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

Lim, Sunhyuk; Mémoli, Facundo; Okutan, Osman Berat

doi:10.2140/agt.2024.24.1019

Algebr. Geom. Topol.

2024

DOI: 10.2140/agt.2024.24.1019

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

Sunhyuk Lim,

Facundo Mémoli,

Osman Berat Okutan

Abstract: Where for metric spaces X and Y morphisms are given by 1-Lipschitz maps W X ! Y , and for persistence modules V and W morphisms are systems of linear maps D . r W V r ! W r / r>0 making all squares commute.

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Gromov–Hausdorff distances from simply connected geodesic spaces to the circle

Rodríguez Martín

2024

Proc. Amer. Math. Soc. Ser. B

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We prove that the Gromov–Hausdorff distance from the circle with its geodesic metric to any simply connected geodesic space is never smaller than π 4 \frac {\pi }{4} . We also prove that this bound is tight through the construction of a simply connected geodesic space E \mathrm {E} which attains the lower bound π 4 \frac {\pi }{4} . We deduce the first statement from a general result that we also establish which gives conditions on how small the Gromov–Hausdorff distance between two geodesic metric spaces ( X , d X ) (X, d_X) and ( Y , d Y ) (Y, d_Y ) has to be in order for π 1 ( X ) \pi _1(X) and π 1 ( Y ) \pi _1(Y) to be isomorphic.

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Gromov–Hausdorff distances from simply connected geodesic spaces to the circle

Rodríguez Martín

2024

Proc. Amer. Math. Soc. Ser. B

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

Abstract: Where for metric spaces X and Y morphisms are given by 1-Lipschitz maps W X ! Y , and for persistence modules V and W morphisms are systems of linear maps D . r W V r ! W r / r>0 making all squares commute.

Cited by 1 publication

References 75 publications

Gromov–Hausdorff distances from simply connected geodesic spaces to the circle

Gromov–Hausdorff distances from simply connected geodesic spaces to the circle

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