Cohomology and K-theory of aperiodic tilings

Abstract

We study the K-theory and cohomology of spaces of aperiodic and repetitive tilings with finite local complexity. Given such a tiling, we build a spectral sequence converging to its K-theory and define a new cohomology (PV cohomology) that appears naturally in the second page of this spectral sequence. This spectral sequence can be seen as a generalization of the Leray-Serre spectral sequence and the PV cohomology generalizes the cohomology of the base space of a Serre fibration with local coefficients in the K-theory of its fiber. We prove that the PV cohomology of such a tiling is isomorphic to the Cech cohomology of its hull. We give examples of explicit calculations of PV cohomology for a class of 1-dimensional tilings (obtained by cut-and-projection of a 2-dimensional lattice).

We also study the groupoid of the transversal of the hull of such tilings and show that they can be recovered: 1) from inverse limit of simpler groupoids (which are quotients of free categories generated by finite graphs), and 2) from an inverse semi group that arises from PV cohomology.

The underslying Delone set of punctures of such tilings modelizes the atomics positions in an aperiodic solid at zero temperature. We also present a study of (classical and harmonic) vibrational waves of low energy on such solids (acoustic phonons). We establish that the energy functional (the "matrix of spring constants" which describes the vibrations of the atoms around their equilibrium positions) behaves like a Laplacian at low energy.