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-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.


Introduction
Topological K-theory has recently been recognized as a useful tool for a classification of topological 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 account. 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 system on the d-dimensional space R d 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 Z 2 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 Z 2 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' The equivariant twists on T d arising from quantum systems on R d , to be explained in Section 2, belong to F 2 H 3 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 2dimensional crystallographic group, a plane symmetry group, a wallpaper group, and so on. It is a subgroup of the Euclidean group R 2 O(2) of isometries of R 2 , and is an extension of a finite group P ⊂ O(2) called the point group by a rank 2 lattice Π ∼ = Z 2 of translations of R 2 : Being a normal subgroup of S, the lattice Π ⊂ R 2 is preserved by the action of P on R 2 through the inclusion P ⊂ O(2) and the standard left action of O(2) on R 2 . This induces the left action of P on the torus T 2 = R 2 /Π that we will consider. Since P is a finite subgroup of O (2), it is the cyclic group Z n of order n or the dihedral group D n = C, σ | C n , σ 2 , 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 = R 2 /Π via P ⊂ O(2) as above. Then, H 3 P (T 2 ; Z) = F 0 H 3 P (T 2 ; Z) = F 1 H 3 P (T 2 ; Z). This cohomology group and its subgroups F p H 3 P (T 2 ; Z) are as in Fig. 1.
Space group S P ori H 3 P (T 2 ; Z) F 2 F 3 E 1,2 The list of H 3 P (T 2 ; Z) and its subgroups F p = F p H 3 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 E p,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. (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 2 H 3 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 2 H 3 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 H 3 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 Z 2 -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 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 H n 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 (2). For n ≤ 3, the P -equivariant cohomology H n P (T 2 ; Z) is as given in Fig. 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 Z 2 , 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 H 1 P (T 2 ; Z 2 ) also admits a filtration Because the degree in question is 1, the degeneration of the Leray-Serre spectral sequence gives the identification which is a direct summand of H 1 P (T 2 ; Z 2 ) and is also computed immediately by using the knowledge of the equivariant cohomology of the space consisting of one point, pt = {one point}, in Section 4.1.
The grading of twists classified by F 1 H 1 P (T 2 ; Z 2 ) = Hom(P, Z 2 ) 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 insulators. The symmetry of KR-theory however concerns Z 2 -actions only, and its use is limited to rather simple cases. To take more general symmetries into account, Freed and Moore introduced 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 → Z 2 . The K-theory of Freed-Moore reduces to the G-equivariant K-theory if φ is trivial, and to the KR-theory if G = Z 2 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 Z 4 -phase.
The knowledge about the twists of the Freed-Moore K-theory has therefore potential importance 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 H 1 G (X; Z 2 )×H 3 G (X; Z φ ) and the ungraded twists by H 3 G (X; Z φ ). Here Z φ denotes a local system for the Borel equivariant cohomology associated to the G-module Z φ such that its underlying group is Z and G acts via φ : G → Z 2 . The cohomology group H n G (X; Z φ ) also admits a filtration 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 → Z 2 .
• In the cases of p2, p4 and p6, the point group P is the cyclic group Z 2m = C | C 2m of even order. We write φ 1 : Z 2m → Z 2 for the unique non-trivial homomorphism given by φ 1 (C) = −1.
• In the other case, the point group P is the dihedral group D n = C, σ | C n , σ 2 , σCσC of degree n and order 2n, and D n is embedded into O(2) so that C is a rotation of R 2 and σ is a reflection. We define φ 0 : D n → Z 2 to be the composition of the inclusion D n → O(2) and det : O(2) → Z 2 . Put differently, φ 0 (C) = 1 and φ 0 (σ) = −1. This provides the unique non-trivial homomorphism D n → Z 2 if n is odd. In the case of even n, we define two more non-trivial homomorphisms φ i : D n → Z 2 by Theorem 1.5. Let P be the point group of one of the 2-dimensional space groups S, acting on . This cohomology group and its subgroups F p H 3 P (T 2 ; Z φ ) are as in Fig. 4. Figure 4. The list of H 3 P (T 2 ; Z φ ) and its subgroups F p = F p H 3 P (T 2 ; Z φ ). The E ∞ -term of the Leray-Serre spectral sequence is related to these subgroups by E p,3−p It should be noticed that the action of the point group P on the torus relevant to an application of the Freed-Moore K-theory to condensed matter physics is the one modified by a non-trivial homomorphism φ : P → Z 2 . 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 H 3 P (T 2 ; Z φ ) and its filtration is similar to the one computing H 3 P (T 2 ; Z). In the computation, the cohomology H n P (T 2 ; Z φ ) 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 . For n ≤ 3, the P -equivariant cohomology H n P (T 2 ; Z φ ) with coefficients in the local system Z φ induced from a non-trivial homomorphism φ : P → Z 2 is as in Fig. 5. 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 decomposition 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 Z φ associated to a homomorphism φ : P → Z 2 . For the cohomology with such a local system, the notion of reduced cohomology does not make sense.

