Index theory and topological phases of aperiodic lattices
The paper studies the interactions between dynamical systems, represented via groupoid operator algebras, and index theory, described using Kasparov's bivariant K-functor. The key motivating example is the dynamical systems associated to Delone sets.
Delone sets give a model of aperiodic atomic configurations, where the material may be amorphous or quasi-crystalline. The atomic configuration does not have a group structure, but key properties can be encoded in a discrete groupoid. We construct bivariant K-cycles, a generalisation of K-theory classes, that encode the dynamics of the aperiodic lattice. For the case of quasi-crystalline configurations, we obtain finer information coming from the long-range aperiodic order of the material.
We then use our constructed K-cycles to define bulk topological phases of aperiodic materials. A bulk-boundary correspondence is also shown, which arises from factorisation properties of the constructed K-cycles.