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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Datum:	2017
1. Verfasser:	Gomi, K.
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Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1486232019-02-19T01:26:55Z Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups Gomi, K. 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. 2017 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/148623 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
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.
format	Article
author	Gomi, K.
spellingShingle	Gomi, K. Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Gomi, K.
author_sort	Gomi, K.
title	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_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_sort	twists on the torus equivariant under the 2-dimensional crystallographic point groups
publisher	Інститут математики НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/148623
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.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT gomik twistsonthetorusequivariantunderthe2dimensionalcrystallographicpointgroups
first_indexed	2025-07-12T19:48:23Z
last_indexed	2025-07-12T19:48:23Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 014, 38 pages 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 E-mail: kgomi@math.shinshu-u.ac.jp URL: http://math.shinshu-u.ac.jp/~kgomi/ Received February 17, 2016, in final form March 03, 2017; Published online March 08, 2017 https://doi.org/10.3842/SIGMA.2017.014 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. Key words: twist; Borel equivariant cohomology; crystallographic group; topological insula- tor 2010 Mathematics Subject Classification: 53C08; 55N91; 20H15; 81T45 1 Introduction Topological K-theory has recently been recognized as a useful tool for a classification of topo- logical insulators in condensed matter physics. In Kitaev’s 10-fold way [17], the usual complex K-theory and also KO or Atiyah’s KR-theory are used. These classifications are in some sense the most simple cases, and a recent study of topological insulators focuses on more complicated cases. Such complicated cases arise when we take the symmetry of quantum systems into ac- count. Then equivariant K-theory and its twisted version naturally fit into the classification scheme of such systems [8]. Actually, as will be explained in Section 2, a certain quantum sys- tem on the d-dimensional space Rd invariant under a space group provides a K-theory class on the d-dimensional torus T d equivariant under the point group of the space group. If the space group is nonsymmorphic, then the equivariant K-class is naturally twisted. In the case of d = 2, such (twisted) equivariant K-theories are computed for the 17 classes of 2-dimensional space groups, in view of the classification of topological crystalline insulators [27, 28]. An outcome of these computations of twisted equivariant K-theories is the discovery of topological insulators which are essentially classified by Z2 but do not require the so-called time-reversal symmetry or the particle-hole symmetry [26]. This type of topological insulators is new in the sense that the known topological insulators essentially classified by Z2 so far require the time-reversal symmetry or the particle-hole symmetry. The understanding of the importance of twisted equivariant K-theory in condensed matter physics leads to a mathematically natural issue: determining the possible ‘twists’ for equivariant K-theory. To explain this issue more concretely, let us recall that twisted K-theory [5, 22] is in some sense a K-theory with ‘local coefficients’. The datum playing the role of a ‘local system’ mailto:kgomi@math.shinshu-u.ac.jp http://math.shinshu-u.ac.jp/~kgomi/ https://doi.org/10.3842/SIGMA.2017.014 2 K. Gomi admits various geometric realizations. In this paper, we realize them by twists in the sense of [7]. If a compact Lie group G acts on a space X, then graded twists on X are classified by the Borel equivariant cohomology H1 G(X;Z2)×H3 G(X;Z). Similarly, ungraded twists are classified by H3 G(X;Z), on which we focus for a moment. (Sometimes H0 G(X;Z) may be included in the twists, but we regard it as the degree of the K-theory.) By definition, the Borel equivariant cohomologyHn G(X;Z) is the usual cohomologyHn(EG×G X;Z) of the Borel construction EG ×G X, which is the quotient of EG × X by the diagonal G-action, where EG is the total space of the universal G-bundle EG→ BG. Associated to the Borel construction is the fibration X → EG ×G X → BG, and hence the Leray–Serre spectral sequence Ep,qr that converges to the graded quotient of a filtration Hn G(X;Z) ⊃ F 1Hn G(X;Z) ⊃ F 2Hn G(X;Z) ⊃ · · · ⊃ Fn+1Hn G(X;Z) = 0. One can interpret F pH3 G(X;Z) ⊂ H3 G(X;Z) geometrically in the classification of twists, and there are four types (see Section 3 for details): (i) Twists which can be represented by group 2-cocycles of G with coefficients in the trivial G-module U(1). These twists are classified by F 3H3 G(X;Z). (ii) Twists which can be represented by group 2-cocycles of G with coefficients in the group C(X,U(1)) of U(1)-valued functions on X regarded as a (right) G-module by pull-back. These twists are classified by F 2H3 G(X;Z). (iii) Twists which can be represented by central extensions of the groupoid X//G. These twists are classified by F 1H3 G(X;Z). (iv) Twists of general type, classified by F 0H3 G(X;Z) = H3 G(X;Z). The equivariant twists on T d arising from quantum systems on Rd, to be explained in Sec- tion 2, belong to F 2H3 P (T d;Z) with P the point group of a d-dimensional space group S, and so are the twists considered in [27]. Now, the mathematical issue is whether the twists arising in this way cover all the possibilities or not. The present paper answers this question in the case of d = 2 by a theorem (Theorem 1.1). To state the theorem, let S be a 2-dimensional space group, which is also known as a 2- dimensional crystallographic group, a plane symmetry group, a wallpaper group, and so on. It is a subgroup of the Euclidean group R2oO(2) of isometries of R2, and is an extension of a finite group P ⊂ O(2) called the point group by a rank 2 lattice Π ∼= Z2 of translations of R2: 1 −→ R2 −→ R2 o O(2) −→ O(2) −→ 1 ∪ ∪ ∪ 1 −→ Π −→ S −→ P −→ 1. Being a normal subgroup of S, the lattice Π ⊂ R2 is preserved by the action of P on R2 through the inclusion P ⊂ O(2) and the standard left action of O(2) on R2. This induces the left action of P on the torus T 2 = R2/Π that we will consider. Since P is a finite subgroup of O(2), it is the cyclic group Zn of order n or the dihedral group Dn = 〈C, σ \|Cn, σ2, σCσC〉 of degree n and order 2n. The classification of 2-dimensional space groups has long been known, and there are 17 types [12, 24], which we label following [23]. Notice that some space groups share the same point group action on T 2, and there arise 13 distinct finite group actions on the torus. These actions realize essentially all the possible finite subgroups in the mapping class group of the torus [20], which is isomorphic to GL(2,Z) as is well known [21]. Theorem 1.1. Let P be the point group of one of the 2-dimensional space groups S, acting on T 2 = R2/Π via P ⊂ O(2) as above. Then, H3 P (T 2;Z) = F 0H3 P (T 2;Z) = F 1H3 P (T 2;Z). This cohomology group and its subgroups F pH3 P (T 2;Z) are as in Fig. 1. Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 3 Space group S P ori H3 P (T 2;Z) F 2 F 3 E1,2 ∞ E2,1 ∞ p1 1 + 0 0 0 0 0 p2 Z2 + 0 0 0 0 0 p3 Z3 + 0 0 0 0 0 p4 Z4 + 0 0 0 0 0 p6 Z6 + 0 0 0 0 0 pm/pg D1 − Z⊕2 2 Z2 0 Z2 Z2 cm D1 − Z2 0 0 Z2 0 pmm/pmg/pgg D2 − Z⊕4 2 Z⊕3 2 Z2 Z2 Z⊕2 2 cmm D2 − Z⊕2 2 Z2 Z2 Z2 0 p3m1 D3 − Z2 0 0 Z2 0 p31m D3 − Z2 0 0 Z2 0 p4m/p4g D4 − Z⊕3 2 Z⊕2 2 Z2 Z2 Z2 p6m D6 − Z⊕2 2 Z2 Z2 Z2 0 Figure 1. The list of H3 P (T 2;Z) and its subgroups F p = F pH3 P (T 2;Z) for the point group P of each space 2-dimensional space group S. The E∞-term of the Leray–Serre spectral sequence is related to these subgroups by Ep,3−p∞ ∼= F p/F p+1. The column “ori” indicates “+” if P preserves the orientation of T 2 and “−” if not. The same actions of point groups on T 2 are grouped in a row. Nonsymmorphic groups are pg, pmg, pgg and p4g. Corollary 1.2. Under the same hypothesis as in Theorem 1.1, (a) All the twists can be represented by central extensions of T 2//P . In particular, there is no non-trivial twist if P preserves the orientation of T 2. (b) If P does not preserve the orientation of T 2, then there are twists which can be represented by central extensions of T 2//P but not by group 2-cocycles of P . (c) The subgroup F 2H3 P (T 2;Z) is generated by the twists represented by: – group 2-cocycle of P with values in C(T 2, U(1)) induced from a nonsymmorphic space group S′ such that the action of its point group P ′ ∼= P on T 2 is the same as P ; and – group 2-cocycle of P with values in U(1). As a result, all the twists classified by F 2H3 P (T 2;Z) are relevant to topological insulators, whereas there actually exist other twists which cannot be realized by group cocycles. At present their roles in condensed matter theory seem to be unknown. Theorem 1.1 follows from case by case computations of the equivariant cohomology H3 P (T 2;Z) and the Leray–Serre spectral sequence. Roughly, there are three methods according to the nature of the point group actions: The first method is applied to the cases where the torus T 2 is the product of circles with P -actions, i.e., the cases of the Z2-actions arising from p2 and pm/pg. In these cases, the equivariant cohomology is computed by means of the splitting of the Gysin exact sequence, as detailed in [10]. The second method is applied to the cases where the point group has no element of order 3. In these cases, the torus T 2 admits an equivariant stable splitting. As a result, the equivariant cohomology of T 2 admits the corresponding splitting, and the Leray–Serre spectral sequence turns out to be trivial. Finally, the third method is applied 4 K. Gomi to the remaining cases. In these cases, we take a P -CW decomposition of T 2 to compute the equivariant cohomology by using the Mayer–Vietoris exact sequence and the exact sequence for a pair, and then study the Leray–Serre spectral sequence. In principle, the third method is the most basic, and hence is applied to all the cases. However, to simplify the computations, we use other methods. These computations contain enough information to determine the equivariant cohomology Hn P (T 2;Z), (n ≤ 2) of the torus with the actions of the possible finite subgroups in the mapping class group GL(2,Z). Theorem 1.3. Let P be the point group of one of the 2-dimensional space groups S, acting on T 2 = R2/Π via P ⊂ O(2). For n ≤ 3, the P -equivariant cohomology Hn P (T 2;Z) is as given in Fig. 2. Space group S P ori H0 P (T 2) H1 P (T 2) H2 P (T 2) H3 P (T 2) p1 1 + Z Z⊕2 Z 0 p2 Z2 + Z 0 Z⊕ Z⊕3 2 0 p3 Z3 + Z 0 Z⊕ Z⊕2 3 0 p4 Z4 + Z 0 Z⊕ Z2 ⊕ Z4 0 p6 Z6 + Z 0 Z⊕ Z6 0 pm/pg D1 − Z Z Z⊕2 2 Z⊕2 2 cm D1 − Z Z Z2 Z2 pmm/pmg/pgg D2 − Z 0 Z⊕4 2 Z⊕4 2 cmm D2 − Z 0 Z⊕3 2 Z⊕2 2 p3m1 D3 − Z 0 Z2 Z2 p31m D3 − Z 0 Z3 ⊕ Z2 Z2 p4m/p4g D4 − Z 0 Z⊕3 2 Z⊕3 2 p6m D6 − Z 0 Z⊕2 2 Z⊕2 2 Figure 2. The list of equivariant cohomology up to degree 3. Note that some specific cases are computed in the literature (e.g., [1, 2, 3]). So far we focused on ungraded twists. To complete the classification of P -equivariant twists on T 2, we need to compute the equivariant first cohomology with coefficients in Z2, which provides the information on ‘gradings’ of a twist. But, the computation is immediately completed by a simple application of the universal coefficient theorem to Theorem 1.3. Notice that the equivariant cohomology H1 P (T 2;Z2) also admits a filtration H1 P ( T 2;Z2 ) = F 0H1 P ( T 2;Z2 ) ⊃ F 1H1 P ( T 2;Z2 ) ⊃ F 2H1 P ( T 2;Z2 ) = 0. Because the degree in question is 1, the degeneration of the Leray–Serre spectral sequence gives the identification F 1H1 P ( T 2;Z2 ) = Hom ( P,Z2 ) = H1 P ( pt;Z2 ) , which is a direct summand of H1 P (T 2;Z2) and is also computed immediately by using the knowl- edge of the equivariant cohomology of the space consisting of one point, pt = {one point}, in Section 4.1. Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 5 Corollary 1.4. Let P be the point group of one of the 2-dimensional space groups S, acting on T 2 = R2/Π via P ⊂ O(2). Then the P -equivariant cohomology H1 P (T 2;Z2) is as in Fig. 3. Space group S P ori H1 P (T 2;Z2) F 1H1 P (T 2;Z2) E1,0 ∞ p1 1 + Z⊕2 2 0 Z⊕2 2 p2 Z2 + Z⊕3 2 Z2 Z⊕2 2 p3 Z3 + 0 0 0 p4 Z4 + Z⊕2 2 Z2 Z2 p6 Z6 + Z2 Z2 0 pm/pg D1 − Z⊕3 2 Z2 Z⊕2 2 cm D1 − Z⊕2 2 Z2 Z2 pmm/pmg/pgg D2 − Z⊕4 2 Z⊕2 2 Z⊕2 2 cmm D2 − Z⊕3 2 Z⊕2 2 Z2 p3m1 D3 − Z2 Z2 0 p31m D3 − Z2 Z2 0 p4m/p4g D4 − Z⊕3 2 Z⊕2 2 Z2 p6m D6 − Z⊕2 2 Z⊕2 2 0 Figure 3. The list of first equivariant cohomology groups with coefficients Z2. The quotient group H1 P (T 2;Z2)/F 1H1 P (T 2;Z2) is denoted with E1,0 ∞ . The grading of twists classified by F 1H1 P (T 2;Z2) = Hom(P,Z2) plays a role in a quantum system with symmetry (see Remark 2.2). However, there are other gradings generally, and their roles in condensed matter theory is unknown. As is mentioned, Atiyah’s KR-theory is also applied to the classification of topological in- sulators. The symmetry of KR-theory however concerns Z2-actions only, and its use is limited to rather simple cases. To take more general symmetries into account, Freed and Moore intro- duced a K-theory which unifies KR-theory and equivariant K-theory [8]. Their K-theory is defined for a space X with an action of a compact Lie group G equipped with a homomorphism φ : G→ Z2. The K-theory of Freed–Moore reduces to the G-equivariant K-theory if φ is trivial, and to the KR-theory if G = Z2 and φ non-trivial. There also exists the notion of twists for the Freed–Moore K-theory. A computation of the twisted Freed–Moore K-theory is carried out in [27], leading to the discovery of a novel Z4-phase. The knowledge about the twists of the Freed–Moore K-theory has therefore potential impor- tance to condensed matter physics as well, and the present paper provides it also in the case where X is the torus T 2 and G is the point group P of a 2-dimensional space group. Notice that the classification of the twists for the Freed–Moore K-theory parallels that of the twists for equivariant K-theory (actually a generalization). In general, the graded twists are classified by H1 G(X;Z2)×H3 G(X;Zφ) and the ungraded twists by H3 G(X;Zφ). Here Zφ denotes a local system for the Borel equivariant cohomology associated to the G-module Zφ such that its underlying group is Z and G acts via φ : G→ Z2. The cohomology group Hn G(X;Zφ) also admits a filtration Hn G(X;Zφ) ⊃ F 1Hn G(X;Zφ) ⊃ F 2Hn G(X;Zφ) ⊃ · · · ⊃ Fn+1Hn G(X;Zφ) = 0. The associated graded quotient is computed by the Leray–Serre spectral sequence, and the subgroups F pH3 G(X;Zφ) ⊂ H3 G(X;Zφ) have geometric interpretations as well (Proposition 5.1). 6 K. Gomi To state our results in the ‘twisted’ case, we introduce the following definition for the point group P of a 2-dimensional space group S that admits a non-trivial homomorphism φ : P → Z2. • In the cases of p2, p4 and p6, the point group P is the cyclic group Z2m = 〈C \|C2m〉 of even order. We write φ1 : Z2m → Z2 for the unique non-trivial homomorphism given by φ1(C) = −1. • In the other case, the point group P is the dihedral group Dn = 〈C, σ \|Cn, σ2, σCσC〉 of degree n and order 2n, and Dn is embedded into O(2) so that C is a rotation of R2 and σ is a reflection. We define φ0 : Dn → Z2 to be the composition of the inclusion Dn → O(2) and det : O(2) → Z2. Put differently, φ0(C) = 1 and φ0(σ) = −1. This provides the unique non-trivial homomorphism Dn → Z2 if n is odd. In the case of even n, we define two more non-trivial homomorphisms φi : Dn → Z2 by{ φ1(C) = −1, φ1(σ) = 1, { φ2(C) = −1, φ2(σ) = −1. Theorem 1.5. Let P be the point group of one of the 2-dimensional space groups S, acting on T 2 = R2/Π via P ⊂ O(2), and φ : P → Z2 a non-trivial homomorphism. Then, H3 P (T 2;Zφ) = F 0H3 P (T 2;Zφ) = F 1H3 P (T 2;Zφ). This cohomology group and its subgroups F pH3 P (T 2;Zφ) are as in Fig. 4. Space group S P φ H3 P (T 2;Zφ) F 2 F 3 E1,2 ∞ E2,1 ∞ p2 Z2 φ1 Z⊕4 2 Z⊕3 2 Z2 Z2 Z⊕2 2 p4 Z4 φ1 Z⊕2 2 Z2 Z2 Z2 0 p6 Z6 φ1 Z⊕2 2 Z2 Z2 Z2 0 pm/pg D1 φ0 Z⊕2 2 Z⊕2 2 Z2 0 Z2 cm D1 φ0 Z2 Z2 Z2 0 0 pmm/pmg/pgg D2 φ0 Z⊕4 2 Z⊕4 2 Z⊕2 2 0 Z⊕2 2 pmm/pmg/pgg D2 φ1, φ2 Z⊕6 2 Z⊕5 2 Z⊕2 2 Z2 Z⊕3 2 cmm D2 φ0 Z⊕2 2 Z⊕2 2 Z⊕2 2 0 0 cmm D2 φ1, φ2 Z⊕4 2 Z⊕3 2 Z⊕2 2 Z2 Z2 p3m1 D3 φ0 Z2 Z2 Z2 0 0 p31m D3 φ0 Z2 Z2 Z2 0 0 p4m/p4g D4 φ0 Z⊕3 2 Z⊕3 2 Z⊕2 2 0 Z2 p4m/p4g D4 φ1, φ2 Z⊕4 2 Z⊕3 2 Z⊕2 2 Z2 Z2 p6m D6 φ0 Z⊕2 2 Z⊕2 2 Z⊕2 2 0 0 p6m D6 φ1 Z⊕3 2 Z⊕2 2 Z⊕2 2 Z2 0 p6m D6 φ2 Z⊕3 2 Z⊕2 2 Z⊕2 2 Z2 0 Figure 4. The list of H3 P (T 2;Zφ) and its subgroups F p = F pH3 P (T 2;Zφ). The E∞-term of the Leray– Serre spectral sequence is related to these subgroups by Ep,3−p∞ ∼= F p/F p+1. It should be noticed that the action of the point group P on the torus relevant to an ap- plication of the Freed–Moore K-theory to condensed matter physics is the one modified by Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 7 a non-trivial homomorphism φ : P → Z2. Some of such modified actions differ from those given by the inclusion P ⊂ O(2), and hence are not covered in Theorem 1.5. The modified actions should be understood in the context of the so-called magnetic space groups (or colour symmetry groups [25]), and the cohomology as well as the K-theory equivariant under the groups deserve to be subjects of a future work. One may notice that there are more twists for the Freed–Moore K-theory in comparison with the twists for equivariant K-theory. At present, we lack such an understanding of twists as in Corollary 1.2(c) in relation with the nonsymmorphic nature of space groups. The method for computing H3 P (T 2;Zφ) and its filtration is similar to the one computing H3 P (T 2;Z). In the computation, the cohomology Hn P (T 2;Zφ) for n ≤ 2 is also determined, as summarized below: Theorem 1.6. Let P be the point group of one of the 2-dimensional space groups S, acting on T 2 = R2/Π via P ⊂ O(2). For n ≤ 3, the P -equivariant cohomology Hn P (T 2;Zφ) with coefficients in the local system Zφ induced from a non-trivial homomorphism φ : P → Z2 is as in Fig. 5. Space group S P φ H0 P (T 2) H1 P (T 2) H2 P (T 2) H3 P (T 2) p2 Z2 φ1 0 Z2 ⊕ Z⊕2 0 Z⊕4 2 p4 Z4 φ1 0 Z2 Z2 Z⊕2 2 p6 Z6 φ1 0 Z2 Z3 Z⊕2 2 pm/pg D1 φ0 0 Z2 ⊕ Z Z2 ⊕ Z Z⊕2 2 cm D1 φ0 0 Z2 ⊕ Z Z Z2 pmm/pmg/pgg D2 φ0 0 Z2 Z⊕3 2 ⊕ Z Z⊕4 2 pmm/pmg/pgg D2 φ1, φ2 0 Z2 ⊕ Z Z⊕2 2 Z⊕6 2 cmm D2 φ0 0 Z2 Z⊕2 2 ⊕ Z Z⊕2 2 cmm D2 φ1, φ2 0 Z2 ⊕ Z Z2 Z⊕4 2 p3m1 D3 φ0 0 Z2 Z⊕2 3 ⊕ Z Z2 p31m D3 φ0 0 Z2 Z3 ⊕ Z Z2 p4m/p4g D4 φ0 0 Z2 Z4 ⊕ Z2 ⊕ Z Z⊕3 2 p4m/p4g D4 φ1, φ2 0 Z2 Z⊕2 2 Z⊕4 2 p6m D6 φ0 0 Z2 Z6 ⊕ Z Z⊕2 2 p6m D6 φ1 0 Z2 Z2 ⊕ Z3 Z⊕3 2 p6m D6 φ2 0 Z2 Z2 Z⊕3 2 Figure 5. The list of equivariant cohomology with local coefficients. Finally, we make comments about the generalizations. To compute cohomology groups of the higher-dimensional tori which are equivariant under space groups, we can in principle apply the three methods in this paper. The first and second methods would be generalized without difficulty. The third method will however get more difficult, because we need a P -CW decom- position of a higher-dimensional torus, which becomes more complicated than decompositions in the 2-dimensional case. As is suggested by Corollary 1.4, there are local systems for the Borel equivariant cohomology other than Zφ associated to a homomorphism φ : P → Z2. For the cohomology with such a local system, the notion of reduced cohomology does not make sense. 8 K. Gomi This prevents us from using the second method based on the equivariant stable splitting of the torus, forcing us to use a P -CW decomposition. The outline of this paper is as follows: In Section 2, we explain how a certain quantum system leads to a twist and defines a twisted K-class, mainly based on a formulation in [8]. At the end of this section, a summary of relationship among some natural actions of point groups on tori is included. In Section 3, we review the Leray–Serre spectral sequence for Borel equivariant cohomology and the notion of twists for equivariant K-theory. The geometric interpretation of the filtration of the degree 3 equivariant cohomology is also provided here, after a general property of the spectral sequence is established. Then, in Section 4, we prove Theorems 1.1 and 1.3. To keep readability of this paper, we provide the detail of computations only in the cases p2, p4m/p4g and p6m. (The detail of the other cases can be found in old versions of arXiv:1509.09194.) Section 5 concerns the equivariant cohomology with the twisted coeffi- cient Zφ. We state direct generalizations of some results in the untwisted case, and then prove Theorems 1.5 and 1.6. To keep readability again, we give the details of the computation only in the case of p6m with φ2. Finally, for convenience, the point group actions of 2-dimensional space groups are listed in Appendix. Throughout, familiarity with basic algebraic topology [4, 11] will be supposed. 2 From quantum systems to twisted K-theory We here illustrate how twisted equivariant K-theory arises from a quantum system with sym- metry, mainly based on a formulation in [8]. (We refer the reader to [29] for a C∗-algebraic approach.) 2.1 Setting Let us consider the following mathematical setting: • A lattice Π ⊂ Π⊗Z R = Rd of rank d. • A subgroup S of the Euclidean group Rd o O(d) of Rd which is an extension of a finite group P ⊂ O(d) by Π: 1 −→ Rd −→ Rd o O(d) −→ O(d) −→ 1 ∪ ∪ ∪ 1 −→ Π −→ S π−→ P −→ 1. • A unitary representation U : P → U(V ) on a finite-dimensional Hermitian vector space V . The group S is nothing but a d-dimensional space group, and P is called the point group of S. When S is the semi-direct product of P and Π, it is called symmorphic, otherwise nonsymmor- phic. Based on the mathematical setting above, we can introduce a quantum system on Rd which has S as its symmetry and V as its internal freedom: • The ‘quantum Hilbert space’ consisting of ‘wave functions’ is the L2-space L2(Rd, V ), on which g ∈ S acts by ψ(x) 7→ (ρ(g)ψ)(x) = U(π(g))ψ(g−1x). • The ‘Hamiltonian’ is a self-adjoint operator H on L2(Rd, V ) invariant under the S-action: H ◦ ρ(g) = ρ(g) ◦ H. A typical form of H is H = ∆ + Φ, where ∆ = ∑ ∂2/∂x2 i is the Laplacian and Φ: Rd → End(V ) is a potential term. Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 9 2.2 Bloch transformation Even if the Hamiltonian H is invariant under the translation of Π, a solution ψ to the ‘time- independent Schrödinger equation’ Hψ = Eψ with E ∈ R is not necessarily S-invariant. The so-called ‘Bloch transformation’ allows us to deal with such a situation. Let Π̂ = Hom(Π, U(1)) denote the Pontryagin dual of the lattice Π, which is often called the ‘Brillouin torus’ in condensed matter physics. We define the space L2 Π(Π̂× Rd, V ) by L2 Π ( Π̂× Rd, V ) = { ψ̂ ∈ L2 ( Π̂× Rd, V ) \| ψ̂(k̂, x+m) = k̂(m)ψ̂(k̂, x) (m ∈ Π)}. We also define transformations B̂ and B, inverse to each other: B̂ : L2 ( Rd, V ) −→ L2 Π ( Π̂× Rd, V ) , (B̂ψ) ( k̂, x ) = ∑ n∈Π k̂(n)−1ψ(x+ n), B : L2 Π ( Π̂× Rd, V ) −→ L2 ( Rd, V ) , (Bψ̂)(x) = ∫ k̂∈Π̂ ψ̂ ( k̂, x ) dk̂. As is described in [8], the space L2 Π(Π̂×Rd, V ) can be identified with the space L2(Π̂, E ⊗V ) of L2-sections of a vector bundle E ⊗ V → Π̂. The infinite-dimensional vector bundle E → Π̂ is given by E = ⋃ k̂∈Π̂ L2 ( Rd/Π,L\|{k̂}×Rd/Π ) , where L → Π̂ × Rd/Π is the Poincaré line bundle, the quotient of the product line bundle Π̂× Rd × C→ Π̂× Rd by the following Π-action Π× ( Π̂× Rd × C ) −→ Π̂× Rd × C, ( m, k̂, x, z ) 7→ ( k̂, x+m, k̂(m)z ) . In summary, we get an identification of L2-spaces L2 ( Rd, V ) ∼= L2 Π ( Π̂× Rd, V ) ∼= L2 ( Π̂, E ⊗ V ) . The Hamiltonian H on L2(Rd, V ) then induces an operator Ĥ on L2 Π(Π̂×Rd, V ) ∼= L2(Π̂, E ⊗ V ) by Ĥ ◦ B̂ = B̂ ◦H. If, for instance, H is of the form H = ∆+Φ, then Ĥ preserves the fiber of E ⊗ V . Generally, this is a consequence of the translation invariance of the Hamiltonian. When the present quantum system is supposed to be an ‘insulator’, a finite number of discrete spectra of Ĥ(k̂) would be confined to a compact region in R as k̂ ∈ Π̂ varies. Then the corresponding eigenfunctions form a finite rank subbundle E ⊂ E ⊗ V , called the ‘Bloch bundle’. The K-class of this vector bundle E → Π̂ is regarded as an invariant of the quantum system under study. 