Cohomology for Anyone

TL;DR

This paper introduces a topological perspective on crystallography, connecting algebraic topology concepts like cohomology to the study of space groups, accessible to students with basic group theory knowledge.

Contribution

It presents a novel approach integrating algebraic topology into crystallography, making advanced mathematical tools accessible to physics and crystallography students.

Findings

Topology relates to global properties of space groups.

Cohomology provides new insights into crystallographic structures.

Approach is accessible without advanced mathematics.

Abstract

Crystallography has proven a rich source of ideas over several centuries. Among the many ways of looking at space groups, N. David Mermin has pioneered the Fourier-space approach. Recently, we have supplemented this approach with methods borrowed from algebraic topology. We now show what topology, which studies global properties of manifolds, has to do with crystallography. No mathematics is assumed beyond what the typical physics or crystallography student will have seen of group theory; in particular, the reader need not have any prior exposure to topology or to cohomology of groups.