Pattern-equivariant functions and cohomology

Kellendonk, Johannes

doi:10.1088/0305-4470/36/21/306

Cited by 45 publications

(70 citation statements)

References 9 publications

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“…Further examples, and methods of calculation of PV cohomology will be investigated in future research. However, as it turns out (see section 4.1), the PV cohomology is isomorphic to other cohomologies used so far on the hull, such as theČech cohomology [1,63], the group cohomology [27] or the pattern equivariant cohomology [42,43,66]. …”

Section: Corollarymentioning

confidence: 98%

“…It has been used to prove that the generators of the d-th cohomology group are in one-to-one correspondence with invariant ergodic probability measures on the hull. Another useful cohomology, the Pattern-Equivariant (PE) cohomology, has been proposed by Kellendonk and Putnam in [42,43] for real coefficients and then generalized to integer coefficients by Sadun [66]. This cohomology has been used for proving that the RuelleSullivan map (associated with an ergodic invariant probability measure on the hull) from theČech cohomology of the hull to the exterior algebra of the dual of R d is a ring homomorphism.…”

Section: Tiling Cohomologiesmentioning

confidence: 99%

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A spectral sequence for theK-theory of tiling spaces

Savinien

Bellissard

2009

Ergod. Th. Dynam. Sys.

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Let T be an aperiodic and repetitive tiling of R d 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 thě Cech cohomology of the hull of T (a compactification of the family of its translates). Main ResultsLet T be an aperiodic and repetitive tiling of R d with finite local complexity (definition 3). The hull Ω is a compactification, with respect to an appropriate topology, of the family of translates of T by vectors of R d (definition 4). The tiles of T are given compatible ∆-complex decompositions (section 4.2), with each simplex punctured, and the ∆-transversal Ξ ∆ is the subset of Ω corresponding to translates of T having the puncture of one of those simplices at the origin 0 R d . The prototile space B 0 (definition 9) is built out of the prototiles of T (translational equivalence classes of tiles) by gluing them together according to the local configurations of their representatives in the tiling. The hull is given a dynamical system structure via the natural action of the group R d on itself by translation [13]. The C * -algebra of the hull is isomorphic to the crossed-productThere is a map p o from the hull onto the prototile space (proposition 1)which, thanks to a lamination structure on Ω (remark 2), resembles (although is not) a fibration with base space B 0 and fiber Ξ ∆ (remark 4). The Pimsner-Voiculescu (PV) *

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Section: Corollarymentioning

confidence: 98%

Section: Tiling Cohomologiesmentioning

confidence: 99%

A spectral sequence for theK-theory of tiling spaces

Savinien

Bellissard

2009

Ergod. Th. Dynam. Sys.

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“…We begin by introducing the notion of a strongly P -equivariant function on R n [Ke03] where P is as in the last section a Delone set of R n of finite local complexity. Roughly speaking, a function f on R n is strongly P -equivariant if there is some constant R such that, if the patterns in P surrounding two points x and y of radius R are equal (after translating by −x and −y, respectively), then f must take the same values at x and y.…”

Section: P -Equivariant Cohomologymentioning

confidence: 99%

“…The Cech cohomology of P is, however, the cohomology of the sub-complex of strongly P -equivariant forms. The latter was therefore simply called the P -equivariant cohomology (of R n ) in [Ke03].…”

Section: As a Consequence Of This Proposition We Havementioning

confidence: 99%

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The Ruelle-Sullivan map for actions of ℝn

Kellendonk

Putnam

2006

Math. Ann.

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The Ruelle Sullivan map for an R n -action on a compact metric space with invariant probability measure is a graded homomorphism between the integer Cech cohomology of the space and the exterior algebra of the dual of R n . We investigate flows on tori to illuminate that it detects geometrical structure of the system. For actions arising from Delone sets of finite local complexity, the existence of canonical transversals and a formulation in terms of pattern equivariant functions lead to the result that the Ruelle Sullivan map is even a ring homomorphism provided the measure is ergodic.

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Operators, Algebras and Their Invariants for Aperiodic Tilings

Kellendonk

2020

Lecture Notes in Mathematics

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Pattern-equivariant functions and cohomology

Cited by 45 publications

References 9 publications

A spectral sequence for theK-theory of tiling spaces

A spectral sequence for theK-theory of tiling spaces

The Ruelle-Sullivan map for actions of ℝn

Operators, Algebras and Their Invariants for Aperiodic Tilings

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