A Spectral Sequence for the K-theory of Tiling Spaces

TL;DR

This paper introduces a spectral sequence that computes the K-theory of aperiodic tiling spaces using a new cohomology called PV, generalizing classical spectral sequences and relating to the cohomology of tiling hulls.

Contribution

It develops a new spectral sequence with PV cohomology for tiling spaces, extending the Serre spectral sequence to aperiodic tilings and linking it to Čech cohomology.

Findings

PV cohomology is isomorphic to Čech cohomology of the tiling hull.

The spectral sequence converges to the K-theory of tiling spaces.

Generalizes the Serre spectral sequence for aperiodic tilings.

Abstract

Let $\Tt$ be an aperiodic and repetitive tiling of $\RM^d$ with finite local complexity. We present a spectral sequence that converges to the $K$-theory of $\Tt$ with $E_2$-page given by a new cohomology that will be called PV in reference to the Pimsner-Voiculescu exact sequence. It is a generalization of the Serre spectral sequence. The PV cohomology of $\Tt$ generalizes the cohomology of the base space of a fibration with local coefficients in the $K$-theory of its fiber. We prove that it is isomorphic to the \v{C}ech cohomology of the hull of $\Tt$ (a compactification of the family of its translates).