2.3 Nonsymmorphic group and twisted K-theory We now take the symmetry into account. From the extension 1 → Π → S π→ P → 1, we can associate a twisted P -equivariant vector bundle on Π̂ to the S-module L2(Rd, V ). This is a version of the so-called ‘Mackey machine’. Recall that the Euclidean group Rd o O(d) is the semi-direct product of the orthogonal group O(d) and the group of translations Rd. Hence a collection of representatives {sp}p∈P of p ∈ P ∼= S/Π in S is expressed as sp = (ap, p) ∈ Rd o O(d) by means of a map a : P → Rd. For p1, p2 ∈ P we put ν(p1, p2) = ap1 + p1ap2 − ap1p2 . 10 K. Gomi Since Π ⊂ S is normal, the action of P ⊂ O(d) on Rd preserves Π ⊂ Rd. Then we have ν(p1, p2) ∈ Π, and ν : P × P → Π is a group 2-cocycle of P with values in Π regarded as a left P -module through the action m 7→ pm of p ∈ P on m ∈ Π. This group 2-cocycle measures the failure for S to be symmorphic. By means of the S-action ρ on L2(Rd, V ), we define an ‘action’ of p ∈ P by ρ(p) : L2 ( Rd, V ) −→ L2 ( Rd, V ) , ρ(p) = ρ((ap, p)), whose explicit formula for ψ ∈ L2(Rd, V ) is given by (ρ(p)ψ)(x) = U(p)ψ(p−1x + ap−1). The Bloch transformation then induces the following ‘action’ of P , ρ̂(p) : L2 Π ( Π̂× Rd, V ) → L2 Π ( Π̂× Rd, V ) , ρ̂(p) ◦ B̂ = B̂ ◦ ρ(p), whose explicit formula for ψ̂ ∈ L2 Π(Π̂×Rd, V ) is (ρ̂(p)ψ̂)(k̂, x) = U(p)ψ̂(p−1k̂, p−1x+ap−1). Here the left P -action on Π̂ is defined by (pk̂)(m) = k̂(p−1m), where p ∈ P acts on m ∈ Π through the inclusion P ⊂ O(d) and the left action of O(d) on Rd. Notice that ρ and ρ̂ can be honest actions of P in the case of symmorphic S, but not in the case of nonsymmorphic S, for the usual composition rule is violated:( ρ̂(p1) ( ρ̂(p2)ψ̂ ))( k̂, ξ ) = ( p−1 2 p−1 1 k̂ )( ν ( p−1 2 , p−1 1 ))( ρ̂(p1p2)ψ̂ )( k̂, ξ ) . To interpret the ‘action’ ρ̂(p) in terms of the vector bundle E ⊗ V through L2 Π(Π̂×Rd, V ) ∼= L2(Π̂, E ⊗ V ), recall that the fiber of E ⊗ V at k̂ ∈ Π̂ is E\|k̂ ⊗ V = L2 ( Rd/Π,L\|{k̂}×Rd/Π ⊗ V ) , and ψ̂ ∈ L2 Π(Π̂× Rd, V ) corresponds to the following section Ψ ∈ L2(Π̂, E ⊗ V ): Ψ ( k̂ ) : Rd/Π −→ L\|{k̂}×Rd/Π ⊗ V, x 7→ [ k̂, x, ψ̂(k̂, x) ] . Define for p ∈ P and k̂ ∈ Π̂ a linear map ρE⊗V ( p; k̂ ) : E\|k̂ ⊗ V −→ E\|pk̂ ⊗ V by the assignment of the sections ρE⊗V ( p; k̂ )([ x 7→ [ k̂, x, ψ̂(k̂, x) ]]) = [ x 7→ [ pk̂, x, U(p)ψ̂ ( k̂, p−1x+ ap−1 )]] . These maps constitute a vector bundle map ρE⊗V (p) : E⊗V → E⊗V covering the action k̂ 7→ pk̂ on Π̂ E ⊗ V ρE⊗V (p)−−−−−→ E ⊗ Vy y Π̂ p−−−−→ Π̂. This is a τ -twisted P -action, in the sense that the formula ρE⊗V ( p1; p2k̂ ) ρE⊗V ( p2; k̂ ) ξ = τ ( p1, p2; k̂ ) ρE⊗V ( p1p2; k̂ ) ξ holds for p1, p2 ∈ P , k̂ ∈ Π̂ and ξ ∈ E\|k̂ ⊗ V . Here τ : P × P × Π̂→ U(1) is defined by τ(p1, p2; k̂) = k̂ ( ν ( p−1 2 , p−1 1 )) , Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 11 and is regarded as a group 2-cocycle of P with its coefficients in the group C(Π̂, U(1)) of U(1)- valued functions on Π̂ thought of as a right P -module through the pull-back under the left action k̂ 7→ pk̂ of p ∈ P on k̂ ∈ Π̂. The map ρE⊗V (p) on the vector bundle induces the transformation on the sections ρE⊗V (p) : L2 ( Π̂, E ⊗ V ) −→ L2 ( Π̂, E ⊗ V ) by (ρE⊗V (p)Ψ)(k̂) = ρE⊗V (p; p−1k̂)Ψ(p−1k̂). One can verify that: if Ψ ∈ L2(Π̂, E ⊗ V ) corre- sponds to ψ̂ ∈ L2 Π(Π̂ × Rd, V ), then ρE⊗V (p)Ψ corresponds to ρ̂(p)ψ̂. Hence the ‘action’ ρ̂(p) on L2 Π(Π̂× Rd, V ) ∼= L2(Π̂, E ⊗ V ) agrees with the one induced from the τ -twisted P -action on E ⊗ V . Now, under the assumption that Ĥ describes an insulator, the Bloch bundle E ⊂ E⊗V inher- its a τ -twisted P -action from E ⊗V . This is a consequence of the invariance of the Hamiltonian under the space group action. Therefore the Bloch bundle, being a τ -twisted P -equivariant vector bundle of finite rank, defines a class in the τ -twisted P -equivariant K-theory Kτ+0 P (Π̂), which is regarded as an invariant of the insulating system under study. As is obvious from the construction, we can apply the construction of the group 2-cocycle τ to symmorphic space groups. However, in the symmorphic case, the cocycle ν and hence τ can be trivialized. So far a linear representation of P on V is considered. We can relax this representation to be a projective representation of P with its group 2-cocycle ω : P ×P → U(1). In this case, the resulting Bloch bundle defines a class in the twisted equivariant K-theory Kτ+ω+0 P (Π̂). Remark 2.1. The phase factor in the composition rule of ρ̂, τR ( k̂; p1, p2 ) = ( p−1 2 p−1 1 k̂ )( ν ( p−1 2 , p−1 1 )) = k̂ ( p1p2ν ( p−1 2 , p−1 1 )) defines a group 2-cocycle of P with coefficients in C(Π̂, U(1)), when regarded as a left P -module by the right action k̂ 7→ k̂p = p−1k̂ of p ∈ P on k̂ ∈ Π̂. The 2-cocycles τ and τR are related by τR(k̂; p1, p2) = τ(p1, p2; (p1p2)−1k̂). This extends to a cochain bijection of group cochains with coefficients in the left/right P -modules C(Π̂, U(1)). Thus, τ and τR have cohomologically the same information. We also remark that τ and τR are respectively cohomologous to the following 2-cocycles: τ ′ ( p1, p2; k̂ ) = ( p1p2k̂ ) (ν(p1, p2))−1, τ ′R ( k̂; p1, p2 ) = k̂(ν(p1, p2))−1. Remark 2.2. Given a homomorphism c : P → Z2, we can impose that the Hamiltonian H and the symmetry ρ(g) with g ∈ S are graded commutative, H ◦ ρ(g) = c(π(g))ρ(g) ◦ H. Then the quantum system with symmetry in question leads to an element of the twisted equivariant K-theory Kτ+c+0 P (Π̂), where the (ungraded) twist τ is now graded by c ∈ H1 P (Π̂;Z2). It should be noticed that the construction of the element uses Karoubi’s formulation of K-theory [13] and requires a reference quantum system. These points of discussion, which will not be detailed in this paper, are implicit in the absence of the graded twist. Remark 2.3. A group 2-cocycle τ can be thought of as the cocycle for a projective represen- tation. Besides the argument in this section, there are other arguments which derive projective representations from quantum systems with symmetry (for example [15, 16]). 2.4 Actions of the point group on the torus To close Section 2, we compare some natural actions of the point group on the torus: Let S be a d-dimensional space group, Π its lattice, and P its point group. 12 K. Gomi (A) By the inclusion P ⊂ O(d) and the standard left action of O(d) on Rd, the point group P acts on Rd, preserving Π ⊂ Rd. Hence the left action of P on Rd descends to give a left action of P on the torus Rd/Π. (B) By the action (A), the point group P acts on the Pontryagin dual Π̂ = Hom(Π, U(1)) of Π from the left: For p ∈ P and k̂ ∈ Π̂, we define pk̂ ∈ Π̂ by (pk̂)(m) = k̂(p−1m) for all m ∈ Π. (C) By the inclusion S ⊂ RdoO(d) and the standard left action of RdoO(d) on Rd, the space group S acts on Rd. The subgroup Π ⊂ S preserves Π ⊂ Rd, so that the point group P ∼= S/Π acts on Rd/Π. The action (A) is what we consider in our main results, and the action (B) is relevant to quantum systems as reviewed in this section. On the one hand, the actions (A) and (B) clearly fix 0 ∈ Rd/Π and 0 ∈ Π̂, respectively, where we regard Π̂ as Hom(Π,R/Z) via R/Z ∼= U(1) and 0 ∈ Π̂ stands for the trivial homomorphism. On the other hand, if (ap, p) ∈ Rd o O(d) is a lift of p ∈ P ⊂ O(d), then the action of p ∈ P on k ∈ Rd/Π in (C) admits the description k 7→ pk + ap. If S is symmorphic, then we can choose ap to be in Π. In this case, the actions (A) and (C) are equivalent. However, if S is nonsymmorphic, then ap cannot be in Π. Thus, in this case, the action of p ∈ P does not fix any point on Rd/Π, so that the actions (A) and (C) are not equivalent. For example, in the case of pg, the action of P = Z2 on the 2-dimensional torus is free, and its quotient is the Klein bottle. To compare the actions (A) and (B), we need to identify Rd/Π with Π̂ = Hom(Π,R/Z), which are topologically d-dimensional tori. In general, such an identification may not be unique. A way to implement the identification is to choose a basis {vj} of the lattice Π ∼= Zd. This choice induces the following identifications of tori inverse to each other: Π̂ −→ Rd/Π, k̂ 7→ ∑ j k̂(vj)vj , Rd/Π −→ Π̂, ∑ j kjvj 7→ [∑ j mjvj 7→ ∑ j mjkj ] . With this identification of tori, the left P -action on Π̂ in (B) induces a right P -action on Rd/Π. Considering the action of p−1 instead of p, we finally get a left action of P on Rd/Π, induced from (B) and the identification Π̂ ∼= Rd/Π. In general, this left action of P on Rd/Π induced from (B) is not equivalent to the action (A). In the 2-dimensional case, their relationship is as follows: Lemma 2.4. Let S be a 2-dimensional space group, PS its point group, ΠS its lattice, and Π̂S the Pontryagin dual of ΠS. (a) We choose a basis {vj} of ΠS to identify Π̂S with R2/ΠS, and let the action in (B) induce an action of PS on R2/ΠS. Then, up to equivalence, this action is independent of the choice of the basis. (b) If S is not p3m1 or p31m, then the action of PS on R2/ΠS induced from (B) is equivalent to the action of PS on R2/ΠS in (A). (c) If S is p3m1 (respectively p31m), then the action of PS on R2/ΠS induced from (B) is equivalent to the action of PS′ on R2/ΠS′ in (A), where S′ is p31m (respectively p3m1). We remark that the space groups p3m1 and p31m share the same lattice and the same point group, as can be seen in Appendix A. Hence we have PS = PS′ and ΠS = ΠS′ in the third item in the lemma above. Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 13 Proof. Considering the action (A), we define ψ(p)`j ∈ Z by pvj = ∑ ` ψ(p)`jv`, and a homo- morphism ψ : PS → GL(2,Z) by ψ(p) = (ψ(p)`j). Since PS is the point group of a 2-dimensional space group S, the homomorphism ψ is injective and its image ψ(PS) is a finite subgroup of GL(2,Z). Let S′ be another 2-dimensional space group with its point group P ′. Choosing a ba- sis of its lattice Π′, we similarly get from the action (A) a homomorphism ψ′ : PS′ → GL(2,Z). If the images ψ(PS) and ψ′(PS′) are conjugate to each other in GL(2,Z), then the actions of PS and PS′ in (A) are equivalent. The action of PS on R2/ΠS induced from (B) also yields an associated homomorphism PS → GL(2,Z). This homomorphism turns out to be the transpose inverse tψ−1 : PS → GL(2,Z), which is again injective and defines a finite sub- group tψ−1(PS) ⊂ GL(2,Z). If we alter the basis {vj}, then tψ−1 changes by a conjugation of a matrix in GL(2,Z). Thus, up to conjugations, the image tψ−1(PS) ⊂ GL(2,Z) is independent of the choice of {vj}, showing (a). Now we can directly verify (b) and (c), by computing the homomorphism ψ based on the explicit basis in Appendix, and comparing the images ψ(PS) and tψ−1(PS) in GL(2,Z). � Another way of identifying Π̂ with Rd/Π is to choose a bilinear form 〈 , 〉 : Π× Π→ Z. We assume that this form is non-degenerate in the sense that the matrix (〈vi, vj〉) is invertible with respect to any basis {vi} of Π. A non-degenerate bilinear form induces an identification of the tori as follows Rd/Π −→ Π̂ = Hom(Π,R/Z), k 7→ [m 7→ 〈m, k〉]. If the bilinear form is P -invariant in the sense that 〈pm, pm′〉 = 〈m,m′〉 for all m,m′ ∈ Π and p ∈ P , then the action (A) on Rd/Π agrees with the action (B) on Π̂ under the induced identification Rd/Π ∼= Π̂. For the 2-dimensional space groups such that Π can be the standard lattice Z2 ⊂ R2, the standard inner product on R2 restricts to give a P -invariant non-degenerate bilinear form. If we choose an orthonormal basis {vj}, then the identifications Rd/Π ∼= Π̂ given by 〈 , 〉 and by {vj} are P -equivariantly the same. In Section 4, we will work with the torus R2/Π with the action (A), and the relation to Π̂ with the action (B) should be understood as above. 3 The Leray–Serre spectral sequence and twists This section gives a geometric interpretation of the filtration of H3 G(X;Z) for the Leray–Serre spectral sequence through types of twists. This is carried out by identifying the Leray–Serre spectral sequence with another natural spectral sequence which computes the Borel equivariant cohomology. Throughout this section, we assume that G is a finite group acting from the left on a ‘rea- sonable’ space X, such as a locally contractible, paracompact and regular topological space as in [7], or a G-CW complex [19]. 3.1 Spectral sequences The Borel equivariant cohomology Hn G(X;Z) is defined to be the (singular) cohomology of the quotient space EG ×G X of EG × X under the diagonal G-action (ξ, x) 7→ (ξg, g−1x), where EG is the total space of the universal G-bundle EG→ BG. Associated to the fibration X → EG×G X → BG is the Leray–Serre spectral sequence Ep,qr =⇒ Hp+q G (X;Z) converging to the graded quotient of a filtration Hn G(X;Z) = F 0Hn G(X;Z) ⊃ F 1Hn G(X;Z) ⊃ · · · ⊃ Fn+1Hn G(X;Z) = 0, 14 K. Gomi that is, Ep,q∞ = F pHp+q G (X;Z)/F p+1Hp+q G (X;Z). The E2-term is given by the group cohomology of G Ep,q2 = Hp group(G;Hq(X;Z)), where the coefficient Hq(X;Z) is regarded as a right G-module by the pull-back action. As a convention of this paper, the group of p-cochains with values in a right G-module M is denoted by Cpgroup(G;M) = C(Gp,M) = {τ : Gp →M}, and the coboundary ∂ : Cpgroup(G;M)→ Cp+1 group(G;M) is given by (∂τ)(g1, . . . , gp+1) = τ(g2, . . . , gp+1) + p∑ i=1 (−1)iτ(g1, . . . , gigi+1, . . . , gp+1) + (−1)p+1τ(g1, . . . , gp)gp+1. As an application of the spectral sequence, we can obtain an identification Hn G(pt;Z) ∼= Hn group(G;Z). (We also have Hn group(G;Z) ∼= Hn−1 group(G;U(1)) for n ≥ 2 by the so-called expo- nential exact sequence.) For a better geometric understanding of the spectral sequence, let us start with the fact that the Borel equivariant cohomology Hn G(X;Z) is isomorphic to the cohomology Hn(G•×X;Z) of a simplicial space G•×X with its coefficients in the constant sheaf Z. This is a consequence of a more general theorem about simplicial space (see [6] for example) together with the fact that the geometric realization \|G• ×X\| of G• ×X is identified with EG×G X. The simplicial space G•×X is associated to the left G-action on X, and consists of a sequence of spaces {Gp ×X}p≥0 together with the face map ∂i : G p ×X → Gp−1 ×X, i = 0, . . . , p, and the degeneracy map si : G p ×X → Gp+1 ×X, i = 0, . . . , p, given by ∂i(g1, . . . , gp, x) =  (g2, . . . , gp, x), i = 0, (g1, . . . , gigi+1, . . . , gp, x), i = 1, . . . , p− 1, (g1, . . . , gp−1, gpx), i = p, si(g1, . . . , gp, x) = (g1, . . . , gi−1, 1, gi, . . . , gp, x). The cohomology Hn(G• × X;Z) is then defined to be the total cohomology of the double complex (Ci(Gj×X;Z), δ, ∂), where (Ci(Gj×X;Z), δ) is the complex computing the cohomology of Gj×X with coefficients in Z and ∂ : Ci(Gj×X;Z)→ Ci(Gj+1×X;Z) is ∂ = j+1∑ i=0 (−1)i∂∗i . The double complex admits a natural filtration {⊕j≥pCi(Gj × X;Z)}p≥0. The associated spectral sequence agrees with the Leray–Serre spectral sequence Ep,qr , since G is finite. Now, let us consider the standard exponential exact sequence of sheaves on the simplicial space 0→ Z→ R→ U(1)→ 0, where R consists of the sheaf of R-valued functions on Gp ×X and U(1) consists of the sheaf of U(1)-valued functions on Gp×X. As in [9, Lemma 4.4], we can readily show that Hn(G• × X;R) = 0 for n > 0. This vanishing together with the associated long exact sequence leads to the following isomorphism for n ≥ 1 Hn(G• ×X;U(1)) ∼= Hn+1(G• ×X;Z). The cohomology Hn(G• × X;U(1)) can be defined exactly in the same way as in the case of Hn(G• ×X;Z) by using a double complex. Therefore we have a spectral sequence ′Ep,qr =⇒ Hp+q(G• ×X;U(1)) converging to the graded quotient of a filtration ′F 0Hn(G• ×X;U(1)) = Hn(G• ×X;U(1)) ⊃ ′F 1Hn(G• ×X;U(1)) ⊃ · · · , Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 15 whose E2-term is ′Ep,q2 = Hp group(G;Hq(X;U(1))), where Hq(X;U(1)) is regarded as a right G-module by pull-back. It is clear that H0(X;U(1)) ∼= C(X,U(1)) and Hn(X;U(1)) ∼= Hn+1(X;Z) for n ≥ 1. Since ′Ep,q2 involves the group cohomol- ogy with coefficients in C(X,U(1)), its computation seems to be more complicated than that of Ep,q2 . However, the spectral sequence is useful from a geometric viewpoint, as will be seen shortly. In view of the exponential exact sequence, the filtrations of Hn+1 G (X;Z) ∼= Hn(G•×X;U(1)) for n ≥ 1 are related as follows Hn(G• ×X;U(1)) = ′F 0Hn ⊃ ′F 1Hn ⊃ · · ·⊃ ′F pHn ⊃ · · · ‖ ↓ ↓ Hn+1 G (X;Z) = F 0Hn+1 ⊃ F 1Hn+1 ⊃ · · ·⊃ F pHn+1 ⊃ · · · . The spectral sequences are related by a map ′Ep,qr → Ep,q+1 r . In particular, the E2-terms ′Ep,02 and Ep,12 are related by a map C(X,U(1))→ H1(X;Z) fitting into the exact sequence 0→ H0(X;Z)→ C(X,R)→ C(X,U(1))→ H1(X;Z)→ 0. As is mentioned, because of the isomorphismHq(X;U(1)) ∼= Hq+1(X;Z), we have ′Ep,q2 ∼= Ep,q+1 2 for q ≥ 1. A more detailed relation between these spectral sequences will be given later under some hypotheses. 3.2 Twists We here recall the definition of twists for equivariant K-theory in [7, 8] for the convenience of the reader. We mainly consider ungraded twists, and refer the reader to [7] for the details about graded twists (see also Remark 3.7). Recall that associated to an action of a finite group G on a space X is the groupoid X//G such that its set of objects is X and the set of morphisms is G×X. Definition 3.1. A central extension (L, τ) of the groupoid X//G consists of the following data: • a Hermitian line bundle L→ G×X, which we write Lg → X for the restriction to {g}×X for each g ∈ G, • unitary isomorphisms of Hermitian line bundles τg,h : h∗Lg ⊗ Lh → Lgh on X for each g, h ∈ G, which we write τg,h(x) : Lg\|hx ⊗ Lh\|x → Lgh\|x for the restriction to x ∈ X. We assume the following diagram is commutative Lg\|hkx ⊗ Lh\|kx ⊗ Lk\|x 1⊗τh,k(x) −−−−−−→ Lg\|hkx ⊗ Lhk\|x τg,h(kx)⊗1 y yτg,hk(x) Lgh\|kx ⊗ Lk\|x τgh,k(x) −−−−−→ Lghk\|x. Notice that if Lg is the product line bundle, then the central extension is just a group 2-cocycle of G with coefficients in C(X,U(1)). Definition 3.2. An isomorphism (K,βg) : (Lg, τg,h)→ (L′g, τ ′ g,h) of central extensions of X//G consists of the following data: 16 K. Gomi • a Hermitian line bundle K → X, • unitary isomorphisms of Hermitian line bundles βg : Lg ⊗K → g∗K ⊗ L′g on X for each g ∈ G, which we write βg(x) : Lg\|x ⊗K\|x → K\|gx ⊗ L′g\|x for the restriction to x ∈ X. We assume the following diagram is commutative Lg\|hx ⊗ Lh\|x ⊗K\|x 1⊗βh(x)−−−−−→ Lg\|hx ⊗K\|hx ⊗ L′h\|x τg,h(x)⊗1 y yβg(hx)⊗1 Lgh\|x ⊗K\|x K\|ghx ⊗ L′g\|hx ⊗ L′h\|x∥∥∥ y1⊗τ ′g,h(x) Lgh\|x ⊗K\|x βgh(x) −−−−→ K\|ghx ⊗ L′gh\|x. The isomorphisms (K,βg) and (K ′, β′g) from (Lg, τg,h) to (L′g, τ ′ g,h) are identified if there is a unitary isomorphism f : K → K ′ making the following diagram commutative Lg\|x ⊗K\|x βg(x)−−−−→ K\|gx ⊗ L′g\|x 1⊗f(x) y yf(gx)⊗1 Lg\|x ⊗K ′\|x β′g(x) −−−−→ K ′\|gx ⊗ L′g\|x. Definition 3.3. An ungraded G-equivariant twist of X, or a twist for short, is a central extension of a groupoid X̃ which has a local equivalence to X//G. A point in this definition is that a twist needs an extra groupoid X̃. A central extension of X//G is a special type of a twist such that X̃ = X//G. Taking the extra groupoids into account, we can introduce a notion of isomorphisms to twists. We refer the reader to [7] for the details of the isomorphisms and the following classification: Proposition 3.4 ([7]). The isomorphisms classes of ungraded G-equivariant twists of X form an abelian group isomorphic to H3 G(X;Z). A key to the classification is the isomorphism H3 G(X;Z) ∼= H2(G• ×X;U(1)). A close look at the proof of the classification leads to: Lemma 3.5. The following holds true: (i) ′F 1H2(G• × X;U(1)) classifies twists represented by central extensions of the groupoid X//G. (ii) ′F 2H2(G•×X;U(1)) classifies twists represented by group 2-cocycles of G with coefficients in the G-module C(X,U(1)). Remark 3.6. In [8], an isomorphism of central extensions of X//G is formulated only by using the product line bundle K = X × C. The reason of the difference in these definitions is that we are considering an isomorphism of central extensions of X//G regarded as twists. By the same reasoning, group cocycles which are not cohomologous to each other can be isomorphic as twists. Remark 3.7. The modification needed to define a graded twist is to replace the Hermitian line bundle L constituting a central extension (L, τ) with a Z2-graded Hermitian line bundle. Since L is of rank 1, its Z2-grading amounts to specifying the degree of L to be even or odd. With the suitable modification of the notion of isomorphisms, we can eventually classify graded twists by H1 G(X;Z2)×H3 G(X;Z). Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 17 3.3 Comparison of two spectral sequences The relation between the spectral sequences Ep,qr and ′Ep,qr can be made more clear under a simple assumption. To present this here, we begin with a key lemma: Recall that the exponential exact sequence of sheaves on X induces a natural exact sequence of right G-modules 0→ H0(X;Z)→ C(X,R)→ C(X,U(1))→ H1(X;Z)→ 0. Let us fold this into a short exact sequence 0→ C(X,R)/H0(X;Z)→ C(X,U(1))→ H1(X;Z)→ 0. In general, this does not split as an exact sequence of G-modules. (Such an example is provided by the circle S1 ⊂ R2 with the action of D2 ⊂ O(2).) Notice that if X is path connected, then H0(X;Z) = Z. Lemma 3.8. If a finite group G acts on a compact and path connected space X fixing a point pt ∈ X, then the following exact sequence of G-modules splits 0→ C(X,R)/Z→ C(X,U(1))→ H1(X;Z)→ 0. Proof. For notational convenience, we use the identification U(1) ∼= R/Z in this proof. Let C(X,pt,R) ⊂ C(X,R) be the subgroup consisting of functions taking 0 at pt. The inclusion ι : pt→ X induces an isomorphism of G-modules C(X,R) −→ C(X,pt,R)⊕ R, f 7→ (f − ι∗f, ι∗f). Similarly, we have an isomorphism C(X,R/Z) ∼= C(X,pt,R/Z)⊕R/Z of G-modules. Thus the exact sequence of G-modules in question is equivalent to 0→ C(X,pt,R)→ C(X,pt,R/Z) δ→ H1(X;Z)→ 0. Since X is supposed to be compact, H1(X;Z) is a free abelian group of finite rank. Let us choose a basis H1(X;Z) ∼= ⊕ i Zai, and also ϕi : X → R/Z such that δϕi = ai and ϕi(pt) = 0. Modifying the splitting ai 7→ ϕi of the exact sequence of abelian groups, we construct a splitting of the exact sequence of G-modules, which will complete the proof. For the modification, we introduce a square matrix A(g) = (Aij(g)) with integer coefficients to each g ∈ G by g∗ai = ∑ j Aij(g)aj . It holds that A(gh) = A(g)A(h). Because of the exact sequence, there are functions f ig ∈ C(X,pt,R) such that the following holds in C(X,pt,R/Z): g∗ϕi = ∑ j Aij(g)ϕj + ( f ig mod Z ) . This can be expressed as g∗Φ = A(g)Φ + Fg by using the vectors Φ = (ϕi) and Fg = (F ig). It then holds that Fgh = A(g)Fh + h∗Fg in C(X,pt,R). Since A(g) is invertible, this is