Abstract
In this paper, we study the expectation values of topological invariants of the Vietoris–Rips complex and Čech complex for a finite set of sample points on a Riemannian manifold. We show that the Betti number and Euler characteristic of the complexes are Lipschitz functions of the scale parameter and that there is an interval such that the Betti curve converges to the Betti number of the underlying manifold.
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Acknowledgements
We are very grateful to the referee for his/her helpful suggestions and very swift work. Taejin Paik and Otto van Koert were supported by National Research Foundation of Korea (NRF) Grant 2022R1A5A6000840 funded by the Korean Government. Taejin Paik was also supported by National Science Foundation of Korea Grant funded by the Korean Government (MSIP) [RS-2022-00165404].
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Paik, T., van Koert, O. Expected invariants of simplicial complexes obtained from random point samples. Arch. Math. 120, 417–429 (2023). https://doi.org/10.1007/s00013-023-01826-5
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DOI: https://doi.org/10.1007/s00013-023-01826-5