Abstract
In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, Čech and witness complexes) built on top of totally bounded metric spaces. Using recent developments in the theory of topological persistence, we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov–Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and Čech complexes built on top of compact spaces.
Similar content being viewed by others
Notes
All vector spaces are taken to be over an arbitrary field \(\mathbf {k}\), fixed throughout this paper.
We use simplicial homology with coefficients in the field \(\mathbf {k}\).
We will usually drop the word ‘intrinsic’ unless we are contrasting it with ‘ambient’.
See [13] chap.1, def 1.2 for a definition of the length of a curve in a metric space.
References
Attali, D., Lieutier, A., Salinas, D.: Vietoris–Rips complexes also provide topologically correct reconstructions of sampled shapes. In: Proceedings of the 27th Annual ACM Symposium on Computational geometry, SoCG ’11, pp. 491–500. ACM, New York, NY, USA (2011). doi:10.1145/1998196.1998276
Bartholdi, L., Schick, T., Smale, N., Smale, S., Baker, A.W.: Hodge theory on metric spaces. Found. Comput. Math. 12(1), 1–48 (2012)
Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence, RI (2001)
Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L., Oudot, S.: Proximity of persistence modules and their diagrams. In: SCG, pp. 237–246 (2009). doi:10.1145/1542362.1542407
Chazal, F., Cohen-Steiner, D., Guibas, L.J., Mémoli, F., Oudot, S.Y.: Gromov–Hausdorff stable signatures for shapes using persistence. Computer Graphics Forum (Proceedings of the SGP 2009) pp. 1393–1403 (2009)
Chazal, F., Oudot, S.Y.: Towards persistence-based reconstruction in euclidean spaces. In: Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, SCG ’08, pp. 232–241. ACM, New York, NY, USA (2008). doi:10.1145/1377676.1377719
Chazal, F., de Silva, V., Glisse, M., Oudot, S.: The structure and stability of persistence modules (2012). ArXiv:1207.3674 [math.AT]
Dowker, C.H.: Homology groups of relations. Ann. Math. 56(1), 84–95 (1952)
Droz, J.M.: A subset of Euclidean space with large Vietoris–Rips homology (2012). ArXiv:1210.4097 [math.GT]
Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society, Providence, RI (2010)
Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discret. Comput. Geom. 28, 511–533 (2002)
Ghys, E., de la Harpe, P.: Sur les groupes hyperboliques d’après Mikhael Gromov, vol. 83. Birkhäuser, Basel (1990)
Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces, 2nd edn. Birkhäuser, Basel (2007)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge, MA (2001). http://www.math.cornell.edu/~hatcher/
Hausmann, J.C.: On the Vietoris–Rips complexes and a cohomology theory for metric spaces. Ann. Math. Stud. 138, 175–188 (1995)
Latschev, J.: Vietoris–Rips complexes of metric spaces near a closed Riemannian manifold. Archiv der Mathematik 77(6), 522–528 (2001)
Munkres, J.R.: Elements of Algebraic Topology. Westview Press, Boulder, CO (1984)
Zomorodian, A., Carlsson, G.: Computing persistent homology. Discret. Comput. Geom. 33(2), 249–274 (2005)
Acknowledgments
The authors are grateful to Steve Smale for fruitful discussions that motivated the results Sect. 5.2, and to J.-M. Droz for suggesting the idea of the proof of Proposition 5.9.
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors gratefully acknowledge the following funding sources for this work: Digiteo project C3TTA (including the Digiteo chair held by the second author); European project CG-Learning (EC contract No. 255827); ANR project GIGA (ANR-09-BLAN-0331-01); DARPA project Sensor Topology and Minimal Planning ‘SToMP’ (HR0011-07-1-0002). The second author is a 2013 Simons Fellow, and is supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.
Rights and permissions
About this article
Cite this article
Chazal, F., de Silva, V. & Oudot, S. Persistence stability for geometric complexes. Geom Dedicata 173, 193–214 (2014). https://doi.org/10.1007/s10711-013-9937-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-013-9937-z