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 coefficient Z φ . 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 φ 2 . Finally, for convenience, the point group actions of 2-dimensional space groups are listed in Appendix.

From quantum systems to twisted K-theory
We here illustrate how twisted equivariant K-theory arises from a quantum system with symmetry, mainly based on a formulation in [8]. (We refer the reader to [29] for a C * -algebraic approach.)

Setting
Let us consider the following mathematical setting: • A subgroup S of the Euclidean group R d O(d) of R d which is an extension of a finite group P ⊂ O(d) by Π: 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 nonsymmorphic.
Based on the mathematical setting above, we can introduce a quantum system on R d which has S as its symmetry and V as its internal freedom: • The 'quantum Hilbert space' consisting of 'wave functions' is the L 2 -space L 2 (R d , V ), on which g ∈ S acts by ψ(x) → (ρ(g)ψ)(x) = U (π(g))ψ(g −1 x).
• The 'Hamiltonian' is a self-adjoint operator H on L 2 (R d , V ) invariant under the S-action:

Bloch transformation
Even if the Hamiltonian H is invariant under the translation of Π, a solution ψ to the 'timeindependent 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. LetΠ = 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 We also define transformationsB and B, inverse to each other: As is described in [8], the space In summary, we get an identification of L 2 -spaces The Hamiltonian H on L 2 (R d , V ) then induces an operatorĤ on L 2 If, for instance, H is of the form H = ∆ + Φ, thenĤ 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 ofĤ(k) would be confined to a compact region in R ask ∈Π 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 →Π is regarded as an invariant of the quantum system under study.

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 onΠ to the S-module L 2 (R d , V ). This is a version of the so-called 'Mackey machine'.
Recall that the Euclidean group R d O(d) is the semi-direct product of the orthogonal group O(d) and the group of translations R d . Hence a collection of representatives {s p } p∈P of p ∈ P ∼ = S/Π in S is expressed as s p = (a p , p) ∈ R d O(d) by means of a map a : P → R d . For Since Π ⊂ S is normal, the action of P ⊂ O(d) on R d preserves Π ⊂ R d . Then we have ν(p 1 , p 2 ) ∈ 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 → 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 L 2 (R d , V ), we define an 'action' of p ∈ P by Bloch transformation then induces the following 'action' of P , Here the left P -action onΠ is defined by (pk)(m) =k(p −1 m), where p ∈ P acts on m ∈ Π through the inclusion P ⊂ O(d) and the left action of O(d) on R d . Notice that ρ andρ 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:

To interpret the 'action'ρ(p) in terms of the vector bundle
Define for p ∈ P andk ∈Π a linear map by the assignment of the sections These maps constitute a vector bundle map ρ E⊗V (p) : E ⊗V → E ⊗V covering the actionk → pk onΠ This is a τ -twisted P -action, in the sense that the formula ρ E⊗V p 1 ; p 2k ρ E⊗V p 2 ;k ξ = τ p 1 , p 2 ;k ρ E⊗V p 1 p 2 ;k ξ holds for p 1 , p 2 ∈ P ,k ∈Π and ξ ∈ E|k ⊗ V . Here τ : P × P ×Π → U (1) is defined by , and is regarded as a group 2-cocycle of P with its coefficients in the group C(Π, U (1)) of U (1)valued functions onΠ thought of as a right P -module through the pull-back under the left action k → pk of p ∈ P onk ∈Π. The map ρ E⊗V (p) on the vector bundle induces the transformation on the sections Now, under the assumption thatĤ describes an insulator, the Bloch bundle E ⊂ E ⊗V inherits 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 K τ +0 P (Π), 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 K τ +ω+0 P (Π).
Remark 2.1. The phase factor in the composition rule ofρ, defines a group 2-cocycle of P with coefficients in C(Π, U (1)), when regarded as a left P -module by the right actionk →kp = p −1k of p ∈ P onk ∈Π. The 2-cocycles τ and τ R are related by τ R (k; p 1 , p 2 ) = τ (p 1 , p 2 ; (p 1 p 2 ) −1k ). This extends to a cochain bijection of group cochains with coefficients in the left/right P -modules C(Π, U (1)). Thus, τ and τ R have cohomologically the same information. We also remark that τ and τ R are respectively cohomologous to the following 2-cocycles: Remark 2.2. Given a homomorphism c : P → Z 2 , 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 K τ +c+0 . 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 representation. Besides the argument in this section, there are other arguments which derive projective representations from quantum systems with symmetry (for example [15,16]).

