# Robustness of Topological Phases on Aperiodic Lattices

## TL;DR

This paper investigates how topological phases behave on aperiodic lattices using advanced mathematical tools like *-homomorphisms and K-theory.

## Contribution

The paper introduces a novel method to analyze topological phases using *-homomorphisms between groupoid and coarse-geometric models.

## Key findings

- Strong topological phases in the groupoid model are detected by position spectral triples.
- Phases from stacking along a Delone set are weak in the coarse-geometric sense.

## Abstract

We study the robustness of topological phases on aperiodic lattices by constructing *-homomorphisms from the groupoid model to the coarse-geometric model of observable C*-algebras. These *-homomorphisms induce maps in K-theory and Kasparov theory. We show that the strong topological phases in the groupoid model are detected by position spectral triples. We show that topological phases coming from stacking along another Delone set are always weak in the coarse-geometric sense.