Maximal equicontinuous factors and cohomology for tiling spaces

TL;DR

This paper investigates the relationship between the cohomology of tiling spaces and their maximal equicontinuous factors, revealing injectivity in degree one and torsion-free cokernels, with examples showing limitations of direct summand decompositions.

Contribution

It provides new insights into the cohomological properties of maximal equicontinuous factors in tiling spaces, including injectivity and torsion-free cokernels, and highlights limitations through examples.

Findings

The induced cohomology map is injective in degree one.

The cokernel of the map is torsion free.

Cohomology of the maximal equicontinuous factor may not be a direct summand.

Abstract

We study the homomorphism induced on cohomology by the maximal equicontinuous factor map of a tiling space. We will see that this map is injective in degree one and has torsion free cokernel. We show by example, however, that the cohomology of the maximal equicontinuous factor may not be a direct summand of the tiling cohomology.