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Following an earlier similar conjecture of Kellendonk and Putnam, Giordano,Putnam and Skau conjectured that all minimal, free ${\mathbb Z}^d$ actions onCantor sets admit "small cocycles." These represent classes in $H^1$ that aremapped to small vectors in ${\mathbb R}^d$ by the Ruelle-Sullivan (RS) map. Weshow that there exist ${\mathbb Z}^d$ actions where no such small cocyclesexist, and where the image of $H^1$ under RS is ${\mathbb Z}^d$. Our methodsinvolve tiling spaces and shape deformations, and along the way we prove arelation between the image of RS and the set of "virtual eigenvalues," i.e.elements of ${\mathbb R}^d$ that become topological eigenvalues of the tilingflow after an arbitrarily small change in the shapes and sizes of the tiles.

Small cocycles, fine torus fibrations, and a ${\mathbb Z}^2$ subshift with neither