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.
K. Gomi (A) By the inclusion P ⊂ O(d) and the standard left action of O(d) on R d , the point group P acts on R d , preserving Π ⊂ R d . Hence the left action of P on R d descends to give a left action of P on the torus R d /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.
, then the action of p ∈ P on k ∈ R d /Π in (C) admits the description k → pk + a p . If S is symmorphic, then we can choose a p to be in Π. In this case, the actions (A) and (C) are equivalent. However, if S is nonsymmorphic, then a p cannot be in Π. Thus, in this case, the action of p ∈ P does not fix any point on R d /Π, so that the actions (A) and (C) are not equivalent. For example, in the case of pg, the action of P = Z 2 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 R d /Π withΠ = 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 {v j } of the lattice Π ∼ = Z d . This choice induces the following identifications of tori inverse to each other: With this identification of tori, the left P -action onΠ in (B) induces a right P -action on R d /Π. Considering the action of p −1 instead of p, we finally get a left action of P on R d /Π, induced from (B) and the identificationΠ ∼ = R d /Π. In general, this left action of P on R d /Π induced from (B) is not equivalent to the action (A). In the 2-dimensional case, their relationship is as follows: (a) We choose a basis {v j } of Π S to identifyΠ S with R 2 /Π S , and let the action in (B) induce an action of P S on R 2 /Π S . Then, up to equivalence, this action is independent of the choice of the basis.
(c) If S is p3m1 (respectively p31m), then the action of P S on R 2 /Π S induced from (B) is equivalent to the action of P S on R 2 /Π S 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 P S = P S and Π S = Π S in the third item in the lemma above.
Proof . Considering the action (A), we define ψ(p) j ∈ Z by pv j = ψ(p) j v , and a homomorphism ψ : P S → GL(2, Z) by ψ(p) = (ψ(p) j ). Since P S is the point group of a 2-dimensional space group S, the homomorphism ψ is injective and its image ψ(P S ) is a finite subgroup of GL(2, Z). Let S be another 2-dimensional space group with its point group P . Choosing a basis of its lattice Π , we similarly get from the action (A) a homomorphism ψ : P S → GL(2, Z). If the images ψ(P S ) and ψ (P S ) are conjugate to each other in GL(2, Z), then the actions of P S and P S in (A) are equivalent. The action of P S on R 2 /Π S induced from (B) also yields an associated homomorphism P S → GL(2, Z). This homomorphism turns out to be the transpose inverse t ψ −1 : P S → GL(2, Z), which is again injective and defines a finite subgroup t ψ −1 (P S ) ⊂ GL(2, Z). If we alter the basis {v j }, then t ψ −1 changes by a conjugation of a matrix in GL(2, Z). Thus, up to conjugations, the image t ψ −1 (P S ) ⊂ GL(2, Z) is independent of the choice of {v j }, 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(P S ) and t ψ −1 (P S ) in GL(2, Z).
Another way of identifyingΠ with R d /Π is to choose a bilinear form , : Π × Π → Z. We assume that this form is non-degenerate in the sense that the matrix ( v i , v j ) is invertible with respect to any basis {v i } of Π. A non-degenerate bilinear form induces an identification of the tori as follows 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 R d /Π agrees with the action (B) onΠ under the induced identification R d /Π ∼ =Π. For the 2-dimensional space groups such that Π can be the standard lattice Z 2 ⊂ R 2 , the standard inner product on R 2 restricts to give a P -invariant non-degenerate bilinear form. If we choose an orthonormal basis {v j }, then the identifications R d /Π ∼ =Π given by , and by {v j } are P -equivariantly the same.
In Section 4, we will work with the torus R 2 /Π with the action (A), and the relation toΠ with the action (B) should be understood as above.

The Leray-Serre spectral sequence and twists
This section gives a geometric interpretation of the filtration of H 3 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 'reasonable' space X, such as a locally contractible, paracompact and regular topological space as in [7], or a G-CW complex [19].

