Topological Crystals

Sunada's work on crystallography emphasizes the role of the "maximal abelian cover" of a graph $X$. This is a covering space of $X$ for which the group of deck transformations is the first homology group $H_1(X,\mathbb{Z})$. An embedding of the maximal abelian cover in a vector space can serve as the pattern for a crystal: atoms are located at the vertices, while bonds lie along the edges. We prove that for any connected graph $X$ without bridges, there is a canonical embedding of the maximal abelian cover of $X$ into the vector space $H_1(X,\mathbb{R})$, called a "topological crystal". Crystals of graphene and diamond are examples of this construction. We prove that any symmetry of a graph lifts to a symmetry of its topological crystal. We also compute the density of atoms in a topological crystal. The key technical tools are a way of decomposing the 1-chain coming from a path in $X$ into manageable pieces, and the work of Bacher, de la Harpe and Nagnibeda on integral cycles and integral cuts.