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

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

Kiyonori GOMI

K. Gomi

Department of Mathematical Sciences, Shinshu University,
3–1–1 Asahi, Matsumoto, Nagano 390-8621, Japan<EMAIL_ADDRESS>http://math.shinshu-u.ac.jp/~kgomi/

Received February 17, 2016, in final form March 03, 2017; Published online
March 08, 2017

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.

twist; Borel equivariant cohomology; crystallographic group; topological
insulator

53C08; 55N91; 20H15; 81T45

##  1 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 $\mathbb{R}^{d}$ 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 $\mathbb{Z}_{2}$ 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 $\mathbb{Z}_{2}$ 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’ 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
$H^{1}_{G}(X;\mathbb{Z}_{2})\times H^{3}_{G}(X;\mathbb{Z})$. Similarly,
ungraded twists are classified by $H^{3}_{G}(X;\mathbb{Z})$, on which we focus
for a moment. (Sometimes $H^{0}_{G}(X;\mathbb{Z})$ may be included in the
twists, but we regard it as the degree of the $K$-theory.)

By definition, the Borel equivariant cohomology $H^{n}_{G}(X;\mathbb{Z})$ is
the usual cohomology $H^{n}(EG\\!\times_{G}\\!$ $X;\mathbb{Z})$ of the Borel
construction $EG\times_{G}X$, which is the quotient of $EG\times X$ by the
diagonal $G$-action, where $EG$ is the total space of the universal $G$-bundle
$EG\to BG$. Associated to the Borel construction is the fibration $X\to
EG\times_{G}X\to BG$, and hence the Leray–Serre spectral sequence
$E_{r}^{p,q}$ that converges to the graded quotient of a filtration

$\displaystyle H^{n}_{G}(X;\mathbb{Z})\supset
F^{1}H^{n}_{G}(X;\mathbb{Z})\supset
F^{2}H^{n}_{G}(X;\mathbb{Z})\supset\cdots\supset
F^{n+1}H^{n}_{G}(X;\mathbb{Z})=0.$

One can interpret $F^{p}H^{3}_{G}(X;\mathbb{Z})\subset
H^{3}_{G}(X;\mathbb{Z})$ 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
$F^{3}H^{3}_{G}(X;\mathbb{Z})$.

  * (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
$F^{2}H^{3}_{G}(X;\mathbb{Z})$.

  * (iii)

Twists which can be represented by central extensions of the groupoid
$X/\\!/G$. These twists are classified by $F^{1}H^{3}_{G}(X;\mathbb{Z})$.

  * (iv)

Twists of general type, classified by
$F^{0}H^{3}_{G}(X;\mathbb{Z})=H^{3}_{G}(X;\mathbb{Z})$.

The equivariant twists on $T^{d}$ arising from quantum systems on
$\mathbb{R}^{d}$, to be explained in Section 2, belong to
$F^{2}H^{3}_{P}(T^{d};\mathbb{Z})$ 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
$\mathbb{R}^{2}\rtimes{\rm O}(2)$ of isometries of $\mathbb{R}^{2}$, and is an
extension of a finite group $P\subset{\rm O}(2)$ called the point group by a
rank $2$ lattice $\Pi\cong\mathbb{Z}^{2}$ of translations of $\mathbb{R}^{2}$:

$\displaystyle\begin{array}[]{c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c}1&\
\longrightarrow&\mathbb{R}^{2}&\ \longrightarrow&\mathbb{R}^{2}\rtimes{\rm
O}(2)&\ \longrightarrow&{\rm O}(2)&\ \longrightarrow&1\\\
&&\cup&&\cup&&\cup&&\\\ 1&\ \longrightarrow&\Pi&\ \longrightarrow&S&\
\longrightarrow&P&\ \longrightarrow&1.\end{array}$