equivalent to A(gh)−1Fgh = A(h)−1Fh +A(h)−1h∗ ( A(g)−1Fg ) . Write \|G\| for the order of G, and put F = 1 \|G\| ∑ g∈G A(g)−1Fg. Taking the average over g ∈ G in the formula above, we get F = A(h)−1Fh +A(h)−1h∗F , which is equivalent to Fg = A(g)F − g∗F . Now g∗(Φ + F ) = A(g)(Φ + F ). Thus, under the expression F = (f i ) by using f i ∈ C(X,pt,R), the assignment ai 7→ ϕi + (f i mod Z) defines a splitting H1(X;Z)→ C(X,pt,R/Z) compatible with the G-module structures. � 18 K. Gomi Lemma 3.9. Let G be a finite group acting on a compact and path connected space X fixing a point pt ∈ X. Then, for n ≥ 1, there is an isomorphism Hn group(G;C(X,U(1))) ∼= Hn group(G;U(1))⊕Hn group ( G;H1(X;Z) ) , where U(1) is the trivial G-module, and H1(X;Z) is regarded as a G-module through the action of G on X. Proof. Lemma 3.8 implies Hn group(G;C(X,U(1))) ∼= Hn group(G;C(X,R)/Z)⊕Hn group ( G;H1(X;Z) ) for all n ≥ 1. By the G-module isomorphism C(X,R)/Z ∼= C(X,pt,R) ⊕ R/Z utilized in Lemma 3.8, we have Hn group(G;C(X,R)/Z) ∼= Hn group(G;C(X,pt,R))⊕Hn group(G;R/Z). Since C(X,pt,R) is a vector space over R, we can prove the vanishing Hn group(G;C(X,pt,R)) = 0 for n ≥ 1 by an average argument as in [9, Lemma 4.4]. � Proposition 3.10. Suppose that a finite group G acts on a compact and path connected space X fixing a point pt ∈ X. Then for r ≥ 2 we have ′Ep,0r ∼= Ep,1r ⊕ Ep+1,0 r , p ≥ 1, ′Ep,qr ∼= Ep,q+1 r , p ≥ 0, q ≥ 1. Proof. Recall that the exponential exact sequence induces the connecting homomorphism δ : Hq(X;U(1)) → Hq+1(X;Z) and this induces a natural homomorphism δ : ′Ep,qr → Ep,q+1 r compatible with the differentials ′dr and dr. In the case of r = 2, the homomorphism δ : ′Ep,q2 → Ep,q+1 2 is bijective for q ≥ 1 and p ≥ 0, and we have ′Ep,02 ∼= Ep,12 ⊕ Ep+1,0 2 for p ≥ 1 as a conse- quence of Lemma 3.9. Notice that, under this isomorphism, δ : ′Ep,02 → Ep,12 for p ≥ 1 restricts to the identity on the direct summand Ep,12 ⊂ ′Ep,02 . Note also that Ep,02 = Ep,0∞ for any p, because Ep,02 = Hp group(G;Z) = Hp(BG;Z) = Hp G(pt;Z) is a direct summand of Hp G(X;Z) ∼= H0 G(pt;Z) ⊕ H̃p G(X;Z), where H̃p G(X;Z) is the reduced cohomology. Thus, for p ≥ 1, the map δ : ′Ep,02 → Ep,12 is the projection onto Ep,12 and the image of the differential ′d2 : ′Ep−2,1 2 → ′Ep,02 is in the direct summand Ep,02 . This leads to ′Ep,03 ∼= Ep,03 ⊕ Ep,13 , p ≥ 1, ′Ep,q3 ∼= Ep,q+1 3 , p ≥ 0, q ≥ 1. The calculation above can be repeated inductively on r. � Corollary 3.11. Let G and X be as in Proposition 3.10. Then, for any n ≥ 1 and p = 0, . . . , n, there is a natural isomorphism ′F pHn(G• ×X;U(1)) ∼= F pHn+1 G (X;Z). In addition, we have a decomposition ′FnHn(G• ×X : U(1)) ∼= FnHn+1 G (X;Z) ∼= En,1∞ ⊕ Fn+1Hn+1 G (X;Z), in which Fn+1Hn+1 G (X;Z) ∼= Hn+1 G (pt;Z) ∼= Hn group(G;U(1)). Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 19 Proof. Put ′F pHn = ′F pHn(G• × X;U(1)) and F pHn+1 = F pHn+1 G (X;Z) for short. The exponential exact sequence induces a homomorphism of short exact sequences 0 −−−−→ ′F p+1Hn −−−−→ ′F pHn −−−−→ ′Ep,n−p∞ −−−−→ 0 δ y δ y δ y 0 −−−−→ F p+1Hn+1 −−−−→ F pHn+1 −−−−→ Ep,n+1−p ∞ −−−−→ 0. In the case of p = n, the diagram above becomes 0 −−−−→ 0 −−−−→ ′FnHn ∼=−−−−→ ′En,0∞ −−−−→ 0 δ y δ y δ y 0 −−−−→ Fn+1Hn+1 −−−−→ FnHn+1 −−−−→ En,1∞ −−−−→ 0. Notice that Fn+1Hn+1 ∼= En+1,0 ∞ ∼= En+1,0 2 since En+1,0 2 ∼= Hn+1 G (pt;Z) must survive into Hn+1 G (X;Z) ∼= Hn+1 G (pt;Z) ⊕ H̃n+1 G (X;Z). Hence FnHn+1 ∼= En+1,0 ∞ ⊕ En,1∞ . If n ≥ 1, then this isomorphism is compatible with the isomorphism ′En,0∞ ∼= En+1,0 ∞ ⊕En,1∞ in Proposition 3.10 through δ, so that ′FnHn δ∼= FnHn+1 ∼= Fn+1Hn+1 ⊕ En,1∞ . For p = n − 1, n − 2, . . . , 1, 0, we know δ : ′Ep,n−p∞ → Ep,n−p+1 ∞ is bijective by Proposition 3.10. Therefore ′F pHn ∼= F pHn+1 inductively. � Combining the above corollary with Lemma 3.5, we get the interpretations of F pH3 G(X;Z) by twists presented in Introduction: Corollary 3.12. Let G and X be as in Proposition 3.10. (i) F 1H3 G(X;Z) classifies twists which can be represented by central extensions of the groupoid X//G. (ii) F 2H3 G(X;Z) classifies twists which can be represented by 2-cocycles of G with coefficients in the G-module C(X,U(1)). (iii) F 3H3 G(X;Z) = H2 group(G;U(1)) classifies twists which can be represented by 2-cocycles of G with coefficients in the trivial G-module U(1). Remark 3.13. The coincidence ′F 1Hn(G•×X;U(1)) = F 1Hn+1(X;Z) in Corollary 3.11 holds true for n ≥ 0 without the assumption that G fixes a point on X. This is because ′E0,n and E0,n+1 are subgroups of Hn(X;U(1)) ∼= Hn+1(X;Z) and it holds that ′F 1Hn(G• ×X;U(1)) = F 1Hn+1(X;Z) = Ker [ f : Hn(G• ×X;U(1)) ∼= Hn+1 G (X;Z)→ Hn+1(X;Z) ] , where f is the homomorphism of “forgetting the group actions”. 4 The proof of Theorems 1.1 and 1.3 Theorems 1.1 and 1.3 are proved here based on case-by-case computations of the equivariant cohomology and the Leray–Serre spectral sequence. Some basic facts that are useful for the computation are summarized first. We then provide the outline of the computations and details of typical cases, p2, p4m/p4g and p6m. Finally, Corollary 1.2 is proved. 20 K. Gomi 4.1 Some generality The cohomology Hn(T 2;Z) of the torus is well known, so that nothing remains to be proven in the case of p1. For the point group P of any 2-dimensional space group, the vanishing H3(T 2;Z) = 0 implies E0,3 ∞ = 0, so that H3 P ( T 2;Z ) = F 0H3 P ( T 2;Z ) = F 1H3 P ( T 2;Z ) . Note that each point group P fixes a point on T 2, so that F 3H3 P ( T 2;Z ) = H3 P (pt;Z) = H3 group(P ;Z) = H2 group(P ;U(1)). Then the main task for the proof of Theorem 1.1 is to compute H3 P (T 2;Z) and F 2H3 P (T 2;Z), since in the case where P is the cyclic group Zn or the dihedral group Dn, the cohomology Hm P (pt;Z) is summarized as follows: P H0 P (pt;Z) H1 P (pt;Z) H2 P (pt;Z) H3 P (pt;Z) Zn Z 0 Zn 0 Dn Z 0 { Z2 (n: odd) Z⊕2 2 (n: even) { 0 (n: odd) Z2 (n: even) The degree 0 part H0 P (pt;Z) = H0(BP ;Z) = Z is clear. Since P is finite, the degree 1 part H1 P (pt;Z) ∼= Hom(P,Z) is trivial. The degree 2 part H2 P (pt;Z) ∼= Hom(P,U(1)) can be seen by the classification of irreducible representations. Finally, the degree 3 part H3 P (pt;Z) ∼= H2 group(P ;U(1)) for P = Zn, Dn can be found in [14]. In the rest of the section, we may use a structure of T 2 as a P -CW complex. In general, for a compact Lie group G, a G-CW complex is an analogue of a CW complex made of G-cells. A d-dimensional G-cell is a G-space of the form G/H × ed, where H ⊂ G is a closed subgroup and ed is the standard d-dimensional cell. The G-action on G/H is the left translation, whereas that on ed is trivial. For the details, we refer the reader to [19]. We later compute a group cohomology via cohomology of a space: Lemma 4.1. Suppose that a finite group G acts on a path connected space Y fixing at least one point pt ∈ Y . Suppose further that Y is a CW complex consisting of only cells of dimension less than or equal to 1. Then the following holds true for all n ≥ 0. Hn group ( G;H1(Y ;Z) ) ∼= H̃n+1 G (Y ;Z), where H̃n+1 G (Y ;Z) stands for the reduced cohomology. Notice that a G-CW complex is naturally a CW complex. Proof. Consider the Leray–Serre spectral sequence Ep,q2 = Hp group(G;Hq(Y ;Z)) =⇒ H∗G(Y ;Z). Note that Hq(Y ;Z) = 0 for q 6= 0, 1. The E2-term Ep,02 = Hp group(G;Z) must survive into the direct summand Hp G(pt;Z) in Hp G(Y ;Z) ∼= Hp G(pt;Z)⊕ H̃p G(Y ;Z). Therefore it must hold that H̃p G(Y ;Z) ∼= Ep−1,1 ∞ = Ep−1,1 2 ∼= Hp−1 group ( G;H1(Y ;Z) ) , which completes the proof. � Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 21 We also prepare a simple lemma about group cohomology: Let G be a finite group, c : G→ Z2 = {±1} a surjective homomorphism, and Z̃ = Zc the G-module such that its underlying group is Z and G acts (from the right) by m 7→ mc(g). A typical example is a finite subgroup P ⊂ O(2) such that P 6⊂ SO(2) with c the composition of the inclusion P → O(2) and the determinant O(2)→ Z2. Lemma 4.2. Let G, c and Z̃ be as above. Then, H0 group(G; Z̃) = 0 and H1 group(G; Z̃) ∼= Z2. Proof. For any n ∈ C0 group(G; Z̃) = Z, its coboundary ∂n : G → Z is (∂n)(g) = n(1 − c(g)). Thus, the assumption that c is surjective implies the vanishing of the 0th cohomology. The inclusion Ker(c) ⊂ G induces an injection on 1-cocycles Z1 group ( G; Z̃ ) → Z1 group ( Ker(c); Z̃ ) = Hom(Ker(c),Z) = 0. Thus, given a group 1-cocycle φ ∈ Z1 group(G; Z̃), it holds that φ(g) = 0 for all g ∈ Ker(c). If g, h 6∈ Ker(c), then the cocycle condition (∂φ)(g, h) = 0 implies φ(g) = φ(h). Therefore φ : G → Z is always of the form φ(g) = n(1 − c(g))/2 for some n ∈ Z. This provides the identification Z1 group(G; Z̃) ∼= Z as well as B1 group(G; Z̃) ∼= 2Z, which completes the proof. � In some cases, the computations of the Leray–Serre spectral sequence are similar, which we summarize as follows: Lemma 4.3. Let G be a finite group acting on the torus T 2 such that: • there is a fixed point pt ∈ T 2, • the G-action does not preserve the orientation of T 2. Then the following holds true about the Leray–Serre spectral sequence: (a) F 2H3 G(T 2;Z) ∼= E2,1 2 ⊕ E3,0 2 , (b) Hn G(T 2;Z) ∼= ⊕ p+q=nE p,q 2 for n ≤ 2. Proof. In the Leray–Serre spectral sequence Ep,q2 = Hp group(G;Hq(T 2;Z)), the coefficient in the group cohomology H0(T 2) ∼= Z is identified with the trivial G-module, and H2(T 2) ∼= Z with the G-module in Lemma 4.2. Then the relevant E2-terms can be summarized as follows: q = 3 0 0 0 0 0 q = 2 0 Z2 q = 1 E0,1 2 E1,1 2 E2,1 2 q = 0 E0,0 2 E1,0 2 E2,0 2 E3,0 2 E4,0 2 Ep,q2 p = 0 p = 1 p = 2 p = 3 p = 4 Since G fixes pt ∈ T 2, we have the decomposition Hn G(T 2;Z) ∼= Hn G(pt;Z) ⊕ H̃n G(T 2;Z), where H̃n G(T 2;Z) is the reduced cohomology. Therefore the E2-term En,02 = Hn group(G;Z) ∼= Hn G(pt;Z) must survive into the direct summand Hn G(pt;Z) in Hn G(T 2;Z). This implies that En,02 = En,0∞ is always a direct summand of the subgroups F pHn G(T 2;Z) ⊂ Hn G(T 2;Z) and that d2 : Ep−2,1 2 → Ep,02 is trivial. As a result, we get E2,1 2 = E2,1 ∞ and the isomorphism (a). Also Ep,q2 = Ep,q∞ for p+ q ≤ 2, and the isomorphism (b) follows. � The degeneration of the spectral sequence in the above lemma can be generalized in some cases. For this aim, the key is the following equivariant stable splitting of T 2 (cf. [8, Theo- rem 11.8]). 22 K. Gomi Lemma 4.4. Suppose a finite group G acts on the torus T 2 = S1 × S1 and • there is a fixed point pt = (x0, y0) ∈ T 2 under the G-action, • G preserves the subspace S1 ∨ S1 = S1 × {y0} ∪ {x0} × S1 ⊂ T 2. Then there is a G-equivariant homotopy equivalence ΣT 2 ' Σ ( S1 ∨ S1 ) ∨ Σ ( T 2/S1 ∨ S1 ) , where Σ stands for the reduced suspension. Proof. The argument of the proof of Proposition 4I.1 [11, p. 467] can be applied to our equiv- ariant case. � We remark that the point groups of the 2-dimensional space groups without elements of order 3 fulfill the assumptions of the lemma above. Lemma 4.5. Under the assumption in Lemma 4.4, we have the following isomorphism of groups for all n ∈ Z Hn G ( T 2;Z ) ∼= Hn G(pt;Z)⊕ H̃n G ( S1 ∨ S1;Z ) ⊕ H̃n G ( T 2/S1 ∨ S1;Z ) . Further, the Leray–Serre spectral sequence for Hn G(T 2;Z) degenerates at E2 and the relevant extension problems are trivial, so that (a) F 2H3 G(T 2;Z) ∼= E2,1 2 ⊕ E3,0 2 , (b) Hn G(T 2;Z) ∼= ⊕ p+q=nE p,q 2 for all n ∈ Z. Proof. The stable splitting in Lemma 4.4 immediately gives the first isomorphism. For the trivial G-space pt, the Leray–Serre spectral sequence clearly degenerates at E2, and we have Hn G(pt;Z) ∼= Hn group ( G;H0(pt;Z) ) . Since H0(pt;Z) ∼= H0(T 2;Z) as G-modules, we get the following identification of the E2- term En,02 of the Leray–Serre spectral sequence for Hn G(T 2;Z) En,02 = Hn group ( G;H0 ( T 2;Z )) ∼= Hn G(pt;Z). For the G-space S1∨S1, we can see, as in the proof of Lemma 4.1, that the Leray–Serre spectral sequence also degenerates at E2 and the extension problems are trivial. Because H1(S1∨S1;Z) ∼= H1(T 2;Z) as G-modules, the E2-term En−1,1 2 of the Leray–Serre spectral sequence for Hn G(T 2;Z) is En−1,1 2 = Hn−1 group ( G;H1 ( T 2;Z )) ∼= H̃n G ( S1 ∨ S1;Z ) . Exactly in the same way, we have En−2,2 2 = Hn−2 group ( G;H2 ( T 2;Z )) ∼= H̃n G ( T 2/S1 ∨ S1;Z ) , since H2(T 2/S1∨S1;Z) ∼= H2(T 2;Z) as G-modules. The first isomorphism now gives Hn G(T 2;Z) ∼= En,02 ⊕ En−1,1 2 ⊕ En−2,2 2 , which also implies the triviality of the spectral sequence. � Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 23 4.2 The outline of computations Theorems 1.1 and 1.3 follow from case by case computations. As mentioned in Section 1, three methods are applicable. 