Spectral sequences
The Borel equivariant cohomology H n G (X; Z) is defined to be the (singular) cohomology of 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 where the coefficient H q (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 As an application of the spectral sequence, we can obtain an identification H n G (pt; (1)) for n ≥ 2 by the so-called exponential exact sequence.) For a better geometric understanding of the spectral sequence, let us start with the fact that the Borel equivariant cohomology H n G (X; Z) is isomorphic to the cohomology H n (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 The simplicial space G • ×X is associated to the left G-action on X, and consists of a sequence of spaces {G p × X} p≥0 together with the face map ∂ i : The cohomology H n (G • × X; Z) is then defined to be the total cohomology of the double complex (C i (G j ×X; Z), δ, ∂), where (C i (G j ×X; Z), δ) is the complex computing the cohomology of G j ×X with coefficients in Z and ∂ : The double complex admits a natural filtration {⊕ j≥p C i (G j × X; Z)} p≥0 . The associated spectral sequence agrees with the Leray-Serre spectral sequence E p,q r , 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 G p × X and U (1) consists of the sheaf of U (1)-valued functions on G p × X. As in [9, Lemma 4.4], we can readily show that H n (G • × X; R) = 0 for n > 0. This vanishing together with the associated long exact sequence leads to the following isomorphism for n ≥ 1 The cohomology H n (G • × X; U (1)) can be defined exactly in the same way as in the case of H n (G • × X; Z) by using a double complex. Therefore we have a spectral sequence converging to the graded quotient of a filtration Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 15 whose E 2 -term is where H q (X; U (1)) is regarded as a right G-module by pull-back. It is clear that H 0 (X; U (1)) ∼ = C(X, U (1)) and H n (X; U (1)) ∼ = H n+1 (X; Z) for n ≥ 1. Since E p,q 2 involves the group cohomology with coefficients in C(X, U (1)), its computation seems to be more complicated than that of E p,q 2 . 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 H n+1 G (X; Z) ∼ = H n (G • × X; U (1)) for n ≥ 1 are related as follows The spectral sequences are related by a map E p,q r → E p,q+1 r . In particular, the E 2 -terms E p,0 2 and E p,1 2 are related by a map C(X, U (1)) → H 1 (X; Z) fitting into the exact sequence As is mentioned, because of the isomorphism H q (X; for q ≥ 1. A more detailed relation between these spectral sequences will be given later under some hypotheses.

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 L g → X for the restriction to {g} × X for each g ∈ G, • unitary isomorphisms of Hermitian line bundles τ g,h : We assume the following diagram is commutative Notice that if L g 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, β g ) : (L g , τ g,h ) → (L g , τ g,h ) of central extensions of X//G consists of the following data: • a Hermitian line bundle K → X, • unitary isomorphisms of Hermitian line bundles β g : L g ⊗ K → g * K ⊗ L g on X for each g ∈ G, which we write β g (x) : L g | x ⊗ K| x → K| gx ⊗ L g | x for the restriction to x ∈ X. We assume the following diagram is commutative The isomorphisms (K, β g ) and (K , β g ) from (L g , τ g,h ) to (L g , τ g,h ) are identified if there is a unitary isomorphism f : K → K making the following diagram commutative 3. An ungraded G-equivariant twist of X, or a twist for short, is a central extension of a groupoidX which has a local equivalence to X//G.
A point in this definition is that a twist needs an extra groupoidX. A central extension of X//G is a special type of a twist such thatX = 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: (ii) F 2 H 2 (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 Z 2 -graded Hermitian line bundle. Since L is of rank 1, its Z 2 -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 H 1

Comparison of two spectral sequences
The relation between the spectral sequences E p,q r and E p,q r 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 Let us fold this into a short exact sequence In general, this does not split as an exact sequence of G-modules. (Such an example is provided by the circle S 1 ⊂ R 2 with the action of D 2 ⊂ O(2).) Notice that if X is path connected, then H 0 (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 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 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 Since X is supposed to be compact, H 1 (X; Z) is a free abelian group of finite rank. Let us choose a basis H 1 (X; Z) ∼ = i Za i , and also ϕ i : X → R/Z such that δϕ i = a i and ϕ i (pt) = 0. Modifying the splitting a i → ϕ i 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) = (A ij (g)) with integer coefficients to each g ∈ G by g * a i = j A ij (g)a j . It holds that A(gh) = A(g)A(h). Because of the exact sequence, there are functions f i g ∈ C(X, pt, R) such that the following holds in C(X, pt, R/Z): This can be expressed as g * Φ = A(g)Φ + F g by using the vectors Φ = (ϕ i ) and F g = (F i g ). It then holds that F gh = A(g)F h + h * F g in C(X, pt, R). Since A(g) is invertible, this is equivalent to Write |G| for the order of G, and put F = 1 |G| g∈G A(g) −1 F g . Taking the average over g ∈ G in the formula above, we get which is equivalent to F g = 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 a i → ϕ i + (f i mod Z) defines a splitting H 1 (X; Z) → C(X, pt, R/Z) compatible with the G-module structures.

