Polyhedral expansions of compacta associated to finite approximations

TL;DR

This paper develops polyhedral inverse sequences based on finite approximations to analyze the shape of compact metric spaces, demonstrating their effectiveness in computing persistent homology and error measurement.

Contribution

It introduces new inverse sequences of polyhedra as HPol-expansions for compacta, enabling explicit computation of persistent homology and error analysis.

Findings

Sequences are proven to be HPol-expansions, validating the General Principle.

Explicit calculations of inverse persistent homology groups are provided.

The method quantifies approximation errors in shape analysis.

Abstract

This paper introduces some inverse sequences of different polyhedra all based on finite approximations of a compact metric space so they can be used to capture the shape type of the original space. It is shown that they are HPol-expansions, proving the so-called General Principle. We use these sequences to compute explicitly some inverse persistent homology groups of a space and measure its errors in the approximation process.