Jean Bellissard, Jean Savinien
A Spectral Sequence for the K-theory of Tiling Spaces
Preprint series: ESI preprints

MSC:

52C20 Tilings in $2$ dimensions, See also {05B45, 51M20}

52C22 Tilings in $n$ dimensions, See also {05B45, 51M20}

55N15 $K$-theory, See also {19Lxx}, {For algebraic $K$-theory, See 18F25, 19-XX}

54H20 Topological dynamics, See also {28Dxx, 34C35, 58Fxx}

Abstract: Let $\Tt$ be an aperiodic and repetitive tiling of $\RM^d$ with finite local com
plexity.
We present a spectral sequence that converges to the $K$-theory of $\Tt$ with pa
ge-$2$ isomorphic to the
Pimsner cohomology of $\Tt$. It is a generalization of Serre spectral sequence to a class of spaces which are not fibered.
The Pimsner cohomology of $\Tt$ generalizes the cohomology of the base space of a fibration with local coefficients in the $K$-theory of its fiber. We prove that it
is isomorphic to the \v{C}ech cohomology of the hull of $\Tt$ (the closure for an appropriate topology of the family of its translates).
Keywords: aperiodic tiling, K-theory