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  1. Home
  2. Maximum Betti Numbers Of Čech Complexes.
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  2. Maximum Betti Numbers Of Čech Complexes.

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Maximum Betti Numbers of Čech Complexes.

Herbert Edelsbrunner1, János Pach2

  • 1ISTA (Institute of Science and Technology Austria), Klosterneuburg, Austria.

Discrete & Computational Geometry
|March 5, 2026

View abstract on PubMed

Summary
This summary is machine-generated.

This study demonstrates that the Upper Bound Theorem for convex polytopes is asymptotically tight for Čech complexes. Researchers constructed point sets in various dimensions that achieve this bound, validating theoretical limits in computational topology.

Keywords:
Alpha complexesBetti numbersComputational topologyDelaunay mosaicsDiscrete geometryExtremal questionsČech complexes

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Area of Science:

  • Computational Topology
  • Geometric Complexity Theory

Background:

  • The Upper Bound Theorem provides a theoretical limit on the Betti numbers of Čech complexes based on the number of points and dimension.
  • Existing bounds are often not tight, limiting practical applications in data analysis and topological inference.

Purpose of the Study:

  • To investigate the asymptotic tightness of the Upper Bound Theorem for Čech complexes.
  • To construct explicit point set configurations that achieve the theoretical upper bound on Betti numbers.

Main Methods:

  • Construction of specific point sets in Euclidean spaces (R^d) of varying dimensions.
  • Analysis of the Čech complex's Betti numbers for these constructed sets at different radii.
  • Comparison of computed Betti numbers with the theoretical upper bound.

Main Results:

  • Demonstrated that the Upper Bound Theorem for Betti numbers of Čech complexes is asymptotically tight.
  • Provided explicit constructions of point sets in even and odd dimensions that achieve this bound.
  • Example: In R^3, a set of 2(n+1) points yielded a first Betti number of (n+1)^2 - 1 and a second Betti number of n^2.

Conclusions:

  • The theoretical upper bound on Betti numbers for Čech complexes is achievable.
  • These findings have implications for understanding the topological complexity of point cloud data.
  • The constructed examples serve as benchmarks for algorithms in topological data analysis.