Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups
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-...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2017 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2017
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/148623 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | 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. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862590331009105920 |
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| author | Gomi, K. |
| author_facet | Gomi, K. |
| citation_txt | 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2025-11-27T04:31:20Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-148623 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-27T04:31:20Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Gomi, K. 2019-02-18T16:44:13Z 2019-02-18T16:44:13Z 2017 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. 1815-0659 2010 Mathematics Subject Classification: 53C08; 55N91; 20H15; 81T45 DOI:10.3842/SIGMA.2017.014 https://nasplib.isofts.kiev.ua/handle/123456789/148623 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. 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. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups Article published earlier |
| spellingShingle | Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups Gomi, K. |
| title | Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups |
| title_full | Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups |
| title_fullStr | Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups |
| title_full_unstemmed | Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups |
| title_short | Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups |
| title_sort | twists on the torus equivariant under the 2-dimensional crystallographic point groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148623 |
| work_keys_str_mv | AT gomik twistsonthetorusequivariantunderthe2dimensionalcrystallographicpointgroups |