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 H n group (G; C(X, U (1))) ∼ = H n group (G; U (1)) ⊕ H n group G; H 1 (X; Z) , where U (1) is the trivial G-module, and H 1 (X; Z) is regarded as a G-module through the action of G on X.
Proof . Lemma 3.8 implies Since C(X, pt, R) is a vector space over R, we can prove the vanishing H n 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 Proof . Recall that the exponential exact sequence induces the connecting homomorphism δ : H q (X; U (1)) → H q+1 (X; Z) and this induces a natural homomorphism δ : E p,q r → E p,q+1 r compatible with the differentials d r and d r . In the case of r = 2, the homomorphism δ : E p,q 2 → E p,q+1 2 is bijective for q ≥ 1 and p ≥ 0, and we have E p,0 2 ∼ = E p, 1 2 ⊕ E p+1,0 2 for p ≥ 1 as a consequence of Lemma 3.9. Notice that, under this isomorphism, δ : E p,0 2 → E p,1 2 for p ≥ 1 restricts to the identity on the direct summand E p,1 2 ⊂ E p,0 2 . Note also that E p,0 2 = E p,0 ∞ for any p, because is a direct summand of H p G (X; Z) ∼ = H 0 G (pt; Z) ⊕H p G (X; Z), whereH p G (X; Z) is the reduced cohomology. Thus, for p ≥ 1, the map δ : E p,0 2 → E p,1 2 is the projection onto E p,1 2 and the image of the differential d 2 : 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 In addition, we have a decomposition Proof . Put F p H n = F p H n (G • × X; U (1)) and F p H n+1 = F p H n+1 G (X; Z) for short. The exponential exact sequence induces a homomorphism of short exact sequences In the case of p = n, the diagram above becomes Combining the above corollary with Lemma 3.5, we get the interpretations of F p H 3 G (X; Z) by twists presented in Introduction: Remark 3.13. The coincidence F 1 H n (G • × X; U (1)) = F 1 H n+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 E 0,n and E 0,n+1 are subgroups of H n (X; U (1)) ∼ = H n+1 (X; Z) and it holds that where f is the homomorphism of "forgetting the group actions".

Some generality
The cohomology H n (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 H 3 (T 2 ; Z) = 0 implies E 0,3 ∞ = 0, so that Note that each point group P fixes a point on T 2 , so that Then the main task for the proof of Theorem 1.1 is to compute H 3 P (T 2 ; Z) and F 2 H 3 P (T 2 ; Z), since in the case where P is the cyclic group Z n or the dihedral group D n , the cohomology H m P (pt; Z) is summarized as follows: (n: even) 0 (n: odd) Z 2 (n: even) The degree 0 part H 0 P (pt; Z) = H 0 (BP ; Z) = Z is clear. Since P is finite, the degree 1 part H 1 P (pt; Z) ∼ = Hom(P, Z) is trivial. The degree 2 part H 2 P (pt; Z) ∼ = Hom(P, U (1)) can be seen by the classification of irreducible representations. Finally, the degree 3 part H 3 P (pt; Z) ∼ = H 2 group (P ; U (1)) for P = Z n , D n 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 × e d , where H ⊂ G is a closed subgroup and e d is the standard d-dimensional cell. The G-action on G/H is the left translation, whereas that on e d is trivial. For the details, we refer the reader to [19].
We later compute a group cohomology via cohomology of a space: 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.
Notice that a G-CW complex is naturally a CW complex.
Proof . Consider the Leray-Serre spectral sequence Note that H q (Y ; Z) = 0 for q = 0, 1. The . Therefore it must hold that 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 → Z 2 = {±1} a surjective homomorphism, andZ = Z c the G-module such that its underlying group is Z and G acts (from the right) by m → mc(g). A typical example is a finite subgroup P ⊂ O(2) such that P ⊂ SO(2) with c the composition of the inclusion P → O(2) and the determinant O(2) → Z 2 . Proof . For any n ∈ C 0 group (G;Z) = 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 Thus, given a group 1-cocycle φ ∈ Z 1 group (G;Z), it holds that φ(g) = 0 for all g ∈ Ker(c). If g, h ∈ 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 Z 1 group (G;Z) ∼ = Z as well as B 1 group (G;Z) ∼ = 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: Proof . In the Leray-Serre spectral sequence E p,q 2 = H p group (G; H q (T 2 ; Z)), the coefficient in the group cohomology H 0 (T 2 ) ∼ = Z is identified with the trivial G-module, and H 2 (T 2 ) ∼ = Z with the G-module in Lemma 4.2. Then the relevant E 2 -terms can be summarized as follows: Since G fixes pt ∈ T 2 , we have the decomposition H n G (T 2 ; Z) ∼ = H n G (pt; Z) ⊕H n G (T 2 ; Z), wherẽ H n G (T 2 ; Z) is the reduced cohomology. Therefore the E 2 -term E n,0 2 = H n group (G; Z) ∼ = H n G (pt; Z) must survive into the direct summand H n G (pt; Z) in H n G (T 2 ; Z). This implies that E n,0 is always a direct summand of the subgroups F p H n G (T 2 ; Z) ⊂ H n G (T 2 ; Z) and that d 2 : E p−2,1 2 → E p,0 2 is trivial. As a result, we get E 2,1 2 = E 2,1 ∞ and the isomorphism (a). Also E p,q 2 = E p,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, Theorem 11.8]).

