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

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 3, but where a proof is known in dimensions 4 only when the Chern character from K 0 ./ to H . ; Q/ lands in H . ; Z/.Computation of the cup product on cohomology often makes it possible to compute the Chern character.We introduce a natural generalization of the gaplabeling 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.