Quantifying the homology of periodic cell complexes

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.