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New results on tilings
via cup products and Chern characters on tiling spaces
Jianlong Liu, Jonathan Rosenberg and Rodrigo Treviño |
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Algebraic & Geometric Topology 26 (2026) 1155–1193
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Abstract |
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We study the cohomology rings of tiling spaces given by cubical substitutions. While there have been many calculations before of cohomology groups of such tiling spaces, the innovation here is that we use computer-assisted methods to compute the cup-product structure. This leads to examples of substitution tilings with isomorphic cohomology groups but different cohomology rings. Part of the interest in studying the cup product comes from Bellissard’s gap-labeling conjecture, which is known to hold in dimensions , but where a proof is known in dimensions only when the Chern character from to lands in . Computation of the cup product on cohomology often makes it possible to compute the Chern character. We introduce a natural generalization of the gap-labeling conjecture, called the equivariant gap-labeling conjecture, which applies to tilings with a finite symmetry group. Again this holds in dimensions , but we are able to show that it fails in general in dimensions . This, plus some of our cup-product calculations, makes it plausible that the gap-labeling conjecture might fail in high dimensions. |
Keywords
aperiodic, tiling, gap-labeling, frequency module, cup
product, Chern character, substitution tiling,
Anderson–Putnam complex, dual complex, Barge–Diamond
complex
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Mathematical Subject Classification
Primary: 19L47, 19L64, 37B52, 55N45
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References |
Publication
Received: 11 October 2024
Revised: 26 March 2025
Accepted: 28 March 2025
Published: 1 April 2026
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Authors |
| Jianlong Liu | |
| Department of Mathematics University of Texas at Austin Austin, TX United States |
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| Jonathan Rosenberg | |
| Department of Mathematics University of Maryland College Park, MD United States |
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| Rodrigo Treviño | |
| Department of Mathematics University of Maryland College Park, MD United States |
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| © 2026 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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