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Random Čech complexes on Riemannian manifolds
Author(s) -
Bobrowski Omer,
Oliveira Goncalo
Publication year - 2019
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20800
Subject(s) - mathematics , riemannian manifold , torus , pure mathematics , euclidean geometry , riemannian geometry , manifold (fluid mechanics) , generalization , topology (electrical circuits) , combinatorics , mathematical analysis , geometry , mechanical engineering , engineering
In this paper we study the homology of a random Čech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M . In particular, we focus on the phase transition for “homological connectivity” where the homology of the complex becomes isomorphic to that of M . The results presented in this paper are an important generalization of [7][O. Bobrowski, 2017], from the flat torus to general compact Riemannian manifolds. In addition to proving the statements related to homological connectivity, the methods we develop in this paper can be used as a framework for translating results for random geometric graphs and complexes from the Euclidean setting into the more general Riemannian one.

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