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

Symmetry Integrability and Geometry-Methods and Applications

article

Kiyonori Gomi¹ 

[1] Department of Mathematical Sciences, Shinshu University

关键词: twist;    Borel equivariant cohomology;    crystallographic group;    topological insulator;   

DOI  :  10.3842/SIGMA.2017.014

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown