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Maximum Betti Numbers of Čech Complexes

Authors Herbert Edelsbrunner , János Pach

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Discrete geometry
  • computational topology
  • Čech complexes
  • Delaunay mosaics
  • Alpha complexes
  • Betti numbers
  • extremal questions

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Abstract

The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². In particular, there is an arrangement of n contruent balls in ℝ³ that enclose a quadratic number of voids, which answers a long-standing open question in computational geometry.

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Herbert Edelsbrunner and János Pach. Maximum Betti Numbers of Čech Complexes. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.53

Author Details

Herbert Edelsbrunner
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
János Pach
  • Rényi Institute of Mathematics, Budapest, Hungary
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria

Funding

The first author is supported by the European Research Council (ERC), grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. {I 02979-N35.} The second author is supported by the European Research Council (ERC), grant "GeoScape" and by the Hungarian Science Foundation (NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.

Acknowledgements

The authors thank Matt Kahle for communicating the question about extremal Čech complexes, Ben Schweinhart for early discussions on the linked circles construction in three dimensions, and Gábor Tardos for helpful remarks and suggestions.

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