Ergodic Theory and Dynamical Systems

Table of Contents - 01 June 2009 - Volume 29, Issue 03  

Research Article

A spectral sequence for the K-theory of tiling spaces

JEAN SAVINIENa1 and JEAN BELLISSARDa1

a1 Georgia Institute of Technology, School of Mathematics, Atlanta, GA 30332-0160, USA (email: savinien@math.gatech.edu, jeanbel@math.gatech.edu)

Abstract

Let 𝒯 be an aperiodic and repetitive tiling of ℝd with finite local complexity. We present a spectral sequence that converges to the K-theory of 𝒯 with page-2 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 𝒯 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 Čech cohomology of the hull of 𝒯 (a compactification of the family of its translates).

(Received May 28 2007)

(Revised April 21 2008)