Pattern-equivariant homology

Walton, James

doi:10.2140/agt.2017.17.1323

Pattern-equivariant homology

Walton, James

Authors

James Walton

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