K. Gomi
Lemma 4.4. Suppose a finite group G acts on the torus T 2 = S 1 × S 1 and • there is a fixed point pt = (x 0 , y 0 ) ∈ T 2 under the G-action, Then there is a G-equivariant homotopy equivalence 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 equivariant 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.
Further, the Leray-Serre spectral sequence for H n G (T 2 ; Z) degenerates at E 2 and the relevant extension problems are trivial, so that 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 E 2 , and we have H n G (pt; Z) ∼ = H n group G; H 0 (pt; Z) .
Since H 0 (pt; Z) ∼ = H 0 (T 2 ; Z) as G-modules, we get the following identification of the E 2term E n,0 2 of the Leray-Serre spectral sequence for H n G (T 2 ; Z) For the G-space S 1 ∨ S 1 , we can see, as in the proof of Lemma 4.1, that the Leray-Serre spectral sequence also degenerates at E 2 and the extension problems are trivial. Because H 1 (S 1 ∨S 1 ; Z) ∼ = H 1 (T 2 ; Z) as G-modules, the E 2 -term E n−1,1 2 of the Leray-Serre spectral sequence for H n G (T 2 ; Z) is Exactly in the same way, we have since H 2 (T 2 /S 1 ∨S 1 ; Z) ∼ = H 2 (T 2 ; Z) as G-modules. The first isomorphism now gives H n G (T 2 ; Z) , which also implies the triviality of the spectral sequence.

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 Z 2 = D 1 acts on the torus T 2 = S 1 × S 1 preserving the direct product structure, so that we can think of T 2 as a stack of certain equivariant circle bundles (Z 2 -equivariant principal circle bundles and/or 'Real' circle bundles 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 S 1 ∨ S 1 , 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 H 3 P (T 2 ; Z). This is carried out by taking a P -CW decomposition 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 H 1 (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 cohomology with its coefficients in H 1 (Y ; Z) through Lemma 4.1. The coefficients H 1 (Y ; Z) and H 1 (T 2 ; Z) are related by a short exact sequence. The associated long exact sequence then computes the group cohomology with coefficients in H 1 (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.

p2
The lattice Π ⊂ R 2 is the standard one Π = Z ⊕ Z and the point group P = Z 2 = {±1} acts on Π and R 2 by (x, y) → (−x, −y). Theorem 4.6 (p2). The Z 2 -equivariant cohomology of T 2 is given as follows Proof . We use the Gysin exact sequence for 'Real' circle bundles in [10]: We write H n Z 2 (X) = H n Z 2 (X; Z) for the equivariant cohomology and H n ± (X) ∼ = H n 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 ofS 1 , whereS 1 = U (1) is the circle with the involution z → z −1 . We can think ofS 1 ×S 1 as the trivial 'Real' circle bundle onS 1 .

K. Gomi
Similarly,S 1 is the trivial 'Real' circle bundle on pt. The Gysin exact sequences for these 'Real' circle bundles are split, and we find As given in [10], the cohomology H n ± (pt) is isomorphic to Z 2 if n > 0 is odd, and is trivial otherwise. We already know H n Z 2 (pt), and get H n Z 2 (T 2 ) easily.

p4m/p4g
The lattice Π = Z 2 ⊂ R 2 is standard. The point group is The D 4 -action on Π and R 2 is given by the following matrix presentation: In the rest of this subsection, we will use the following notations to indicate elements in D 4 : The closure of a fundamental domain is {s(1, 0) + t(0, 1) ∈ R 2 | 0 ≤ s, t ≤ 1}. Then we find that the D 4 -action on T 2 = R 2 /Π satisfies the assumptions in Lemma 4.4, in which pt = (0, 0) and To apply Lemma 4.5, we compute the cohomology of Y : Lemma 4.7. The equivariant cohomology of Y is as follows: Proof . We use the Mayer-Vietoris exact sequence: Cover Y by invariant subspaces U and V with the following D 4 -equivariant homotopy equivalences We can summarize the equivariant cohomology of these spaces in low degrees as follows: In the Mayer-Vietoris exact sequence

