Quantifying the homology of periodic cell complexes

TL;DR

The paper addresses how to quantify the homology of spatially periodic cell complexes by relating the infinite periodic cover $K$ to its finite quotient $q(K)$. For periodic graphs, it introduces weighted quotient graphs with translation-vectors that recover $H_0$ and $H_1$ via deck-translation data and commutator lifts, including a complete treatment of the maximal abelian cover and explicit cycle generators. For higher-dimensional complexes, the authors show that no simple edge-weight analogue exists; they develop a heuristic based on the Mayer–Vietoris spectral sequence to distinguish true cycles of $K$ from toroidal cycles arising from periodic boundary conditions, and propose finite approximations to compute homology in practice. The MVSS framework enables parallel, local computations on unit cells and provides a principled approach to identify toroidal versus non-toroidal features, with applications to crystal structures and periodic point patterns. The work outlines open questions on generalizing weights to higher dimensions, possible k-weighted quotient spaces, and extensions to persistence, setting a path for robust analysis of periodic topological spaces.

Abstract

A periodic cell complex, $K$, has a finite representation as the quotient space, $q(K)$, consisting of equivalence classes of cells identified under the translation group acting on $K$. We study how the Betti numbers and cycles of $K$ are related to those of $q(K)$, first for the case that $K$ is a graph, and then higher-dimensional cell complexes. When $K$ is a $d$-periodic graph, it is possible to define $\mathbb{Z}^d$-weights on the edges of the quotient graph and this information permits full recovery of homology generators for $K$. The situation for higher-dimensional cell complexes is more subtle and studied in detail using the Mayer-Vietoris spectral sequence.

Figures (7)

• Figure 1: A section of two interwoven cubical lattices along integer and half integer coordinates in $\mathbb{R}^3$ (with connected components coloured green and purple) and its weighted quotient graph with respect to a basis of translations in the directions $t_1 = (-\frac{1}{2},\frac{1}{2},\frac{1}{2})$, $t_2 = (\frac{1}{2},-\frac{1}{2},\frac{1}{2})$ and $t_3 = (\frac{1}{2},\frac{1}{2},-\frac{1}{2})$.
• Figure 2: Left: A section of the Kagome pattern with translational basis for $T$ shown in blue. Right: A choice of weighted quotient graph with edge weights given with respect to $T$ and the given choice of vertex representatives.
• Figure 3: A section of the 2-periodic graph $K_A$ (left) and its WQG with edge weights $(1,1)$, $(2,0)$ and $(0,3)$ (right). The vertices of $K_A$ are at all points with integer coordinates and the translation group $T \cong \mathbb{Z}^2$ has the standard basis.
• Figure 4: A section of the Kagome pattern $K_G$ and its WQG $q(K_G)$ with edge labels for a choice representative in the cover. Refer to Figure \ref{['fig:kagomeWQG']} for the edge weights.
• Figure 5: A section of a $2$-periodic graph and its weighted quotient graph with respect to translations by $\mathbb{Z}^2$. In red is a cycle satisfying Case 1 of Lemma \ref{['lem:four-cases']}, in orange a cycle satisfying Case 2, in green a chain satisfying Case 3, and in blue a chain satisfying Case 4.

Theorems & Definitions (33)

• Theorem 1: HatcherAT, p.166
• Theorem 2: HatcherAT, p.67
• Theorem 3: HatcherAT, p.72
• Theorem 4: sunada2012topological, p.78
• Definition 1
• Definition 2
• Lemma 5
• proof
• Theorem 6
• proof