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SIGMA, 2017, Volume 13, 014, 38 pages (Mi sigma1214)  

This article is cited in 6 scientific papers (total in 6 papers)

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

Kiyonori Gomi

Department of Mathematical Sciences, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan

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.

Keywords: twist; Borel equivariant cohomology; crystallographic group; topological insulator.

Funding Agency Grant Number
Japan Society for the Promotion of Science KAKENHI No. JP15K04871
This work is supported by JSPS KAKENHI Grant Number JP15K04871.


DOI: https://doi.org/10.3842/SIGMA.2017.014

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Bibliographic databases:

ArXiv: 1509.09194
MSC: 53C08; 55N91; 20H15; 81T45
Received: February 17, 2016; in final form March 3, 2017; Published online March 8, 2017
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Citation: Kiyonori Gomi, “Twists on the Torus Equivariant under the $2$-Dimensional Crystallographic Point Groups”, SIGMA, 13 (2017), 014, 38 pp.

Citation in format AMSBIB
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\by Kiyonori~Gomi
\paper Twists on the Torus Equivariant under the $2$-Dimensional Crystallographic Point Groups
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\yr 2017
\vol 13
\papernumber 014
\totalpages 38
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\crossref{https://doi.org/10.3842/SIGMA.2017.014}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Po H.Ch., Vishwanath A., Watanabe H., “Complete Theory of Symmetry-Based Indicators of Band Topology”, Nat. Commun., 8 (2017), 50  crossref  isi
    2. Shiozaki K., Sato M., Gomi K., “Topological Crystalline Materials: General Formulation, Module Structure, and Wallpaper Groups”, Phys. Rev. B, 95:23 (2017), 235425  crossref  isi
    3. H. Ch. Po, H. Watanabe, A. Vishwanath, “Fragile topology and Wannier obstructions”, Phys. Rev. Lett., 121:12 (2018), 126402  crossref  isi  scopus
    4. Gomi K., Thiang G.Ch., “Crystallographic Bulk-Edge Correspondence: Glide Reflections and Twisted Mod 2 Indices”, Lett. Math. Phys., 109:4 (2019), 857–904  crossref  mathscinet  zmath  isi  scopus
    5. Gomi K., Thiang G.Ch., “Crystallographic T-Duality”, J. Geom. Phys., 139 (2019), 50–77  crossref  mathscinet  isi  scopus
    6. Else V D., Po H.Ch., Watanabe H., “Fragile Topological Phases in Interacting Systems”, Phys. Rev. B, 99:12 (2019), 125122  crossref  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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