Persistent Local Systems of Periodic Spaces

TL;DR

This work develops a robust, computable framework for recovering the persistent homology of spatially periodic spaces from finite quotient data. By leveraging cellular bisheaves, epification/monofication, and the resulting persistent local systems, the authors classify toroidal cycles and connect them to the ambient periodic topology through a principled lifting to the covering space. They prove embedding theorems for 1- and d-periodic cases, showing that the relevant toroidal information embeds into a canonical PLS and can be accessed via monodromy matrices, enabling polynomial-time computation of the canonical PLS. The proposed algorithms for constructing bisheaves and performing isobisheafification render the theory computationally viable and scalable, with explicit procedures for extracting toroidal-cycle data from real data. Overall, the paper provides a complete, practical persistence-theoretic treatment of periodic spaces, bridging covering-space theory, monodromy, and scalable TDA for applications in materials science, cosmology, and beyond.

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.

Figures (9)

• Figure 1: A 2-periodic point cloud, with the $r$-offset (for $r=0.08$) shaded in gray. Horizontal and vertical lines indicate the lattice structure. The dashed red circle and dashed yellow path indicate 1-cycles, the latter appearing only after imposing periodic boundary conditions on $[0,1]^2$.
• Figure 2: A graph $K$ in $\mathbb{R}^2$ with vertex set $\mathbb{Z}\times\{0,\pm1\}$. $K$ is 1-periodic (see Definition \ref{['def:periodic']}) and for each $n\in \mathbb{N}$, the quotient space $K/n\mathbb{Z}$ has a natural projection to the 1-sphere $\mathbb{S}^1$ defined by $(x,y)\mapsto e^{2\pi i x/n}$.
• Figure 3: Three examples of 2-dimensional 2-periodic spaces (implied to extend out indefinitely) in $\mathbb{R}^3$ which project onto a standard 2-torus $\mathbb{T}^2$ in their quotient spaces with respect to translations of $\mathbb{Z}^2\times 0$. Left: $\mathbb{R}^2\times 0$, where the red lines project onto toroidal 1-cycles and the purple surface projects onto a toroidal 2-cycle. Center: disjoint copies of $\mathbb{T}^2$, where the quotient action projects the green circles onto non-toroidal 1-cycles, and the blue surfaces onto a non-toroidal 2-cycles. Right: disjoint infinite cylinders, where the quotient action projects the purple surfaces onto a toroidal 2-cycle, the red lines onto a toroidal 1-cycle, and the green circles onto a non-toroidal 1-cycle.
• Figure 4: An periodic simplicial complex over $\mathbb{S}^1$.
• Figure 5: Successive subdivisions of the triangulation in Figure \ref{['fig:periodic-boundary']} so that the preimage of the highlighted vertices in $\mathbb{S}^1$ are subcomplexes.

Theorems & Definitions (42)

• Definition 1
• Example 1
• Definition 2
• Definition 3
• Example 2
• Definition 4
• Definition 5
• Definition 6
• Example 3
• Remark 1