Index theory and topological phases of aperiodic lattices

Abstract

We examine the noncommutative index theory associated to the dynamics of a Delone set and the corresponding transversal groupoid. Our main motivation comes from the application to topological phases of aperiodic lattices and materials, and applies to invariants from tilings as well. Our discussion concerns semifinite index pairings, factorisation properties of Kasparov modules and the construction of unbounded Fredholm modules for lattices with finite local complexity.

52 pages, Section 1.6 added and other minor improvements. To appear in Annales Henri Poincaré