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Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2017, том 13
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Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups
Інший ID:
2010 Mathematics Subject Classification: 53C08; 55N91; 20H15; 81T45
DOI:10.3842/SIGMA.2017.014
DOI:10.3842/SIGMA.2017.014
Посилання:
Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups / K. Gomi // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 29 назв. — англ.
Підтримка:
I would like to thank K. Shiozaki and M. Sato for valuable discussions. I would also thank
G.C. Thiang, D. Tamaki, anonymous referees and an editor for helpful criticisms and comments.
This work is supported by JSPS KAKENHI Grant Number JP15K04871.
Дата:
2017
Переглядів:
414
Завантажень:
222
Анотація:
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.
