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We study the cohomology rings of tiling spaces $\Omega$ given by cubicalsubstitutions. While there have been many calculations before of cohomologygroups of such tiling spaces, the innovation here is that we usecomputer-assisted methods to compute the cup-product structure. This leads toexamples of substitution tilings with isomorphic cohomology groups butdifferent cohomology rings. Part of the interest in studying the cup productcomes from Bellissard's gap-labeling conjecture, which is known to hold indimensions $\le 3$, but where a proof is known in dimensions $\ge 4$ only whenthe Chern character from $K^0(\Omega)$ to $H^*(\Omega,\mathbb{Q})$ lands in$H^*(\Omega,\mathbb{Z})$. Computation of the cup product on cohomology oftenmakes it possible to compute the Chern character. We introduce a naturalgeneralization of the gap-labeling conjecture, called the equivariantgap-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 failsin general in dimensions $\ge 4$. This, plus some of our cup productcalculations, makes it plausible that the gap-labeling conjecture might fail inhigh dimensions.

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