A Spectral Sequence for the K-theory of Tiling Spaces

Details A Spectral Sequence for the K-theory of Tiling Spaces A Spectral Sequence for the K-theory of Tiling Spaces

2007-05-17

arxiv.org/abs/0705.2483v2

Mathematics

K-Theory and Homology

40pages, 1 figure

Scientific paper

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).

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