An index theorem to solve the gap-labeling conjecture for the pinwheel tiling

TL;DR

This paper establishes a new index theorem linking the K0-group of the pinwheel tiling's C*-algebra to cohomological data, advancing the proof of the gap-labeling conjecture for this tiling.

Contribution

It introduces an index theorem that relates the K0-group of the pinwheel tiling's C*-algebra to cohomology, enabling explicit computation of the gap-labeling.

Findings

K0-group of the C*-algebra is Z + Z^6 plus a cohomological group

The trace map's image of Z is zero, and of Z^6 is within patch frequencies

Application of Connes' measured index theorem connects K0-image to a computable cohomological formula

Abstract

In this paper, we study the K0-group of the C?-algebra associated to a pinwheel tiling. We prove that it is given by the sum of Z + Z^6 with a cohomological group. The C?-algebra is endowed with a trace that induces a linear map on its K0-group. We then compute explicitly the image, under this map, of the summand Z+Z^6, showing that the image of Z is zero and the image of Z^6 is included in the module of patch frequencies of the pinwheel tiling. We finally prove that we can apply the measured index theorem due to A. Connes to relate the image of the last summand of the K0-group to a cohomological formula which is more computable. This is the first step in the proof of the gap-labeling conjecture for the pinwheel tiling.