2024/09/27 by Liu, Jianlong, Rosenberg, Jonathan, Treviño, Rodrigo
#19L47 #19L64 #37B52 #55N45 #Algebraic Topology (math.AT) #Dynamical Systems (math.DS) #FOS: Mathematics #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.2409.18789
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 ≤ 3, but where a proof is known in dimensions ≥ 4 only when the Chern character from K0(Ω) to H^*(Ω,ℚ) lands in H^*(Ω,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 ≤ 3, but we are able to show that it fails in general in dimensions ≥ 4. This, plus some of our cup product calculations, makes it plausible that the gap-labeling conjecture might fail in high dimensions.