Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups

TL;DR

This paper classifies the types of twists in equivariant topological K-theory for 2D tori under all crystallographic point group actions, computing equivariant cohomology and spectral sequences.

Contribution

It provides a comprehensive classification of twists for torus actions by 2D space group point groups, including explicit cohomology calculations.

Findings

Classified four types of twists for torus with crystallographic group actions.

Computed equivariant cohomology up to degree three for all cases.

Analyzed twists in relation to Freed-Moore K-theory with local coefficients.

Abstract

A twist is a datum playing a role of a local system for topological $K$-theory. In equivariant setting, twists are classified into four types according to how they are realized geometrically. This paper lists the possible types of twists for the torus with the actions of the point groups of all the 2-dimensional space groups (crystallographic groups), or equivalently, the torus with the actions of all the possible finite subgroups in its mapping class group. This is carried out by computing Borel's equivariant cohomology and the Leray-Serre spectral sequence. As a byproduct, the equivariant cohomology up to degree three is determined in all cases. The equivariant cohomology with certain local coefficients is also considered in relation to the twists of the Freed-Moore $K$-theory.