# On the relationship between topological and geometric defects

## TL;DR

This paper reviews the formal classification of topological defects in solids, contrasting them with geometric defects, and discusses how symmetry-breaking conditions influence their formation and impact material properties.

## Contribution

It provides a clear distinction between topological and geometric defects, with formal definitions, examples, and implications for material properties based on symmetry considerations.

## Key findings

- Topological defects are classified using homotopy theory.
- Symmetry-breaking conditions determine defect formation.
- Topological defects influence material properties significantly.

## Abstract

The study of topology in solids is undergoing a renaissance following renewed interest in the properties of ferroic domain walls as well as recent discoveries regarding topological insulators and skyrmionic lattices. Each of these systems possess a property that is `protected' in a symmetry sense, and is defined rigorously using a branch of mathematics known as topology. In this article we review the formal definition of topological defects as they are classified in terms of homotopy theory, and discuss the precise symmetry-breaking conditions that lead to their formation. We distinguish topological defects from geometric defects, which arise from the details of the stacking or structure of the material but are not protected by symmetry. We provide simple material examples of both topological and geometric defect types, and discuss the implications of the classification on the resulting material properties.