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2017 Pattern-equivariant homology
James Walton
Algebr. Geom. Topol. 17(3): 1323-1373 (2017). DOI: 10.2140/agt.2017.17.1323

Abstract

Pattern-equivariant (PE) cohomology is a well-established tool with which to interpret the Čech 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 the Čech 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.

Citation

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James Walton. "Pattern-equivariant homology." Algebr. Geom. Topol. 17 (3) 1323 - 1373, 2017. https://doi.org/10.2140/agt.2017.17.1323

Information

Received: 10 February 2014; Revised: 22 September 2016; Accepted: 25 October 2016; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1373.52027
MathSciNet: MR3677930
Digital Object Identifier: 10.2140/agt.2017.17.1323

Subjects:
Primary: 52C23
Secondary: 37B50, 52C22, 55N05

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.17 • No. 3 • 2017
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