1. In the cases of p2 and pm/pg, the point group Z2 = D1 acts on the torus T 2 = S1 × S1 preserving the direct product structure, so that we can think of T 2 as a stack of certain equivariant circle bundles (Z2-equivariant principal circle bundles and/or ‘Real’ circle bun- dles in the sense of [10]). For such circle bundles, we can use the Gysin exact sequence to compute the equivariant cohomology, as detailed in [10]. In particular, in the cases of p2 and pm/pg, the Gysin exact sequences are split, and the computations are very simple. (The computation by using the Gysin sequence is also valid for cm, even though the sequence is non-split.) In the case of p2, we do not need to compute the Leray–Serre spectral sequence, since the third cohomology is trivial. In the case of pm/pg, the spectral sequence can be computed directly. 2. In the cases of p4, cm, pmm/pmg/pgg, cmm and p4m/p4g, we can verify that the action of the point group on T 2 satisfies the assumptions of Lemma 4.4, by inspecting the explicit presentation in Appendix A. Hence we can apply Lemma 4.5 to the computation of the equivariant cohomology and the spectral sequence. In this application, the only non-trivial part is the equivariant cohomology of the invariant subspace S1 ∨ S1, which we compute by using the Mayer–Vietoris exact sequence. 3. In the cases of p3, p6, p3m1, p31m and p6m, the computation can be divided into two parts. One part is to compute H3 P (T 2;Z). This is carried out by taking a P -CW decom- position of T 2, and by using the Mayer–Vietoris exact sequence and the exact sequence for a pair. The other part is to compute the Leray–Serre spectral sequence. In this part, we need to know the group cohomology with coefficients in H1(T 2;Z). For this aim, we take an invariant subspace Y ⊂ T 2 of one dimension. The equivariant cohomology of Y is computed by using Mayer–Vietoris sequence, which allows us to know the group cohomol- ogy with its coefficients in H1(Y ;Z) through Lemma 4.1. The coefficients H1(Y ;Z) and H1(T 2;Z) are related by a short exact sequence. The associated long exact sequence then computes the group cohomology with coefficients in H1(T 2;Z). Depending on the cases, one of these parts happens to be enough to complete the computation. In the cases of p2, p4m/p4g and p6m, the detail of the computation is provided in the following subsections. The details for the other cases can be found in old versions of arXiv:1509.09194. 4.3 p2 The lattice Π ⊂ R2 is the standard one Π = Z ⊕ Z and the point group P = Z2 = {±1} acts on Π and R2 by (x, y) 7→ (−x,−y). Theorem 4.6 (p2). The Z2-equivariant cohomology of T 2 is given as follows n = 0 n = 1 n = 2 n = 3 Hn Z2 (T 2;Z) Z 0 Z⊕ Z⊕3 2 0 Proof. We use the Gysin exact sequence for ‘Real’ circle bundles in [10]: We write Hn Z2 (X) = Hn Z2 (X;Z) for the equivariant cohomology and Hn ±(X) ∼= Hn Z(X;Z(1)) for a variant of the equivariant cohomology, which can be formulated by the equivariant cohomology with local coefficients. The torus T 2 is the product of two copies of S̃1, where S̃1 = U(1) is the circle with the involution z 7→ z−1. We can think of S̃1 × S̃1 as the trivial ‘Real’ circle bundle on S̃1. 24 K. Gomi Similarly, S̃1 is the trivial ‘Real’ circle bundle on pt. The Gysin exact sequences for these ‘Real’ circle bundles are split, and we find Hn Z2 ( T 2 ) ∼= Hn Z2 ( S̃1 ) ⊕Hn−1 ± ( S̃1 ) ∼= Hn Z2 (pt)⊕Hn−1 ± (pt)⊕Hn−1 ± (pt)⊕Hn−2 Z2 (pt). As given in [10], the cohomology Hn ±(pt) is isomorphic to Z2 if n > 0 is odd, and is trivial otherwise. We already know Hn Z2 (pt), and get Hn Z2 (T 2) easily. � 4.4 p4m/p4g The lattice Π = Z2 ⊂ R2 is standard. The point group is P = D4 = 〈C4, σx \|C4 4 , σ 2 x, σxC4σxC4〉. The D4-action on Π and R2 is given by the following matrix presentation: C4 = ( 0 −1 1 0 ) , σx = ( −1 0 0 1 ) . In the rest of this subsection, we will use the following notations to indicate elements in D4: 1, C4, C2 = C2 4 , C−1 4 = C3 4 , σx, σd = σxC4, σy = C2σx, σ′d = C4σx. The closure of a fundamental domain is {s(1, 0) + t(0, 1) ∈ R2 \| 0 ≤ s, t ≤ 1}. Then we find that the D4-action on T 2 = R2/Π satisfies the assumptions in Lemma 4.4, in which pt = (0, 0) and S1 ∨ S1 ∼= Y = ((R⊕ 0)/(Z⊕ 0)) ∨ ((0⊕ R)/(0⊕ Z)). To apply Lemma 4.5, we compute the cohomology of Y : Lemma 4.7. The equivariant cohomology of Y is as follows: n = 0 n = 1 n = 2 n = 3 Hn D4 (Y ;Z) Z 0 Z⊕3 2 Z⊕2 2 Proof. We use the Mayer–Vietoris exact sequence: Cover Y by invariant subspaces U and V with the following D4-equivariant homotopy equivalences U ' pt, V ' D4/D (v) 2 , U ∩ V ' D4/Z (v) 2 , where D (v) 2 = {1, C2, σx, σy} ∼= D2 and Z(v) 2 = {1, σy} ∼= Z2. We can summarize the equivariant cohomology of these spaces in low degrees as follows: n = 3 Z2 ⊕ Z2 0 n = 2 Z⊕2 2 ⊕ Z⊕2 2 Z2 n = 1 0 0 n = 0 Z⊕ Z Z Hn D4 (Y ) Hn D4 (U)⊕Hn D4 (V ) Hn D4 (U ∩ V ) In the Mayer–Vietoris exact sequence · · · → Hn D4 (Y )→ Hn D4 (U)⊕Hn D4 (V ) ∆→ Hn D4 (U ∩ V )→ Hn+1 D4 (Y )→ · · · , Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 25 the map ∆: Hn D4 (U)⊕Hn D4 (V ) → Hn D4 (U ∩ V ) is expressed as ∆(u, v) = j∗U (u)− j∗V (v), where jU : U ∩ V → U and jV : U ∩ V → V are the inclusions. Under the natural identifications H2 D4 (V ) ∼= H2 D (v) 2 (pt) ∼= Hom ( D (v) 2 , U(1) ) , H2 D4 (U ∩ V ) ∼= H2 Z(v) 2 (pt) ∼= Hom ( Z(v) 2 , U(1) ) , the map j∗U agrees with the homomorphism induced from the inclusion Z(v) 2 → D (v) 2 . This implies that j∗U is surjective, and so is ∆ in degree 2. Clearly, ∆: H0 D4 (U)⊕H0 D4 (V )→ H0 D4 (U ∩ V ) is identified with the homomorphism Z ⊕ Z → Z given by (m,n) 7→ m − n. Hence we can solve the Mayer–Vietoris exact sequence for {U, V } to get the result claimed in this lemma. � Theorem 4.8 (p4m/p4g). The D4-equivariant cohomology of T 2 in low degrees is as follows: n = 0 n = 1 n = 2 n = 3 Hn D4 (T 2;Z) Z 0 Z⊕3 2 Z⊕3 2 We also have F 2H3 D4 (T 2;Z) ∼= Z⊕2 2 . Proof. In the Leray–Serre spectral sequence Ep,q2 = Hp group(D4;Hq(T 2;Z)), the D4-modules H0(T 2), H1(T 2) and H2(T 2) are identified with the trivial D4-module Z, H1(Y ) and the D4- module Z̃ in Lemma 4.2, respectively. Using Lemmas 4.1 and 4.2, we can summarize the E2-terms as follows: q = 3 0 0 0 0 q = 2 0 Z2 q = 1 0 Z2 Z2 q = 0 Z 0 Z⊕2 2 Z2 Ep,q2 p = 0 p = 1 p = 2 p = 3 Now the proof is completed by Lemma 4.5. � 4.5 p6m We let Π = Za ⊕ Zb ⊂ R2 be the lattice spanned by a = ( 1 0 ) and b = ( 1/2√ 3/2 ) . The point group P is D6 = 〈C, σ1 \|C6, σ2 1, σ1Cσ1C〉 = {1, C, C2, C3, C4, C5, σ1, σ2, σ3, σ4, σ5, σ6}, where σ` = C`−1σ1. This group acts on Π and R2 through the inclusion D6 ⊂ O(2) defined by C = ( 1/2 − √ 3/2√ 3/2 1/2 ) , σ1 = ( 1 0 0 −1 ) . If we use the identifications a = 1 and b = τ = exp 2πi/6 under R2 = C, then the actions of C ∈ D6 and σ1 are given by the multiplication by τ and the complex conjugation, respectively. The closure of a fundamental domain is {sa+ tb \| 0 ≤ s, t ≤ 1} or equivalently {s+ tτ \| 0 ≤ s, t ≤ 1}. We decompose this region to define a D6-CW decomposition of T 2 as follows: 0-cell 1-cell 2-cell ẽ0 0 = pt ẽ1 01 = (D6/{1, σ1})× e1 ẽ2 = D6 × e2 ẽ0 1 = D6/{1, C3, σ1, σ4} ẽ1 02 = (D6/{1, σ2})× e1 ẽ0 2 = D6/{1, C2, C4, σ2, σ4, σ6} ẽ1 12 = (D6/{1, σ4})× e1 26 K. Gomi • (0-cell) The 0-cell ẽ0 0 = (D6/D6) × e0 = pt is the unique fixed point on T 2. The other 0-cells are defined as follows: ẽ0 1 = { 1 2 , τ 2 , 1 + τ 2 } ∼= ( D6/ { 1, C3, σ1, σ4 }) × e0, ẽ0 2 = { 1 + τ 3 , 2(1 + τ) 3 } ∼= ( D6/ { 1, C2, C4, σ2, σ4, σ6 }) × e0. • • • • • • • (1-cell) For 0 ≤ i < j ≤ 2, the 1-cell ẽ1 ij consists of the six segments connecting ẽ0 i and ẽ0 j . They are of the forms ẽ1 01 = (D6/{1, σ1}) × e1, ẽ1 02 = (D6/{1, σ2}) × e1 and ẽ1 12 = (D6/{1, σ4})× e1. ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦ • (2-cell) The 2-cell ẽ2 = D6 × e2 consists of the twelve small triangular regions surrounded by the 1-cells. Let Y ⊂ T 2 be the invariant subspace Y = ẽ0 0 ∪ ẽ0 1 ∪ ẽ1 01. Lemma 4.9. The equivariant cohomology of Y is given as follows: n = 0 n = 1 n = 2 n = 3 Hn D6 (Y ;Z) Z 0 Z⊕3 2 Z⊕2 2 Proof. We can find D6-invariant subspaces U and V in Y which have the following equivariant homotopy equivalences U ' ẽ0 0 = pt, V ' ẽ0 1 = D6/D2, U ∩ V ' ẽ1 01 ' D6/Z (1) 2 , where D2 = {1, C3, σ1, σ4} and Z(1) 2 = {1, σ1}. The equivariant cohomology groups of these spaces can be summarized as follows: n = 3 Z2 ⊕ Z2 0 n = 2 Z⊕2 2 ⊕ Z⊕2 2 Z2 n = 1 0 0 n = 0 Z⊕ Z Z Hn D6 (Y ) Hn D6 (U)⊕Hn D6 (V ) Hn D6 (U ∩ V ) In the Mayer–Vietoris exact sequence · · · → Hn D6 (Y )→ Hn D6 (U)⊕Hn D6 (V ) ∆→ Hn D6 (U ∩ V )→ Hn+1 D6 (Y )→ · · · , Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 27 the homomorphism ∆ is expressed as ∆(u, v) = j∗U (u) − j∗V (v) with jU : U ∩ V → U and jV : U ∩ V → V the inclusions. This immediately determines H0 D6 (Y ) ∼= Z and H1 D6 (Y ) = 0. To complete the proof, we recall the identifications H2 D6 (U) ∼= Hom(D6, U(1)) ∼= Z⊕2 2 , H2 D6 (V ) ∼= Hom(D2, U(1)) ∼= Z⊕2 2 , H2 D6 (U ∩ V ) ∼= Hom ( Z(1) 2 , U(1) ) ∼= Z2, under which j∗U and j∗V are induced from the inclusions D2 → D6 and Z(1) 2 → D6. As a basis of H2 D6 (U) we can choose the following 1-dimensional representations ρi : D6 → U(1) of D6 ρ1 : { C 7→ 1, σ1 7→ −1, ρ2 : { C 7→ −1, σ1 7→ 1. Similarly, we can choose the following 1-dimensional representations ρ′i of D2 = {1, σ1, C 3, σ4} as a basis of H2 D6 (V ) ρ′1 : { C3 7→ 1, σ1 7→ −1, ρ′2 : { C3 7→ −1, σ1 7→ 1. Now, we can see H2 D6 (Y ) ∼= Ker ∆ ∼= Z⊕3 2 , and it has the following basis {(ρ1, ρ ′ 1), (ρ2, ρ ′ 2), (0, ρ′2)} ⊂ Hom(D6, U(1))⊕Hom(D4, U(1)). We can also see that ∆ is surjective, and H3 D6 (Y ) ∼= Z⊕2 2 . � Let X1 be the 1-skeleton of the D6-CW complex T 2. Lemma 4.10. H3 D6 (X1;Z) ∼= Z⊕2 2 . Proof. We cover X1 = ẽ0 0 ∪ ẽ0 1 ∪ ẽ0 2 ∪ ẽ1 01 ∪ ẽ1 02 ∪ ẽ1 12 by invariant subspaces U ′ and V ′ which admit the following equivariant homotopy equivalences U ′ ' Y, V ′ ' ẽ0 2 = D6/D3, U ′ ∩ V ′ ' ẽ1 02 t ẽ1 12 ' D6/Z (2) 2 tD6/Z (4) 2 , where D3 = {1, C2, C4, σ2, σ4, σ6}, Z(2) 2 = {1, σ2} and Z(4) 2 = {1, σ4}. The equivariant cohomol- ogy groups of these spaces are summarized as follows: n = 3 Z⊕2 2 ⊕ 0 0 n = 2 Z⊕3 2 ⊕ Z2 Z2 ⊕ Z2 n = 1 0 0 n = 0 Z⊕ Z Z⊕ Z Hn D6 (X1) Hn D6 (U ′)⊕Hn D6 (V ′) Hn D6 (U ′ ∩ V ′) The homomorphism ∆ in the Mayer–Vietoris exact sequence · · · → H2 D6 (X1)→ H2 D6 (U ′)⊕H2 D6 (V ′) ∆→ H2 D6 (U ′ ∩ V ′)→ H3 D6 (X1)→ · · · is expressed as ∆(u, v) = j∗U ′(u)− j∗V ′(v) by using the inclusions jU ′ : U ′∩V ′ → U ′ and jV ′ : U ′∩ V ′ → V ′. An inspection proves that j∗U ′ agrees with the composition of the following two homomorphisms: 28 K. Gomi (i) the inclusion that follows from the calculation of H2 D6 (Y ) in Lemma 4.9 H2 D6 (U ′) ∼= H2 D6 (Y ) −→ Hom(D6, U(1))⊕Hom(D2, U(1)). (ii) the direct sum i∗2 ⊕ i∗4 of the homomorphisms i∗2 : Hom(D6, U(1))→ Hom ( Z(2) 2 , U(1) ) , i∗4 : Hom(D2, U(1))→ Hom ( Z(4) 2 , U(1) ) , induced from the inclusions i2 : Z(2) 2 → D6 and Z(4) 2 → D2. Then, using the basis presented in the calculation of H2 D6 (Y ), we find j∗U ′(ρ1, ρ ′ 1) = (ρ, ρ), j∗U ′(ρ2, ρ ′ 2) = (ρ, ρ), j∗U ′(0, ρ ′ 