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arXiv · 0705.2483

A Spectral Sequence for the K-theory of Tiling Spaces

Jean Savinien·Jean Bellissard
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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 Čech cohomology of the hull of $\Tt$ (a compactification of the family of its translates).

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Jean Savinien, Jean Bellissard. 2009-06-05. A Spectral Sequence for the K-theory of Tiling Spaces. https://arxiv.org/abs/0705.2483

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