Maximum Betti Numbers of Čech Complexes.

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.