2) = (0, ρ), where ρ : Z2 → Z2 is the identity map generating Hom(Z2, U(1)) ∼= Z2. Hence j∗U ′ as well as ∆ are surjective, and H3 D6 (X1) ∼= H3 D6 (Y ) ∼= Z⊕2 2 . � Theorem 4.11 (p6m). H3 D6 (T 2;Z) ∼= Z⊕2 2 . Proof. The relevant part of the exact sequence for the pair (T 2, X1) is H3 D6 ( T 2, X1 ) → H3 D6 ( T 2 ) → H3 D6 (X1)→ H4 D6 ( T 2, X1 ) . By means of the excision axiom, we have Hn D6 (T 2, X1) ∼= Hn−2(pt). Therefore we get H3 D6 (T 2) ∼= H3 D6 (X1) ∼= Z⊕2 2 . � Let Ẑ = Zφ1 be the D6-module such that its underlying group is Z and D6 acts via the homomorphism φ1 : D6 → Z2 given by φ1(C) = −1 and φ1(σ1) = 1. Lemma 4.12. There is an exact sequence of D6-modules 0→ H1 ( T 2;Z ) → H1(Y ;Z) π→ Ẑ→ 0 admitting a module homomorphism s : Ẑ→ H1(Y ;Z) such that π ◦ s = 3. Proof. Let η1, η2 ∈ H1(T 2) be the homology classes of the loops going along the vectors 1 and τ respectively in the fundamental domain, which form a basis of H1(T 2) ∼= Z2. Also, let γ1, γ2, γ3 ∈ H1(Y ) be the homology classes of loops along 1, τ and τ − 1, which form a basis of H1(Y ) ∼= Z3. The inclusion map i : Y → T 2 relates these bases by i∗γ1 = η1, i∗γ2 = η2 and i∗γ3 = η2 − η1. The actions of C ∈ D6 and σ1 on these bases are { C∗η1 = η2, C∗η2 = η2 − η1, { σ1∗η1 = η1, σ1∗η2 = η1 − η2,  C∗γ1 = γ2, C∗γ2 = γ3, C∗γ3 = −γ1,  σ1∗γ1 = γ1, σ1∗γ2 = −γ3, σ1∗γ3 = −γ2. Let {h1, h2} ⊂ H1(T 2) and {g1, g2, g3} ⊂ H1(Y ) be dual to the homology bases. They are related by i∗h1 = g1 − g3 and i∗h2 = g2 + g3, and the induced D6-actions are as follows. { C∗h1 = −h2, C∗h2 = h1 + h2, { σ∗1h1 = h1 + h2, σ∗1h2 = −h2,  C∗g1 = −g3, C∗g2 = g1, C∗g3 = g2,  σ∗1g1 = g1, σ∗1g2 = −g3, σ∗1g3 = −g2. These expressions allow us to prove that the cokernel of the homomorphism i∗ : H1(T 2)→ H2(Y ) is isomorphic to Ẑ, yielding the exact sequence. The homomorphism s : Ẑ→ H1(Y ) is given by s(1) = g1 − g2 + g3. � Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 29 Lemma 4.13. Hn group(D6;H1(T 2;Z)) = 0 for n = 0, 1, 2. Proof. We use the long exact sequence in group cohomology induced from the exact sequence 0 → H1(T 2) → H1(Y ) π→ Ẑ → 0 in coefficients. By Lemmas 4.1, 4.9 and 5.6 to be given in Section 5, we get the following: n = 2 Z2 Z2 n = 1 Z2 Z2 n = 0 0 0 Hn group(D6;H1(T 2)) Hn group(D6;H1(Y )) Hn group(D6; Ẑ) It is clear that H0 group(D6;H1(T 2)) = 0. The homomorphism in group cohomology induced from π : H1(Y ) → Ẑ is surjective in degree 1 and 2, because π ◦ s = 3. This leads to the remaining vanishing. � Theorem 4.14 (p6m). The following holds true: (a) F 2H3 D6 (T 2;Z) ∼= Z2, (b) the D6-equivariant cohomology of T 2 in low degrees is as follows: H0 D6 ( T 2;Z ) ∼= Z, H1 D6 ( T 2;Z ) = 0, H2 D6 ( T 2;Z ) ∼= Z⊕2 2 . Proof. In the E2-term of the Leray–Serre spectral sequence Ep,q2 = Hp group(D6;Hq(T 2;Z)), the coefficient H0(T 2) is identified with the trivial D6-module Z, and H2(T 2) with Z̃ as in Lemma 4.2. The group cohomology with coefficients in H1(T 2) is already computed, and that in Z̃ is also computed in Lemma 4.2. The E2-terms are summarized as follows: q = 3 0 0 0 0 q = 2 0 Z2 q = 1 0 0 0 q = 0 Z 0 Z⊕2 2 Z2 Ep,q2 p = 0 p = 1 p = 2 p = 3 This list and Lemma 4.3 lead to the theorem. � 4.6 The proof of Corollary 1.2 The only non-trivial point in the corollary is (c), which we prove here. Let P be the point group of one of the 2-dimensional space groups. We can assume that P does not preserve the orientation of T 2. Then we have F 2H3 P ( T 2;Z ) ∼= E2,1 2 ⊕ E3,0 2 by Lemma 4.3, in which the direct summands are E2,1 2 = H2 group ( P ;H1 ( T 2;Z )) , E3,0 2 = H3 group(P ;Z) ∼= H2 group(P ;U(1)). Thus, it suffices to prove that the group cocycles τ induced from the nonsymmorphic 2- dimensional space groups as in Section 2 generate E2,1 2 . Recall from Section 2 that the group 2-cocycle ν ∈ Z2 group(P ; Π) measures the failure for a space group S to be a semi-direct product of its point group P and the lattice Π, where Π is regarded as a left P -module naturally. In other words, S is nonsymmorphic if and only if [ν] ∈ H2 group(P ; Π) is non-trivial. 30 K. Gomi Lemma 4.15. Let Π and P be the lattice and the point group of a d-dimensional space group S. Then there is an isomorphism of groups H2 group(P ; Π) ∼= H2 group ( P ;H1 ( Π̂;Z )) . In particular, this factors through the homomorphisms H2 group(P ; Π) −→ H2 group ( P ;C ( Π̂, U(1) )) given by the assignment of the cocycles ν 7→ τ in Section 2 and H2 group ( P ;C ( Π̂, U(1) )) −→ H2 group ( P ;H1 ( Π̂;Z )) induced from the natural surjection δ : C(Π̂, U(1))→ H1(Π̂;Z). Proof. Instead of the left P -action on the Pontryagin dual Π̂ = Hom(Π, U(1)) defined in Section 2, we consider the natural right action k̂(m) 7→ k̂(pm) of p ∈ P on k̂ ∈ Π̂, from which the left action originates. This choice of the actions does not affect the group cohomology. The right P -action on Π̂ induces by pull-back a left P -action on the cohomology H1(Π̂;Z). Thus, the isomorphism of the group cohomologies will be established once we see H1(Π̂;Z) ∼= Π as left P -modules. In general, for each element m ∈ Π ⊂ Rd = V , the path [0, 1] → V , (t 7→ tm) defines a loop in V/Π. This induces an isomorphism of left P -modules Π ∼= H1(V/Π;Z). By the universal coefficient theorem, the dual Π∗ = Hom(Π,Z) of Π is identified with the first homology group of V/Π as a right P -module: Π∗ = Hom(Π,Z) ∼= Hom(H1(V/Π;Z),Z) ∼= H1(V/Π;Z). Considering the dual space V ∗ = Hom(V,R) and its lattice Π∗ instead, we similarly get an isomorphism of left P -modules Π ∼= H1(V ∗/Π∗;Z). Since there is a natural isomorphism of tori V ∗/Π∗ → Π̂ = Hom(Π, U(1)) with right P -actions, the isomorphism of the group cohomologies is proved. The factoring of the isomorphism can be verified by a direct inspection. � Now, in the case of pm/pg, the nonsymmorphic group pg defines the non-trivial element of H2 group(Z2; Π) ∼= Z2 through ν, and the element corresponds by Lemma 4.15 to the non- trivial element of H2 group(Z2;H1(T 2;Z)) ∼= Z2 represented by the group 2-cocycle τ induced from pg. The same holds true in the case of p4m/p4g. In the case of pmm/pmg/pgg, we have H2 group(D2; Π) ∼= H2 group(D2;H1(T 2;Z)) ∼= Z2 ⊕ Z2. In view of the classification of 2- dimensional space groups ([12]), the non-trivial elements (−1, 1) and (−1,−1), with respect to a basis of Z2 ⊕ Z2, correspond to the nonsymmorphic groups pmg and pgg respectively. (The nonsymmorphic group corresponding to (1,−1) is equivalent to pmg through an affine transformation preserving the lattice.) Therefore H2 group(D2;H1(T 2;Z)) ∼= Z2 ⊕Z2 is generated by the group 2-cocycles induced from the nonsymmorphic groups. 5 The twisted case This section concerns the equivariant cohomology with local coefficients. We start with some remarks about the Leray–Serre spectral sequence, focusing on the similarities and the differences with the case of the usual Borel equivariant cohomology. We then summarize tools for computa- tion in the version adapted to the case with local coefficients. After that, we prove Theorems 1.5 and 1.6: As in the untwisted case, the full computation is lengthy, and the details are only pro- vided in the case of p6m. Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 31 5.1 The Leray–Serre spectral sequence For a finite group G and a homomorphism φ : G→ Z2, we define Zφ to be the G-module Zφ such that its underlying group is Z and G acts via φ. Suppose that G acts on a space X. Associated to the fibration X → EG×G X → BG is the long exact sequence of homotopy groups: · · · → πn(X)→ πn(EG×G X)→ πn(BG)→ · · · . Since π1(BG) ∼= G, we get a homomorphism π1(EG×GX)→Z2 by composing π1(EG×G X)→G with φ : G→ Z2. The homomorphism defines a local system on the Borel construction EG×GX, which we denote with the same notation Zφ. By using this local system, we define the G- equivariant cohomology with local coefficients Hn G(X;Zφ) = Hn(EG×G X;Zφ). Of course, if φ is trivial, then the cohomology above recovers the usual equivariant cohomology with integer coefficients Z. For Hn G(X;Zφ), we also have the Leray–Serre spectral sequence Ep,qr converging to the graded quotient of a filtration Hn G(X;Zφ) = F 0Hn G(X;Zφ) ⊃ F 1Hn G(X;Zφ) ⊃ F 2Hn G(X;Zφ) ⊃ · · · . Its E2-term is again given by the group cohomology Ep,q2 = Hp group(G;Hq(X;Z)⊗ Zφ), where Hq(X;Z) ⊗ Zφ is the tensor product of the G-modules Hq(X;Z) and Zφ, namely, its underlying group is Hq(X;Z)⊗ Zφ = Hq(X;Z)⊗ Z ∼= Hq(X;Z), and g ∈ G acts on x ∈ Hq(X;Z) by x 7→ φ(g) · g∗x. As in the usual equivariant cohomology, Hn G(X;Zφ) can be identified with a sheaf cohomology of the simplicial space G• ×X. (See for instance the appendix of [10] in the case of G = Z2.) This interpretation leads to the classification of the twists for the Freed–Moore K-theory [8], whose definition is similar to the one given in Section 3.2 (see [18]). In terms of the Borel equivariant cohomology, the graded twists are classified by H1 G(X;Z2) × H3 G(X;Zφ) and the ungraded twists by H3 G(X;Zφ). The results in Section 3.3 can be generalized to the equivariant cohomology with coefficients in Zφ, and we get the following geometric interpretation generalizing Corollary 3.12: Proposition 5.1. Let G be a finite group acting on a compact and path connected space X with at least one fixed point, and φ : G→ Z2 a homomorphism. (i) F 1H3 G(X;Zφ) classifies (ungraded) twists which can be represented by φ-twisted central extensions of the groupoid X//G. (ii) F 2H3 G(X;Zφ) classifies (ungraded) twists which can be represented by 2-cocycles of G with coefficients in the G-module C(X,U(1))φ, where C(X,U(1))φ = C(X,U(1)) is the group of U(1)-valued functions on X and g ∈ G acts on f : X → U(1) by f 7→ g∗fφ(g). (iii) F 3H3 G(X;Zφ) = H2 group(G;U(1)φ) classifies (ungraded) twists which can be represented by 2-cocycles of G with coefficients in the G-module U(1)φ, where U(1) = U(1)φ as a group and g ∈ G acts on u ∈ U(1) by u 7→ uφ(g). 32 K. Gomi 5.2 Tools As long as we are concerned with the local system Zφ associated to a homomorphism φ : G→ Z2, the reduced cohomology H̃n G(X;Zφ) makes sense for a G-space X with a fixed point pt ∈ X, and we have the direct sum decomposition Hn G(X;Zφ) ∼= Hn G(pt;Zφ)⊕H̃n G(X;Zφ). Then we can generalize the proof of Lemma 4.1 to show: Lemma 5.2. Suppose that a finite group G acts on a path connected space Y fixing at least one point pt ∈ Y . Suppose further that Y is a CW complex consisting of only cells of dimension less than or equal to 1. Then, for any homomorphism φ : G → Z2 and n ≥ 0, the following holds true: Hn group ( G;H1(Y ;Z)⊗ Zφ ) ∼= H̃n+1 G (Y ;Zφ). Similarly, we can generalize Lemma 4.5 as follows: Lemma 5.3. Suppose a finite group G acts on the torus T 2 = S1 × S1 and • there is a fixed point pt = (x0, y0) ∈ T 2 under the G-action, • G preserves the subspace S1 ∨ S1 = S1 × {y0} ∪ {x0} × S1 ⊂ T 2. Then there is the following isomorphism of groups for any homomorphism φ : G → Z2 and all n ∈ Z Hn G ( T 2;Zφ ) ∼= Hn G(pt;Zφ)⊕ H̃n G ( S1 ∨ S1;Zφ ) ⊕ H̃n G ( T 2/S1 ∨ S1;Zφ ) . Further, the Leray–Serre spectral sequence for Hn G(T 2;Zφ) degenerates at E2 and the relevant extension problems are trivial, so that (a) F 2H3 G(T 2;Zφ) ∼= E2,1 2 ⊕ E3,0 2 , (b) Hn G(T 2;Zφ) ∼= ⊕ p+q=nE p,q 2 for all n ∈ Z. Besides the generalizations above, we will use the following universal coefficient theorem in the sequel: Lemma 5.4. For any φ : G→ Z2, there is a split exact sequence of groups 0→ Hn G(X;Zφ)⊗ Z2 → Hn G(X;Z2)→ Tor ( Hn+1 G (X;Zφ),Z2 ) → 0. Proof. For any homomorphism φ : G→ Z2, let (Z2)φ be the G-module such that its underlying group is Z2 and its G-action is given by φ : G → Z2. This G-module (Z2)φ agrees with the trivial G-module Z2, even if φ is non-trivial. Then, looking at the cochain complexes defining the equivariant cohomology, the usual proof of the universal coefficient