How big is a tiling's return module?

TL;DR

This paper investigates the relationship between the rank of a tiling's return module and its first Čech cohomology, establishing bounds and conditions under which they are equal, based on tile geometry and patch size.

Contribution

It demonstrates that for generic tile shapes and large patches, the return module's rank equals the first Čech cohomology's rank, linking geometric and topological properties.

Findings

The return module's rank depends on tile geometry, not topology.

For large patches, the return module's rank is bounded above by cohomology rank.

In generic cases, the return module's rank equals the cohomology rank.

Abstract

The rank of a tiling's return module depends on the geometry of its tiles and is not a topological invariant. However, the rank of the first \v Cech cohomology $\check H^1(\Omega)$ gives upper and lower bounds for the size of the return module. For all sufficiently large patches, the rank of the return module is at most the same as the rank of the cohomology. For a generic choice of tile shapes and an arbitrary reference patch, the rank of the return module is at least the rank of $\check H^1(\Omega)$. Therefore, for generic tile shapes and sufficiently large patches, the rank of the return module is equal to the rank of $\check H^1(\Omega)$.