Revisiting the topological classification of defects in crystals

TL;DR

This paper introduces a comprehensive topological framework for classifying defects in crystals using algebraic topology tools, highlighting both its applications and limitations in various geometric settings.

Contribution

It develops a general topological classification method for crystal defects and critically examines its assumptions and inconsistencies in lattice contexts.

Findings

Application of homotopy and cohomology groups to classify defects.

Explicit calculations for various geometric structures.

Identification of methodological limitations in crystal lattice classification.

Abstract

A general theory of topological classification of defects is introduced. We illustrate the application of tools from algebraic topology, including homotopy and cohomology groups, to classify defects including several explicit calculations for crystals in $\mathbb{R}^2$, $S^2$, 2-dimensional cylinder, 2-dimensional annulus, and 2-tori. A set of physically motivated assumptions is formulated in order to justify the classification process and also to expose certain inherent inconsistencies in the considered methodology, particularly for crystal lattices.