theorem leads to the lemma. Another proof is to use the Thom isomorphism, which unwinds the local coefficients: Let Rφ → X be the G-equivariant real line bundle on X whose underlying bundle is X ×R and the action of g ∈ G on (x, r) ∈ X × R is (x, r) 7→ (gx, φ(g)r). The Thom isomorphism theorem then provides Hn G(X;Aφ) ∼= Hn+1 G (D,S;A), where A is Z2 or Z, and D ⊂ Rφ and S ⊂ Rφ are the unit disk bundle and the unit sphere bundle, respectively. Then the usual universal coefficient theorem leads to the present lemma. � To compute the equivariant cohomology Hn G(X;Zφ), we usually need the cohomology of the point Hn G(pt;Zφ). This cohomology is identified with the group cohomology Hn group(G;Zφ) by the degeneration of the Leray–Serre spectral sequence, but its direct computation is not realistic except for the simplest cases (cf. Lemma 4.2). A useful way to compute it is: Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 33 Lemma 5.5. For any φ : G→ Z2, there are natural exact sequences · · · → Hn−1 G (pt;Zφ)→ Hn G(pt;Z) i∗→ Hn Kerφ(pt;Z)→ Hn G(pt;Zφ)→ · · · , · · · → Hn−1 G (pt;Z)→ Hn G(pt;Zφ) i∗→ Hn Kerφ(pt;Z)→ Hn G(pt;Z)→ · · · , where i∗ is induced from the inclusion i : Kerφ→ G. Proof. Let G act on Rφ = R via φ : G → Z2. We can regard Rφ as a G-equivariant real line bundle on pt. We have the Thom isomorphisms Hn G(pt;Zφ) ∼= Hn+1 G (D,S;Z), Hn G(pt;Z) ∼= Hn+1 G (D,S;Zφ), where D and S are the unit interval in Rφ and its boundary, respectively. Note that D is equiv- ariantly contractible. Note also that S ∼= G/Kerφ as a G-space. Thus, we have isomorphisms Hn G(D;Zφ) ∼= Hn G(pt;Zφ), Hn G(S;Zφ) ∼= Hn Kerφ(pt;Z). Substituting these isomorphisms and the Thom isomorphisms into the exact sequences for the pair (D,S), we complete the proof. � By means of the lemma above, we get: Lemma 5.6. Let P be Z2m, D2m−1 or D2m with m ≥ 1. The P -equivariant cohomology of the point with coefficients in Zφ, Hn P (pt;Zφ) ∼= Hn group(P ;Zφ), in low degrees is given as follows: P φ H0 P (pt;Zφ) H1 P (pt;Zφ) H2 P (pt;Zφ) H3 P (pt;Zφ) Z2m φ1 0 Z2 0 Z2 D2m−1 φ0 0 Z2 Z2m−1 Z2 D2m φ0 0 Z2 Z2m Z⊕2 2 D2m φ1, φ2 0 Z2 Z2 Z⊕2 2 Proof. In the case of D2m with m even and φ 6= φ0, the first exact sequence in Lemma 5.5 leads to H0 D2m (pt;Zφ) = 0 and H1 D2m (pt;Zφ) ∼= Z2. This computation also shows that H2 D2m (pt;Zφ) contains Z2 as a subgroup. Here, applying the universal coefficient theorem to Hn P (pt;Z), we compute the cohomology with coefficients in Z2 to have H1 D2m (pt;Z2) ∼= Z⊕2 2 . If we apply the universal coefficient theorem in Lemma 5.4 to Hn P (pt;Zφ), then H1 D2m (pt;Z2) ∼= Z2 ⊕ Tor ( H2 D2m (pt;Zφ),Z2 ) . Thus the consistency of these computations implies H2 D2m (pt;Zφ) ∼= Z2. Based on this result, the second sequence in Lemma 5.5 suggests that H3 D2m (pt;Zφ) is either Z⊕2 2 or Z4. If we apply the universal coefficient theorem to Hn P (pt;Z), then H2 D2m (pt;Z2) ∼= Z⊕3 2 . If we compute this cohomology applying Lemma 5.4 to Hn P (pt;Zφ), then H2 D2m (pt;Z2) ∼= Z2 ⊕ Tor ( H3 D2m (pt;Zφ),Z2 ) . Therefore we conclude that H3 D2m (pt;Zφ),Z2) ∼= Z⊕2 2 by the consistency. In the other cases, a combined use of the two exact sequences in Lemma 5.5 determines the group Hn P (pt;Zφ) without difficulty. � 34 K. Gomi 5.3 The proof of Theorems 1.5 and 1.6 Theorems 1.5 and 1.6 again follow from case-by-case computations. To these cases, we can apply the methods in the proof of Theorems 1.1 and 1.3. However, in some cases, only the possibility of a cohomology group is suggested by an exact sequence. In this case, we apply an argument used in the proof of Lemma 5.6: We compute the cohomology with coefficients in Z2 applying the universal coefficient theorem to the result in Theorem 1.3. Then the consistency with Lemma 5.4 eventually determines the cohomology in question. In the following, we carry out the computation in the case of p6m with φ = φ2. Let Y ⊂ T 2 be the D6-invariant subspace given in Section 4.5. Lemma 5.7. The D6-equivariant cohomology of Y with coefficients in Zφ2 in low degrees is as follows: n = 0 n = 1 n = 2 n = 3 Hn D6 (Y ;Zφ2) 0 Z2 Z⊕2 2 Z⊕3 2 Proof. To use the Mayer–Vietoris sequence, we cover Y by D6-invariant subspaces U and V which have the following D6-equivariant homotopy equivalences U ' pt, V ' D6/D2, U ∩ V ' D6/Z2, where D2 = {1, C3, σ1, σ4} ⊂ D6 and Z2 = {1, σ1} ⊂ D6. We see Hn D6 (V ;Zφ2) ∼= Hn D2 (pt;Zφ2), Hn D6 (U ∩ V ;Zφ2) ∼= Hn Z2 (pt;Zφ1). The equivariant cohomology groups in low degrees can be summarized as follows: n = 3 Z⊕2 2 ⊕ Z⊕2 2 Z2 n = 2 Z2 ⊕ Z2 0 n = 1 Z2 ⊕ Z2 Z2 n = 0 0⊕ 0 0 Hn D6 (Y ;Zφ2) Hn D6 (U t V ;Zφ2) Hn D6 (U ∩ V ;Zφ2) We have H0 D6 (Y ;Zφ2) = 0 clearly, and H1 D6 (Y ;Zφ2) is either Z2 or Z⊕2 2 . Applying the universal coefficient theorem to Lemma 4.9, we find H0 D6 (Y ;Z2) ∼= Z2. This result must be consistent with the computation of H0 D6 (Y ;Z2) by using Lemma 5.4, which leads to H1 D6 (Y ;Zφ2) ∼= Z2. Solving the Mayer–Vietoris exact sequence, we then find H2 D6 (Y ;Zφ2) ∼= Z⊕2 2 . We also find that H3 D6 (Y ;Zφ2) is either Z⊕3 2 or Z⊕4 2 . Computing again the cohomology with Z2-coefficients in two ways, we conclude that H3 D6 (Y ;Zφ2) ∼= Z⊕3 2 . � Lemma 5.8. There is an exact sequence of D6-modules 0→ H1 ( T 2;Z ) ⊗ Zφ2 → H1(Y ;Z)⊗ Zφ2 π→ Zφ0 → 0 admitting a module homomorphism s : Zφ0 → H1(Y ;Z)⊗ Zφ2 such that π ◦ s = 3. Proof. The proof of Lemma 4.12 can be adapted to this case. � Lemma 5.9. Hn group(D6;H1(T 2;Z)⊗ Zφ2) = 0 for n = 0, 1, 2. Proof. We use the long exact sequence of group cohomology induced from the short exact sequence of coefficients. Notice that Hn group ( D6;H1(Y ;Z)⊗ Zφ2 ) ∼= H̃n+1 D6 (Y ;Zφ2). Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 35 The relevant cohomology can be summarized as follows: 2 Z2 Z6 1 Z2 Z2 0 0 0 n Hn group(D6;H1(T 2)⊗ Zφ2) Hn group(D6;H1(Y )⊗ Zφ2) Hn group(D6;Zφ0) By s : Zφ0 → H1(Y ;Z)⊗ Zφ2 , the group cohomology is determined as stated. � Theorem 5.10 (p6m with φ2). The D6-equivariant cohomology of T 2 with coefficients in Zφ2 in low degrees is given as follows: n = 0 n = 1 n = 2 n = 3 Hn D6 (T 2;Zφ2) 0 Z2 Z2 Z⊕3 2 We also have: F 2H3 D6 (T 2;Zφ2) ∼= F 3H3 D6 (T 2;Zφ2) ∼= Z⊕2 2 . Proof. In the E2-term of the Leray–Serre spectral sequence, we have the following iden- tifications Hn group ( D6;H0 ( T 2;Z ) ⊗ Zφ2 ) ∼= Hn D6 (pt;Zφ2), Hn group ( D6;H2 ( T 2;Z ) ⊗ Zφ2 ) ∼= Hn D6 (pt;Zφ1). We can summarize the E2-terms as follows: q = 3 0 0 0 0 q = 2 0 Z2 q = 1 0 0 0 q = 0 0 Z2 Z2 Z⊕2 2 Ep,q2 p = 0 p = 1 p = 2 p = 3 Because En,02 must survive into the direct summand Hn D6 (pt;Zφ2) of the cohomology Hn D6 (T 2;Zφ2), we get the degeneration Ep,q2 = Ep,q∞ for p + q ≤ 2, and the relevant extension problems are readily solved. We also have H3 D6 (T 2;Zφ2) ∼= Z⊕2 2 ⊕ E 1,2 ∞ , where E1,2 ∞ ⊂ E1,2 2 = Z2 is either Z2 or 0. By computing the cohomology with Z2-coefficients in two ways, we conclude that E1,2 ∞ = E1,2 2 = Z2. � A The list of 2-dimensional space groups Here is a list of the lattices Π and the point groups P of the 2-dimensional space groups S. In the nonsymmorphic case, the map a : P → R2 in Section 2 is also presented. A.1 Oblique, rectangular and square lattices For p1, p2, p4, pm, pg, pmm, pmg, pgg, p4m and p4g, we can take the lattice Π ⊂ R2 to be the standard lattice Π = Z2. • (p1) The point group is trivial. • (p2) The point group Z2 = 〈C \|C2〉 acts on Π and R2 through the matrix C = ( −1 0 0 −1 ) . 36 K. Gomi • (p4) The point group Z4 = 〈C \|C4〉 acts on Π and R2 through C = ( 0 −1 1 0 ) . • (pm/pg) The point group D1 = 〈σ \|σ2〉 acts on Π and R2 through( −1 0 0 1 ) . In the case of pg, the map a : D1 → R2 is given by a1 = ( 0 0 ) , aσ = ( 0 1/2 ) . • (pmm/pmg/pgg) The point group is D2 ∼= Z2 × Z2. We let the following matrices σx and σy generate D2, and act on Π and R2 σx = ( −1 0 0 1 ) , σy = ( 1 0 0 −1 ) . In the case of pmg, the map a : D2 → R2 is given by a1 = ( 0 0 ) , aσx = ( 0 1/2 ) , aσy = ( 0 0 ) , aσxσy = ( 0 1/2 ) . In the case of pgg, the map a : D2 → R2 is given by a1 = ( 0 0 ) , aσx = ( 0 1/2 ) , aσy = ( 1/2 0 ) , aσxσy = ( 1/2 1/2 ) . • (p4m/p4g) The point group is D4 = 〈C4, σx \|C4 4 , σ 2 x, σxC4σxC4〉, which acts on Π and R2 through the following matrix presentation C4 = ( 0 −1 1 0 ) , σx = ( −1 0 0 1 ) . In the case of p4g, the map a : D4 → R2 is as follows: p 1 C4 C2 4 C3 4 σx σd σy σ′d ap [ 0 0 ] [ 0 1 2 ] [ 1 2 1 2 ] [ 1 2 0 ] [ 0 1 2 ] [ 0 0 ] [ 1 2 0 ] [ 1 2 1 2 ] In the above, we set σd = σxC4, σy = C2 4σx and σ′d = C4σx. A.2 Rhombic lattices For cm and cmm, the lattice is Π = Za⊕ Zb ⊂ R2, where a = ( 1 1 ) , b = ( −1 1 ) . • (cm) The point group D1 = 〈σ\|σ2〉 acts on Π and R2 by σ = ( −1 0 0 1 ) . • (cmm) The point group is D2 ∼= Z2 × Z2. The following matrices σx and σy generate D2, and define the D4-action on Π and R2 σx = ( −1 0 0 1 ) , σy = ( 1 0 0 −1 ) . Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 37 A.3 Hexagonal lattices For p3, p6, p3m1, p31m and p6m, the lattice Π = Za⊕ Zb ⊂ R2 is spanned by a = ( 1 0 ) , b = ( 1/2√ 3/2 ) . • (p3) The point group Z3 = 〈C \|C3〉 acts on Π and R2 through C = ( −1/2 − √ 3/2√ 3/2 −1/2 ) . • (p6) The point group Z6 = 〈C \|C6〉 acts on Π and R2 through C = ( 1/2 − √ 3/2√ 3/2 1/2 ) . • (p3m1) The point group is D3 = 〈C, σx \|C3, σ2 x, σxCσxC〉. We let D3 act on Π and R2 through the inclusion D3 ⊂ O(2) given by C = ( −1/2 − √ 3/2√ 3/2 −1/2 ) , σx = ( −1 0 0 1 ) . • (p31m) The point group is D3 = 〈C, σy \|C3, σ2 y , σyCσyC〉. We let D3 act on Π and R2 through the inclusion D3 ⊂ O(2) given by C = ( −1/2 − √ 3/2√ 3/2 −1/2 ) , σy = ( 1 0 0 −1 ) . • (p6m) The point group is D6 = 〈C, σ1 \|C6, σ2 1, σ1Cσ1C〉. We let D6 act on Π and R2 through the inclusion D6 ⊂ O(2) given by C = ( 1/2 − √ 3/2√ 3/2 1/2 ) , σ1 = ( 1 0 0 −1 ) . Acknowledgements I would like to thank K. 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Henri Poincaré 17 (2016), 757–794, arXiv:1406.7366. https://doi.org/10.1007/BFb0065364 https://doi.org/10.1112/jtopol/jtr019 http://arxiv.org/abs/0711.1906 https://doi.org/10.1007/s00023-013-0236-x http://arxiv.org/abs/1208.5055 http://arxiv.org/abs/math.DG/0307373 https://doi.org/10.1007/s00220-014-2153-3 http://arxiv.org/abs/1310.8446 https://doi.org/10.2307/2322930 https://doi.org/10.1007/978-3-540-79890-3 https://doi.org/10.1088/1742-6596/597/1/012048 https://doi.org/10.1007/s00023-015-0443-8 http://arxiv.org/abs/1208.3266 https://doi.org/10.1063/1.3149495 http://arxiv.org/abs/0901.2686 https://doi.org/10.1142/S0129167X16500580 http://arxiv.org/abs/1511.05312 https://doi.org/10.1090/cbms/091 https://doi.org/10.1090/cbms/091 https://doi.org/10.1017/S1446788700033097 https://doi.org/10.1017/S1446788700033097 https://doi.org/10.2307/2320063 https://doi.org/10.2307/3617798 https://doi.org/10.1112/blms/16.3.209 https://doi.org/10.1103/PhysRevB.91.155120 http://arxiv.org/abs/1502.03265 https://doi.org/10.1103/PhysRevB.93.195413 http://arxiv.org/abs/1511.01463 http://arxiv.org/abs/1701.08725 https://doi.org/10.1007/s00023-015-0418-9 http://arxiv.org/abs/1406.7366 1 Introduction 2 From quantum systems to twisted K-theory 2.1 Setting 2.2 Bloch transformation 2.3 Nonsymmorphic group and twisted K-theory 2.4 Actions of the point group on the torus 3 The Leray–Serre spectral sequence and twists 3.1 Spectral sequences 3.2 Twists 3.3 Comparison of two spectral sequences 4 The proof of Theorems 1.1 and 1.3 4.1 Some generality 4.2 The outline of computations 4.3 p2 4.4 p4m/p4g 4.5 p6m 4.6 The proof of Corollary 1.2 5 The twisted case 5.1 The Leray–Serre spectral sequence 5.2 Tools 5.3 The proof of Theorems 1.5 and 1.6 A The list of 2-dimensional space groups A.1 Oblique, rectangular and square lattices A.2 Rhombic lattices A.3 Hexagonal lattices References

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

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