New results on tilings via cup products and Chern characters on tiling spaces

TL;DR

This paper uses computer-assisted methods to analyze the cohomology rings of tiling spaces, revealing differences in ring structures despite identical cohomology groups, and explores implications for the gap-labeling conjecture.

Contribution

It introduces computational techniques to determine cup-product structures in tiling space cohomology and examines their impact on the gap-labeling conjecture in various dimensions.

Findings

Identified examples of tilings with isomorphic cohomology groups but different cohomology rings.

Showed the equivariant gap-labeling conjecture holds in dimensions ≤ 3 but fails in higher dimensions.

Suggested potential failure of the gap-labeling conjecture in high-dimensional tilings.

Abstract

We study the cohomology rings of tiling spaces $\Omega$ 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 $K^0(\Omega)$ to $H^*(\Omega,\mathbb{Q})$ lands in $H^*(\Omega,\mathbb{Z})$. 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.