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

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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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.

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\Bibitem{Gom17} \by Kiyonori~Gomi \paper Twists on the Torus Equivariant under the $2$-Dimensional Crystallographic Point Groups \jour SIGMA \yr 2017 \vol 13 \papernumber 014 \totalpages 38 \mathnet{http://mi.mathnet.ru/sigma1214} \crossref{https://doi.org/10.3842/SIGMA.2017.014} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000396322200001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85014827494} 

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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
2. Shiozaki K., Sato M., Gomi K., “Topological Crystalline Materials: General Formulation, Module Structure, and Wallpaper Groups”, Phys. Rev. B, 95:23 (2017), 235425
3. H. Ch. Po, H. Watanabe, A. Vishwanath, “Fragile topology and Wannier obstructions”, Phys. Rev. Lett., 121:12 (2018), 126402
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
5. Gomi K., Thiang G.Ch., “Crystallographic T-Duality”, J. Geom. Phys., 139 (2019), 50–77
6. Else V D., Po H.Ch., Watanabe H., “Fragile Topological Phases in Interacting Systems”, Phys. Rev. B, 99:12 (2019), 125122
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