the inclusions. Under the natural identifications
the map j * U agrees with the homomorphism induced from the inclusion Z 2 . This implies that j * U is surjective, and so is ∆ in degree 2. Clearly, ∆ : Proof . In the Leray-Serre spectral sequence E p,q 2 = H p group (D 4 ; H q (T 2 ; Z)), the D 4 -modules H 0 (T 2 ), H 1 (T 2 ) and H 2 (T 2 ) are identified with the trivial D 4 -module Z, H 1 (Y ) and the D 4 -moduleZ in Lemma 4.2, respectively. Using Lemmas 4.1 and 4.2, we can summarize the E 2 -terms as follows: Now the proof is completed by Lemma 4.5.

p6m
We let Π = Za ⊕ Zb ⊂ R 2 be the lattice spanned by a = 1 This group acts on Π and R 2 through the inclusion D 6 ⊂ O(2) defined by If we use the identifications a = 1 and b = τ = exp 2πi/6 under R 2 = C, then the actions of C ∈ D 6 and σ 1 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 D 6 -CW decomposition of T 2 as follows:

K. Gomi
• (0-cell) The 0-cellẽ 0 0 = (D 6 /D 6 ) × e 0 = pt is the unique fixed point on T 2 . The other 0-cells are defined as follows: Lemma 4.9. The equivariant cohomology of Y is given as follows: Proof . We can find D 6 -invariant subspaces U and V in Y which have the following equivariant homotopy equivalences The equivariant cohomology groups of these spaces can be summarized as follows: In the Mayer-Vietoris exact sequence the homomorphism ∆ is expressed as ∆(u, v) = j * U (u) − j * V (v) with j U : U ∩ V → U and j V : U ∩ V → V the inclusions. This immediately determines H 0 D 6 (Y ) ∼ = Z and H 1 D 6 (Y ) = 0. To complete the proof, we recall the identifications under which j * U and j * V are induced from the inclusions D 2 → D 6 and Z (1) 2 → D 6 . As a basis of H 2 D 6 (U ) we can choose the following 1-dimensional representations ρ i : Similarly, we can choose the following 1-dimensional representations ρ i of D 2 = {1, σ 1 , C 3 , σ 4 } as a basis of H 2 D 6 (V ) , and it has the following basis We can also see that ∆ is surjective, and H 3 Let X 1 be the 1-skeleton of the D 6 -CW complex T 2 .
Proof . We cover X 1 =ẽ 0 0 ∪ẽ 0 1 ∪ẽ 0 2 ∪ẽ 1 01 ∪ẽ 1 02 ∪ẽ 1 12 by invariant subspaces U and V which admit the following equivariant homotopy equivalences where D 3 = {1, C 2 , C 4 , σ 2 , σ 4 , σ 6 }, Z The equivariant cohomology groups of these spaces are summarized as follows: The homomorphism ∆ in the Mayer-Vietoris exact sequence is expressed as ∆(u, v) = j * U (u) − j * V (v) by using the inclusions j U : U ∩ V → U and j V : U ∩ V → V . An inspection proves that j * U agrees with the composition of the following two homomorphisms: K. Gomi (i) the inclusion that follows from the calculation of H 2 D 6 (Y ) in Lemma 4.9 2 → D 2 . Then, using the basis presented in the calculation of H 2 D 6 (Y ), we find where ρ : Z 2 → Z 2 is the identity map generating Hom(Z 2 , U (1)) ∼ = Z 2 . Hence j * U as well as ∆ are surjective, and H 3 Proof . The relevant part of the exact sequence for the pair (T 2 , X 1 ) is By means of the excision axiom, we have H n D 6 (T 2 , X 1 ) ∼ = H n−2 (pt). Therefore we get H 3 LetẐ = Z φ 1 be the D 6 -module such that its underlying group is Z and D 6 acts via the homomorphism φ 1 : D 6 → Z 2 given by φ 1 (C) = −1 and φ 1 (σ 1 ) = 1. Proof . Let η 1 , η 2 ∈ H 1 (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 H 1 (T 2 ) ∼ = Z 2 . Also, let γ 1 , γ 2 , γ 3 ∈ H 1 (Y ) be the homology classes of loops along 1, τ and τ − 1, which form a basis of H 1 (Y ) ∼ = Z 3 . The inclusion map i : Y → T 2 relates these bases by i * γ 1 = η 1 , i * γ 2 = η 2 and i * γ 3 = η 2 − η 1 . The actions of C ∈ D 6 and σ 1 on these bases are Let {h 1 , h 2 } ⊂ H 1 (T 2 ) and {g 1 , g 2 , g 3 } ⊂ H 1 (Y ) be dual to the homology bases. They are related by i * h 1 = g 1 − g 3 and i * h 2 = g 2 + g 3 , and the induced D 6 -actions are as follows.
These expressions allow us to prove that the cokernel of the homomorphism i * : is isomorphic toẐ, yielding the exact sequence. The homomorphism s :Ẑ → H 1 (Y ) is given by Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 29 Lemma 4.13. H n group (D 6 ; H 1 (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 → H 1 (T 2 ) → H 1 (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: It is clear that H 0 group (D 6 ; H 1 (T 2 )) = 0. The homomorphism in group cohomology induced from π : H 1 (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: -equivariant cohomology of T 2 in low degrees is as follows: Proof . In the E 2 -term of the Leray-Serre spectral sequence E p,q 2 = H p group (D 6 ; H q (T 2 ; Z)), the coefficient H 0 (T 2 ) is identified with the trivial D 6 -module Z, and H 2 (T 2 ) withZ as in Lemma 4.2. The group cohomology with coefficients in H 1 (T 2 ) is already computed, and that inZ is also computed in Lemma 4.2. The E 2 -terms are summarized as follows: This list and Lemma 4.3 lead to the theorem.

