Persistent Local Systems of Periodic Spaces

TL;DR

This paper develops a framework using bisheaves and persistent local systems to recover and classify the homology of periodic spaces from finite quotient samples, enabling practical analysis of crystalline and cosmological data.

Contribution

It introduces a computationally feasible persistence theory for periodic spaces and algorithms to identify toroidal cycles from finite quotient data.

Findings

Complete classification of toroidal and non-toroidal cycles in periodic complexes.

A polynomial time algorithm for computing persistent local systems.

Framework applicable to real-world crystalline and cosmological datasets.

Abstract

The topology of periodic spaces has attracted a lot of interest in recent years in order to study and classify crystalline structures and other large homogeneous data sets, such as the distribution of galaxies in cosmology. In practice, these objects are studied by taking a finite sample and introducing periodic boundary conditions, however this introduces and removes many subtle homological features. Here, build on the work of Onus and Robins (2022) and Onus and Skraba (2023) to investigate whether one can recover the (persistent) homology of a periodic cell complex $K$ from a finite quotient space $G$ of equivalence classes under translations. In particular, we search for a computationally friendly method to identify all ''toroidal cycles'' of $G$ which do not lift to cycles in $K$. We show that all toroidal and non-toroidal cycles of $G$ of arbitrary homology degree can be completely classified for $K$ of arbitrary periodicity using the recently developed machinery of bisheaves and persistent local systems. In doing so, we also introduce a framework for a computationally viable persistence theory of periodic spaces. Finally, we outline algorithms for how to apply our results to real data, including a polynomial time algorithm for calculating the canonical persistent local system attributed to a given bisheaf.