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A spectral sequence for the K-theory of tiling spaces

Published online by Cambridge University Press:  01 June 2009

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

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Type
Research Article
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Ergodic Theory and Dynamical Systems , Volume 29 , Issue 3 , June 2009 , pp. 997 - 1031
Copyright
Copyright © Cambridge University Press 2009

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