[Paper Review] A Spectral Sequence for the K-theory of Tiling Spaces

Jean Savinien, Jean Bellissard|ArXiv.org|May 17, 2007

Quasicrystal Structures and Properties27 references4 citations

TL;DR

This paper introduces a spectral sequence converging to the K-theory of tiling C*-algebras, with its E2 page given by a new cohomology—PV cohomology—generalizing the Pimsner-Voiculescu exact sequence. It establishes that PV cohomology of a repetitive, aperiodic tiling with finite local complexity is isomorphic to the Čech cohomology of the tiling's hull, providing a canonical, topological framework for computing K-theory in non-fibrational dynamical systems.

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).

Motivation & Objective

• To develop a canonical spectral sequence for computing the K-theory of C*-algebras associated with aperiodic, repetitive tilings of R^d with finite local complexity.
• To define and characterize a new cohomology theory—PV cohomology—that generalizes the Pimsner-Voiculescu sequence in the context of tiling spaces.
• To establish an isomorphism between PV cohomology and the Čech cohomology of the hull of the tiling, thereby linking K-theory computation to topological invariants of the hull.
• To provide a generalization of the Serre spectral sequence for laminations that are not fibrations, using a cofiltration of C*-algebras associated with the tiling's structure.

Proposed method

• Constructs a spectral sequence using a cofiltration of the crossed-product C*-algebra C(Ω) ⋊ R^d via ideals derived from the CW decomposition of the torus T^d.
• Defines the PV cohomology H^r_{PV}(B_0; K^s(Ξ_Δ)) as the E2 page of the spectral sequence, with local coefficients in the K-theory of the transversal Ξ_Δ.
• Establishes the spectral sequence's convergence to K_{r+s+d}(C(Ω) ⋊ R^d) via the Thom-Connes isomorphism, which identifies this K-theory with the topological K-theory of the hull Ω.
• Uses the Pimsner-Voiculescu complex d_{PV} = ∑_{i=1}^d (α_{i*} - 1) ⊗ e_i ∧ on K_*(A) ⊗ Λ^* Z^d to model the differential in the spectral sequence.
• Proves the isomorphism between the spectral sequence associated with the filtration of ideals and the one from the cofiltration of quotients, via exact couples and commutative diagrams in K-theory.
• Demonstrates that the PV cohomology is isomorphic to the Čech cohomology of the hull Ω by exploiting the lamination structure and the dynamical system on the hull.

Experimental results

Research questions

• RQ1 Can a spectral sequence be constructed to compute the K-theory of the C*-algebra of a tiling space that generalizes the Serre spectral sequence for non-fibrational laminations?
• RQ2 How can the Pimsner-Voiculescu exact sequence be generalized to Z^d-actions on C*-algebras arising from tilings with finite local complexity?
• RQ3 Is there a cohomology theory that captures the K-theory of tiling C*-algebras in terms of the base space of a dynamical system and local coefficients in the K-theory of the fiber?
• RQ4 Does the PV cohomology of a tiling space coincide with the Čech cohomology of its hull, thereby providing a topological interpretation of the spectral sequence's E2 page?
• RQ5 Can the spectral sequence be constructed canonically without relying on a choice of Z^d action, thus offering a more intrinsic invariant than prior constructions?

Key findings

• The spectral sequence converges to K_{r+s+d}(C(Ω) ⋊ R^d), with E2 page isomorphic to H^r_{PV}(B_0; K^s(Ξ_Δ)), providing a systematic tool for K-theory computation.
• PV cohomology is isomorphic to the Čech cohomology of the hull Ω, establishing a deep link between K-theory and topological invariants of the tiling space.
• The construction generalizes the Serre spectral sequence to laminations that are not fibrations, extending its applicability to aperiodic tiling systems.
• The spectral sequence is independent of the choice of Z^d action, offering a more canonical and intrinsic method than prior approaches based on inverse limits or Cantor set actions.
• The Pimsner-Voiculescu complex d_{PV} = ∑_{i=1}^d (α_{i*} - 1) ⊗ e_i ∧ provides the differential on the E2 page, generalizing the 1D case to higher dimensions.
• The isomorphism between the filtration and cofiltration spectral sequences confirms the robustness of the construction and validates the use of exact couples in K-theory computations.