James Walton
Pattern-equivariant homology
Walton, James
Authors
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
Walton, J. (2017). Pattern-equivariant homology. Algebraic & geometric topology, 17(3), 1323-1373. https://doi.org/10.2140/agt.2017.17.1323
Journal Article Type | Article |
---|---|
Acceptance Date | Sep 21, 2016 |
Online Publication Date | Jul 17, 2017 |
Publication Date | Jul 17, 2017 |
Deposit Date | Feb 21, 2017 |
Publicly Available Date | Aug 8, 2017 |
Journal | Algebraic and Geometric Topology |
Print ISSN | 1472-2747 |
Electronic ISSN | 1472-2739 |
Publisher | Mathematical Sciences Publishers (MSP) |
Peer Reviewed | Peer Reviewed |
Volume | 17 |
Issue | 3 |
Article Number | 1323-1373 |
Pages | 1323-1373 |
DOI | https://doi.org/10.2140/agt.2017.17.1323 |
Files
Published Journal Article
(777 Kb)
PDF
You might also like
Statistics of patterns in typical cut and project sets
(2018)
Journal Article
Perfectly ordered quasicrystals and the Littlewood conjecture
(2018)
Journal Article
A characterization of linearly repetitive cut and project sets
(2018)
Journal Article
Cohomology of rotational tiling spaces
(2017)
Journal Article
Gaps problems and frequencies of patches in cut and project sets
(2016)
Journal Article
Downloadable Citations
About Durham Research Online (DRO)
Administrator e-mail: dro.admin@durham.ac.uk
This application uses the following open-source libraries:
SheetJS Community Edition
Apache License Version 2.0 (http://www.apache.org/licenses/)
PDF.js
Apache License Version 2.0 (http://www.apache.org/licenses/)
Font Awesome
SIL OFL 1.1 (http://scripts.sil.org/OFL)
MIT License (http://opensource.org/licenses/mit-license.html)
CC BY 3.0 ( http://creativecommons.org/licenses/by/3.0/)
Powered by Worktribe © 2024
Advanced Search