The cohomology and $K$-theory of commuting homeomorphisms of the Cantor set

Abstract

Given a $\mathbb{Z}^d$ homeomorphic action, $\alpha$, on the Cantor set, $X$, we consider the higher order continuous integer valued dynamical cohomology, $H^*(X,\alpha)$. We also consider the dynamical $K$-theory of the action, the $K$-theory of the crossed product $C^*$-algebra $C(X)\times_{\alpha}\mathbb{Z}^d$. We show that these two invariants are essentially equivalent. We also show that they only take torsion free values. Our work links the two invariants via a third invariant which is based on topological complex $K$-theory evaluated on an associated mapping torus.

A spectral sequence for theK-theory of tiling spaces

“…By an argument using the Thom-Connes isomorphism [21] the K-theory of C(Ω) ⋊ R d is isomorphic to the topological K-theory of Ω (with a shift in dimension by d), and the above theorem can thus be seen formally as a generalization of the Serre spectral sequence [67] for a certain class of laminations Ξ ∆ ֒→ Ω → B 0 , which are foliated spaces [52] but not fibrations. This result brings a different point of view on a problem solved earlier by Hunton and Forrest in [27]. They built a spectral sequence for the K-theory of a crossed product C * -algebra of a Z d -action on a Cantor set.…”

“…Further examples, and methods of calculation of PV cohomology will be investigated in future research. However, as it turns out (see section 4.1), the PV cohomology is isomorphic to other cohomologies used so far on the hull, such as theČech cohomology [1,63], the group cohomology [27] or the pattern equivariant cohomology [42,43,66]. …”

“…The spectral sequence used by Forrest and Hunton in [27] in the case A = C(X) where X is the Cantor set coincides with the PV spectral sequence. The present paper generalizes this construction for tilings by replacing the classifying space T d , by the prototile space B 0 .…”

“…The first use of spectral sequences in computing the set of gap labels on the case of the 2D-octagonal tiling [10] was followed by a proof of the Gap Labeling Theorem for 3D-quasicrystals [12]. Finally, the work by Forrest and Hunton [27] used a classical spectral sequence to compute the full K-theory of the C * -algebra for an action of Z d on the Cantor set. It made possible the computation of the K-theory and the cohomology of the Hull for quasicrystals in two and three dimensions [28,29].…”

The Noncommutative Geometry of Aperiodic Solids

“…(31) The following result has been proved in [24] which is a an extenxion of the Forrest-Hunton theorem 21 [53] Theorem 23 Let L be a repetitive Delone set of finite type in R Let now P be an R d -invariant measure on Ω. It then defines a canonical probability on X, called the transverse measure induced by P [36,41].…”