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 Thus, it suffices to prove that the group cocycles τ induced from the nonsymmorphic 2dimensional space groups as in Section 2 generate E 2,1 2 . Recall from Section 2 that the group 2-cocycle ν ∈ Z 2 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] ∈ H 2 group (P ; Π) is non-trivial. In particular, this factors through the homomorphisms given by the assignment of the cocycles ν → τ in Section 2 and induced from the natural surjection δ : C(Π, U (1)) → H 1 (Π; Z).
Proof . Instead of the left P -action on the Pontryagin dualΠ = Hom(Π, U (1)) defined in Section 2, we consider the natural right actionk(m) →k(pm) of p ∈ P onk ∈Π, from which the left action originates. This choice of the actions does not affect the group cohomology. The right P -action onΠ induces by pull-back a left P -action on the cohomology H 1 (Π; Z). Thus, the isomorphism of the group cohomologies will be established once we see H 1 (Π; Z) ∼ = Π as left P -modules. In general, for each element m ∈ Π ⊂ R d = V , the path [0, 1] → V , (t → tm) defines a loop in V /Π. This induces an isomorphism of left P -modules Π ∼ = H 1 (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: Considering the dual space V * = Hom(V, R) and its lattice Π * instead, we similarly get an isomorphism of left P -modules Since there is a natural isomorphism of tori V * /Π * →Π = 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 H 2 group (Z 2 ; Π) ∼ = Z 2 through ν, and the element corresponds by Lemma 4.15 to the nontrivial element of H 2 group (Z 2 ; H 1 (T 2 ; Z)) ∼ = Z 2 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 In view of the classification of 2dimensional space groups ( [12]), the non-trivial elements (−1, 1) and (−1, −1), with respect to a basis of Z 2 ⊕ Z 2 , 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 H 2 group (D 2 ; H 1 (T 2 ; Z)) ∼ = Z 2 ⊕ Z 2 is generated by the group 2-cocycles induced from the nonsymmorphic groups.

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 computation 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 provided in the case of p6m.

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 Z 2 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 = φ 2 . Let Y ⊂ T 2 be the D 6 -invariant subspace given in Section 4.5.
Proof . The proof of Lemma 4.12 can be adapted to this case.
Proof . We use the long exact sequence of group cohomology induced from the short exact sequence of coefficients. Notice that Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 35 The relevant cohomology can be summarized as follows: By s : Z φ 0 → H 1 (Y ; Z) ⊗ Z φ 2 , the group cohomology is determined as stated.
Theorem 5.10 (p6m with φ 2 ). The D 6 -equivariant cohomology of T 2 with coefficients in Z φ 2 in low degrees is given as follows: We also have: In the E 2 -term of the Leray-Serre spectral sequence, we have the following identifications H n group D 6 ; H 0 T 2 ; Z ⊗ Z φ 2 ∼ = H n D 6 (pt; Z φ 2 ), H n group D 6 ; H 2 T 2 ; Z ⊗ Z φ 2 ∼ = H n D 6 (pt; Z φ 1 ).

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 → R 2 in Section 2 is also presented.
• (p4m/p4g) The point group is D 4 = C 4 , σ x | C 4 4 , σ 2 x , σ x C 4 σ x C 4 , which acts on Π and R 2 through the following matrix presentation In the case of p4g, the map a : D 4 → R 2 is as follows: In the above, we set σ d = σ x C 4 , σ y = C 2 4 σ x and σ d = C 4 σ x .
Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups 37