Cohomology of rotational tiling spaces

Walton, James J.

doi:10.1112/blms.12098

Cited by 4 publications

(18 citation statements)

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“…The above examples show that ePE Poincaré duality can fail in the presence of non-trivial rotational symmetry. We consider the discrepancy between the ePE homology and ePE cohomology to be a feature of interest, and it will be of relevance in forthcoming work on the cohomology of Euclidean hulls [42]. However,…”

Section: Restoring Poincaré Dualitymentioning

confidence: 99%

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Pattern-equivariant homology

Walton

2017

Algebr. Geom. Topol.

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Pattern-equivariant homology JAMES J WALTONPattern-equivariant (PE) cohomology is a well-established tool with which to interpret theČech cohomology groups of a tiling space in a highly geometric way. We consider homology groups of PE infinite chains and establish Poincaré duality between the PE cohomology and PE homology. The Penrose kite and dart tilings are taken as our central running example; we show how through this formalism one may give highly approachable geometric descriptions of the generators of theČech cohomology of their tiling space. These invariants are also considered in the context of rotational symmetry. Poincaré duality fails over integer coefficients for the "ePE homology groups" based upon chains which are PE with respect to orientation-preserving Euclidean motions between patches. As a result we construct a new invariant, which is of relevance to the cohomology of rotational tiling spaces. We present an efficient method of computation of the PE and ePE (co)homology groups for hierarchical tilings.52C23; 37B50, 52C22, 55N05 IntroductionIn the past few decades a rich class of highly ordered patterns has emerged whose central examples, despite lacking global translational symmetries, exhibit intricate internal structure, imbuing these patterns with properties akin to those enjoyed by periodically repeating patterns. The field of aperiodic order aims to study such patterns, and to establish connections between their properties, and their constructions, to other fields of mathematics and the natural sciences. PE cohomology allows for an intuitive description of theČech cohomology L H . / of tiling spaces. Over R coefficients the PE cochain groups may be defined using PE differential forms [24], and over general abelian coefficients, when the tiling has a cellular structure, with PE cellular cochains; see Sadun [37]. Rather than just providing a reflection of topological invariants of tiling spaces, on the contrary, these PE invariants are of principal relevance to aperiodic structures and their connections with other fields in their own right; see, for example, Kelly and Sadun's use of them [29] in a topological proof of theorems of Kesten and Oren regarding the discrepancy of irrational rotations. It is perhaps more appropriate to view the isomorphism between L H . / and the PE cohomology as an elegant interpretation of the PE cohomology, rather than vice versa, of theoretical and computational importance.In this paper we introduce the pattern-equivariant homology gro...

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Section: Restoring Poincaré Dualitymentioning

confidence: 99%

“…Furthermore, the method provides explicit generators in terms of pattern-equivariant chains. The result for the ePE homology of the Penrose tiling, along with precise descriptions of the generators of the ePE homology, will be essential in forthcoming work [42].…”

Section: 2mentioning

confidence: 99%

Pattern-equivariant homology

Walton

2017

Algebr. Geom. Topol.

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“…Nevertheless, with a few notable exceptions, such as [4,21,25,26,33,36], and see also [3], the topological study to date has largely been confined to the analogue of the translational symmetries for aperiodic patterns. This is perhaps surprising as many of the most interesting examples display apparent strong rotational or reflective organisation.…”

Section: Introductionmentioning

confidence: 99%

“…Just as Ω, the continuous, or what we shall now refer to as the translational, hull, denoting it by Ω t , can be considered as a certain completion of the space of translates of the pattern, the rotational hull Ω r is the corresponding completion of the set of all Euclidean motions of T . The space Ω r has been considered before, but by and large only for 2-dimensional patterns [5,30,36].…”

Section: Introductionmentioning

confidence: 99%

Aperiodicity, rotational tiling spaces and topological space groups

Hunton,

Walton

2018

Preprint

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We study the rotational structures of aperiodic tilings in Euclidean space of arbitrary dimension using topological methods. Classical topological approaches to the study of aperiodic patterns have largely concentrated just on translational structures, studying an associated space, the continuous hull, here denoted Ωt. In this article we consider two further spaces Ωr and ΩG (the rotational hulls) which capture the full rigid motion properties of the underlying patterns. We develop new S-MLD invariants derived from the homotopical and cohomological properties of these spaces demonstrating their computational as well as theoretical utility. We compute these invariants for a variety of examples, including a class of 3-dimensional aperiodic patterns, as well as for the space of periodic tessellations of R 3 by unit cubes. We show that the classical space group of symmetries of a periodic pattern may be recovered as the fundamental group of our space ΩG, and similarly for those patterns associated to quasicrystals, the crystallographers' aperiodic space group may be recovered as a quotient of our fundamental invariant.

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“…Pattern Equivariant cohomology [15,22] has proved a useful alternative approach, yielding the same algebraic invariant as the Cech theory, but in a way that elements can be realised in terms of geometric patches of T . A homological variant [25] shows that tiling spaces satisfy a Poincaré duality property analogous to that of manifolds, and has offered computational advantage, for example in the study of spaces remembering the symmetries of T [26].…”

mentioning

confidence: 99%