt.2024.24.1019

Bibliography

1	M Adamaszek, H Adams, The Vietoris–Rips complexes of a circle, Pacific J. Math. 290 (2017) 1 MR3673078
2	M Adamaszek, H Adams, F Frick, Metric reconstruction via optimal transport, SIAM J. Appl. Algebra Geom. 2 (2018) 597 MR3871057
3	M Adamaszek, H Adams, E Gasparovic, M Gommel, E Purvine, R Sazdanovic, B Wang, Y Wang, L Ziegelmeier, Vietoris–Rips and Čech complexes of metric gluings, from: "34th international symposium on computational geometry" (editors B Speckmann, C D Tóth), Leibniz Int. Proc. Inform. 99, Schloss Dagstuhl. Leibniz-Zent. Inform. (2018) MR3824247
4	M Adamaszek, H Adams, S Reddy, On Vietoris–Rips complexes of ellipses, J. Topol. Anal. 11 (2019) 661 MR3999516
5	H Adams, J Bush, J Mirth, Operations on metric thickenings, from: "Proceedings of the 3rd annual international applied category theory conference 2020" (editors D I Spivak, J Vicary), Electron. Proc. Theor. Comput. Sci. 333, EPTCS (2021) 261 MR4259620
6	H Adams, B Coskunuzer, Geometric approaches to persistent homology, SIAM J. Appl. Algebra Geom. 6 (2022) 685 MR4522864
7	H Adams, F Mémoli, M Moy, Q Wang, The persistent topology of optimal transport based metric thickenings, preprint (2021) arXiv:2109.15061
8	N Aronszajn, P Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956) 405 MR0084762
9	S Awodey, Category theory, 52, Oxford Univ. Press (2010) MR2668552
10	G Azumaya, Corrections and supplementaries to my paper concerning Krull–Remak–Schmidt’s theorem, Nagoya Math. J. 1 (1950) 117 MR0037832
11	U Bauer, Ripser, C++ code (2015)
12	U Bauer, M Lesnick, Induced matchings of barcodes and the algebraic stability of persistence, from: "Computational geometry" (editors S W Cheng, O Devillers), ACM (2014) 355 MR3382316
13	A Blumberg, M Lesnick, Universality of the homotopy interleaving distance, Trans. Amer. Math. Soc. 376 (2023) 8269 MR4669297
14	M Bonk, O Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000) 266 MR1771428
15	D Burago, Y Burago, S Ivanov, A course in metric geometry, 33, Amer. Math. Soc. (2001) MR1835418
16	G Carlsson, Topology and data, Bull. Amer. Math. Soc. 46 (2009) 255 MR2476414
17	G Carlsson, V de Silva, Zigzag persistence, Found. Comput. Math. 10 (2010) 367 MR2657946
18	G Chaparro Sumalave, Vietoris–Rips complexes of the circle and the torus, master’s thesis, Universidad de los Andes (2016)
19	F Chazal, D Cohen-Steiner, L J Guibas, F Mémoli, S Y Oudot, Gromov–Hausdorff stable signatures for shapes using persistence, Computer Graphics Forum 28 (2009) 1393
20	F Chazal, W Crawley-Boevey, V de Silva, The observable structure of persistence modules, Homology Homotopy Appl. 18 (2016) 247 MR3575998
21	F Chazal, S Y Oudot, Towards persistence-based reconstruction in Euclidean spaces, from: "Computational geometry" (editor M Teillaud), ACM (2008) 232 MR2504289
22	F Chazal, V de Silva, M Glisse, S Oudot, The structure and stability of persistence modules, Springer (2016) MR3524869
23	F Chazal, V de Silva, S Oudot, Persistence stability for geometric complexes, Geom. Dedicata 173 (2014) 193 MR3275299
24	S Chowdhury, F Mémoli, A functorial Dowker theorem and persistent homology of asymmetric networks, J. Appl. Comput. Topol. 2 (2018) 115 MR3873182
25	W Crawley-Boevey, Decomposition of pointwise finite-dimensional persistence modules, J. Algebra Appl. 14 (2015) MR3323327
26	J Curry, The fiber of the persistence map for functions on the interval, J. Appl. Comput. Topol. 2 (2018) 301 MR3927355
27	C J A Delfinado, H Edelsbrunner, An incremental algorithm for Betti numbers of simplicial complexes on the 3–sphere, Comput. Aided Geom. Design 12 (1995) 771 MR1365107
