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

TL;DR

The paper addresses the gap-labeling problem for tiling spaces by examining the integral cohomology ring and the Chern character in both non-equivariant and equivariant settings. It introduces a computational pipeline for cubical substitutions to compute cup products via a dual complex, revealing that cohomology groups can coincide while the ring structure differs, and it produces explicit 4D examples where the Chern character fails to be integral. Extending to symmetry, the authors formulate an equivariant gap-labeling conjecture and show it holds in dimensions ≤3 but fails in dimension 4, with concrete aperiodic counterexamples. Overall, the work suggests the gap-labeling phenomenon may fail in high dimensions and highlights the value of cup-product data and equivariant analysis in understanding tiling spaces and their spectra.

Abstract

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.

Figures (8)

• Figure 1: A substitution rule with even expansion applied to a patch of size $2^d$ representing a top-dimensional cell (center tile shaded) returns a patch (light gray lines) whose center tiles (shaded) has collars (black lines) that are not actual patches, therefore is not cellular. One needs to compose the substitution with a homotopy $h$ to produce a cellular map.
• Figure 2: Substitution rule for an expansion $5$ two-dimensional cubical substitution that has the cohomology groups of a product, but does not have the ring structure of a product.
• Figure 3: Substitution rule for an expansion $4$ two-dimensional cubical substitution that has the cohomology groups of a product, but does not have the ring structure of a product.
• Figure 4: Patches from Examples \ref{['ex:2']} and \ref{['ex:4']}, respectively. That their tiling spaces are not homeomorphic can only be detected through the cup product.
• Figure 5: A two-dimensional analogue of the main example in section \ref{['subsec:ex']}. We begin with a one-dimensional checkerboard substitution on $8$ prototiles (drawn vertically in the first map; only the first supertile is drawn, the rest are obtained by $+1\mod 8$), then extending it along the remaining coordinate (horizontally) by another (different) checkerboard (second map). The resulting tiling has rotational symmetry about axis the pattern is extended along (horizontal here).

Theorems & Definitions (15)

• Remark 2.1
• Definition 2.1
• Conjecture 2.1: Equivariant gap-labeling conjecture
• Example 2.1
• Remark 2.2
• Theorem 2.1
• proof
• Corollary 2.1
• Remark 2.3
• Example 3.1: Chair tiling