New results on tilings via cup products and Chern characters on tiling spaces
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| Format: | Preprint |
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2024
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| author | Liu, Jianlong Rosenberg, Jonathan Treviño, Rodrigo |
| author_facet | Liu, Jianlong Rosenberg, Jonathan Treviño, Rodrigo |
| contents | We study the cohomology rings of tiling spaces $Ω$ 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 $\le 3$, but where a proof is known in dimensions $\ge 4$ only when the Chern character from $K^0(Ω)$ to $H^*(Ω,\mathbb{Q})$ lands in $H^*(Ω,\mathbb{Z})$. 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 $\le 3$, but we are able to show that it fails in general in dimensions $\ge 4$. This, plus some of our cup product calculations, makes it plausible that the gap-labeling conjecture might fail in high dimensions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_18789 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New results on tilings via cup products and Chern characters on tiling spaces Liu, Jianlong Rosenberg, Jonathan Treviño, Rodrigo Dynamical Systems Algebraic Topology K-Theory and Homology 37B52, 19L64, 55N45, 19L47 We study the cohomology rings of tiling spaces $Ω$ 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 $\le 3$, but where a proof is known in dimensions $\ge 4$ only when the Chern character from $K^0(Ω)$ to $H^*(Ω,\mathbb{Q})$ lands in $H^*(Ω,\mathbb{Z})$. 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 $\le 3$, but we are able to show that it fails in general in dimensions $\ge 4$. This, plus some of our cup product calculations, makes it plausible that the gap-labeling conjecture might fail in high dimensions. |
| title | New results on tilings via cup products and Chern characters on tiling spaces |
| topic | Dynamical Systems Algebraic Topology K-Theory and Homology 37B52, 19L64, 55N45, 19L47 |
| url | https://arxiv.org/abs/2409.18789 |