28	T K Dey, F Mémoli, Y Wang, Topological analysis of nerves, Reeb spaces, mappers, and multiscale mappers, from: "33rd international symposium on computational geometry" (editors B Aronov, M J Katz), Leibniz Int. Proc. Inform. 77, Schloss Dagstuhl. Leibniz-Zent. Inform. (2017) 36 MR3685708
29	T tom Dieck, Partitions of unity in homotopy theory, Compositio Math. 23 (1971) 159 MR0293625
30	A W M Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. in Math. 53 (1984) 321 MR0753872
31	A Dress, K T Huber, J Koolen, V Moulton, A Spillner, Basic phylogenetic combinatorics, Cambridge Univ. Press (2012) MR2893879
32	L E Dubins, G Schwarz, Equidiscontinuity of Borsuk–Ulam functions, Pacific J. Math. 95 (1981) 51 MR0631658
33	J Dugundji, Absolute neighborhood retracts and local connectedness in arbitrary metric spaces, Compositio Math. 13 (1958) 229 MR0113217
34	H Edelsbrunner, J Harer, Persistent homology: a survey, from: "Surveys on discrete and computational geometry" (editors J E Goodman, J Pach, R Pollack), Contemp. Math. 453, Amer. Math. Soc. (2008) 257 MR2405684
35	H Edelsbrunner, J L Harer, Computational topology: an introduction, Amer. Math. Soc. (2010) MR2572029
36	H Edelsbrunner, D Letscher, A Zomorodian, Topological persistence and simplification, Discrete Comput. Geom. 28 (2002) 511 MR1949898
37	H Federer, Geometric measure theory, 153, Springer (1969) MR0257325
38	A Fomenko, D Fuchs, Homotopical topology, 273, Springer (2016) MR3497000
39	P Frosini, A distance for similarity classes of submanifolds of a Euclidean space, Bull. Austral. Math. Soc. 42 (1990) 407 MR1083277
40	P Frosini, Measuring shapes by size functions, from: "Intelligent robots and computer vision, X: Algorithms and techniques" (editor D P Casasent), SPIE Proceedings 1607, SPIE (1992) 122
41	P Frosini, C Landi, Size functions and morphological transformations, Acta Appl. Math. 49 (1997) 85 MR1482881
42	H Gakhar, J A Perea, Künneth formulae in persistent homology, preprint (2019) arXiv:1910.05656
43	M Gameiro, Y Hiraoka, I Obayashi, Continuation of point clouds via persistence diagrams, Phys. D 334 (2016) 118 MR3545973
44	E Gasparovic, M Gommel, E Purvine, R Sazdanovic, B Wang, Y Wang, L Ziegelmeier, A complete characterization of the one-dimensional intrinsic Čech persistence diagrams for metric graphs, from: "Research in computational topology" (editors E W Chambers, B T Fasy, L Ziegelmeier), Assoc. Women Math. Ser. 13, Springer (2018) 33 MR3905000
45	R Ghrist, A Muhammad, Coverage and hole-detection in sensor networks via homology, from: "Fourth international symposium on information processing in sensor networks" (editor A Savvides), IEEE (2005) 254
46	M Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983) 1 MR0697984
47	M Gromov, Hyperbolic groups, from: "Essays in group theory" (editor S M Gersten), Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 MR0919829
48	M Gromov, Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser (2007) MR2307192
49	A Hatcher, Algebraic topology, Cambridge Univ. Press (2002) MR1867354
50	J C Hausmann, On the Vietoris–Rips complexes and a cohomology theory for metric spaces, from: "Prospects in topology" (editor F Quinn), Ann. of Math. Stud. 138, Princeton Univ. Press (1995) 175 MR1368659
51	S T Hu, Theory of retracts, Wayne State Univ. Press (1965) 234 MR0181977
52	J R Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964) 65 MR0182949
53	P Joharinad, J Jost, Topological representation of the geometry of metric spaces, preprint (2020) arXiv:2001.10262
