Topological Invariants of Tiling Spaces

Abstract

Given some aperiodic tiling (of Euclidean space, say), a fruitful approach to understanding its properties is to associate to it a moduli space of “locally isomorphic” tilings, and to then study the topology of this “tiling space”. A common topological invariant to consider in this context is the Čech cohomology. I will describe how, using a Poincaré Duality like result, one may describe these groups in a very geometric way using cellular chains (although non-compactly supported ones) of the Euclidean space which are “pattern equivariant” (PE) with respect to the tiling. I will show how, with this perspective, one may give a simple method to compute these groups for hierarchical tilings. If time allows, I will also discuss the rotationally invariant PE complexes, which seem to capture extra information about the rotationally invariant tilings in the tiling space to the Čech cohomology groups. These groups can be incorporated into a spectral sequence converging to the cohomology of the tiling space of rigid motions of a tiling.