A Spectral Sequence for the K-theory of Tiling Spaces

2007/05/17 by Jean Savinien, Savinien, Jean, Jean Bellissard +1
Materials Science · Mathematics · #Quasicrystal Structures and Properties #Mathematical Dynamics and Fractals #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.0705.2483

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 E₂-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 Čech cohomology of the hull of \Tt (a compactification of the family of its translates).

Citations

• Metallic Phase with Long-Range Orientational Order and No Translational Symmetry

Related

• The K-theory cochains of H-spaces and height 1 chromatic homotopy theory
• Cohomology of Hierarchical Tilings
• The slice spectral sequence for the C₄ analog of real K-theory
• Spectral sequences of a Morse shelling
• The homotopy Leray spectral sequence