54	M Katz, The filling radius of two-point homogeneous spaces, J. Differential Geom. 18 (1983) 505 MR0723814
55	M Katz, Diameter-extremal subsets of spheres, Discrete Comput. Geom. 4 (1989) 117 MR0973541
56	M Katz, On neighborhoods of the Kuratowski imbedding beyond the first extremum of the diameter functional, Fund. Math. 137 (1991) 161 MR1110030
57	M Katz, The rational filling radius of complex projective space, Topology Appl. 42 (1991) 201 MR1137947
58	M G Katz, Systolic geometry and topology, 137, Amer. Math. Soc. (2007) MR2292367
59	M Kılıç, Ş Koçak, Tight span of subsets of the plane with the maximum metric, Adv. Math. 301 (2016) 693 MR3539386
60	U Lang, Injective hulls of certain discrete metric spaces and groups, J. Topol. Anal. 5 (2013) 297 MR3096307
61	M Lesnick, The theory of the interleaving distance on multidimensional persistence modules, Found. Comput. Math. 15 (2015) 613 MR3348168
62	S Lim, F Mémoli, O B Okutan, Vietoris–Rips persistent homology, injective metric spaces, and the filling radius, preprint (2020) arXiv:2001.07588
63	S Lim, F Mémoli, Z Smith, The Gromov–Hausdorff distance between spheres, Geom. Topol. 27 (2023) 3733 MR4674839
64	L Liu, The mapping properties of filling radius and packing radius and their applications, Differential Geom. Appl. 22 (2005) 69 MR2106377
65	F Mémoli, A distance between filtered spaces via tripods, preprint (2017) arXiv:1704.03965
66	F Mémoli, O B Okutan, Quantitative simplification of filtered simplicial complexes, Discrete Comput. Geom. 65 (2021) 554 MR4212978
67	F Mémoli, O B Okutan, Q Wang, Metric graph approximations of geodesic spaces, preprint (2018) arXiv:1809.05566
68	J R Munkres, Elements of algebraic topology, Addison-Wesley (1984) MR0755006
69	A Nabutovsky, Linear bounds for constants in Gromov’s systolic inequality and related results, Geom. Topol. 26 (2022) 3123 MR4540902
70	P Niyogi, S Smale, S Weinberger, Finding the homology of submanifolds with high confidence from random samples, Discrete Comput. Geom. 39 (2008) 419 MR2383768
71	J A Perea, Persistent homology of toroidal sliding window embeddings, from: "2016 IEEE international conference on acoustics, speech and signal processing" (editors M Dong, T F Zheng), IEEE (2016) 6435
72	J A Perea, Künneth formuale in persistent homology, (2018)
73	V Robins, Towards computing homology from finite approximations, Topology Proc. 24 (1999) 503 MR1876386
74	M Schmahl, Structure of semi-continuous q–tame persistence modules, Homology Homotopy Appl. 24 (2022) 117 MR4404959
75	C Semple, M Steel, Phylogenetics, 24, Oxford Univ. Press (2003) MR2060009
76	V de Silva, G Carlsson, Topological estimation using witness complexes, from: "Symposium on point-based graphics 2004" (editors M Gross, H Pfister, M Alexa, S Rusinkiewicz), The Eurographics Association (2004) 157
77	V de Silva, R Ghrist, Coverage in sensor networks via persistent homology, Algebr. Geom. Topol. 7 (2007) 339 MR2308949
78	E H Spanier, Algebraic topology, Springer (1981) MR0666554
79	A Verri, C Uras, P Frosini, M Ferri, On the use of size functions for shape analysis, Biol. Cybern. 70 (1993) 99
80	L Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann. 97 (1927) 454 MR1512371
81	Ž Virk, 1–dimensional intrinsic persistence of geodesic spaces, J. Topol. Anal. 12 (2020) 169 MR4080099
82	Ž Virk, Rips complexes as nerves and a functorial Dowker-nerve diagram, Mediterr. J. Math. 18 (2021) MR4218370
83	F H Wilhelm Jr., On the filling radius of positively curved manifolds, Invent. Math. 107 (1992) 653 MR1150606
84	T Yokota, On the filling radius of positively curved Alexandrov spaces, Math. Z. 273 (2013) 161 MR3010155