Being a normal subgroup of $S$, the lattice $\Pi\subset\mathbb{R}^{2}$ is
preserved by the action of $P$ on $\mathbb{R}^{2}$ through the inclusion
$P\subset{\rm O}(2)$ and the standard left action of ${\rm O}(2)$ on
$\mathbb{R}^{2}$. This induces the left action of $P$ on the torus
$T^{2}=\mathbb{R}^{2}/\Pi$ that we will consider. Since $P$ is a finite
subgroup of ${\rm O}(2)$, it is the cyclic group $\mathbb{Z}_{n}$ of order $n$
or the dihedral group $D_{n}=\langle C,\sigma\,|\,C^{n},\sigma^{2},\sigma
C\sigma C\rangle$ 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 ${\rm GL}(2,\mathbb{Z})$ 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 $T^{2}=\mathbb{R}^{2}/\Pi$ via $P\subset{\rm O}(2)$ as above. Then,
$H^{3}_{P}(T^{2};\mathbb{Z})=F^{0}H^{3}_{P}(T^{2};\mathbb{Z})=F^{1}H^{3}_{P}(T^{2};\mathbb{Z})$.
This cohomology group and its subgroups $F^{p}H^{3}_{P}(T^{2};\mathbb{Z})$ are
as in Fig. 1.

$\begin{array}[]{|c|c|c||c|c|c||c|c|}\hline\cr\mbox{Space group
$S$}&P&\mbox{ori}&H^{3}_{P}(T^{2};\mathbb{Z})&F^{2}&F^{3}&E^{1,2}_{\infty}&E^{2,1}_{\infty}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p1}}&1&+&0&0&0&0&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p2}}&\mathbb{Z}_{2}&+&0&0&0&0&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p3}}&\mathbb{Z}_{3}&+&0&0&0&0&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p4}}&\mathbb{Z}_{4}&+&0&0&0&0&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p6}}&\mathbb{Z}_{6}&+&0&0&0&0&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{pm}/{pg}}&D_{1}&-&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}&0&\mathbb{Z}_{2}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{cm}}&D_{1}&-&\mathbb{Z}_{2}&0&0&\mathbb{Z}_{2}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{pmm}/{pmg}/{pgg}}&D_{2}&-&\mathbb{Z}_{2}^{\oplus
4}&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}^{\oplus
2}\tsep{2pt}\bsep{2pt}\\\ \hline\cr\mbox{{cmm}}&D_{2}&-&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p3m1}}&D_{3}&-&\mathbb{Z}_{2}&0&0&\mathbb{Z}_{2}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p31m}}&D_{3}&-&\mathbb{Z}_{2}&0&0&\mathbb{Z}_{2}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p4m}/{p4g}}&D_{4}&-&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p6m}}&D_{6}&-&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\end{array}$ Figure 1: The list of $H^{3}_{P}(T^{2};\mathbb{Z})$ and
its subgroups $F^{p}=F^{p}H^{3}_{P}(T^{2};\mathbb{Z})$ for the point group $P$
of each space $2$-dimensional space group $S$. The $E_{\infty}$-term of the
Leray–Serre spectral sequence is related to these subgroups by
$E^{p,3-p}_{\infty}\cong F^{p}/F^{p+1}$. 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 $F^{2}H^{3}_{P}(T^{2};\mathbb{Z})$ 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 $S^{\prime}$ such that the action of its point group
$P^{\prime}\cong P$ 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 $F^{2}H^{3}_{P}(T^{2};\mathbb{Z})$
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 $H^{3}_{P}(T^{2};\mathbb{Z})$ 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
$\mathbb{Z}_{2}$-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 $H^{n}_{P}(T^{2};\mathbb{Z})$, ($n\leq 2$) of the torus with the
actions of the possible finite subgroups in the mapping class group ${\rm
GL}(2,\mathbb{Z})$.

######  Theorem 1.3.

Let $P$ be the point group of one of the $2$-dimensional space groups $S$,
acting on $T^{2}=\mathbb{R}^{2}/\Pi$ via $P\subset{\rm O}(2)$. For $n\leq 3$,
the $P$-equivariant cohomology $H^{n}_{P}(T^{2};\mathbb{Z})$ is as given in
Fig. 2.

$\begin{array}[]{|c|c|c||c|c|c|c|}\hline\cr\mbox{Space group
$S$}&P&\mbox{ori}&H^{0}_{P}(T^{2})&H^{1}_{P}(T^{2})&H^{2}_{P}(T^{2})&H^{3}_{P}(T^{2})\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p1}}&1&+&\mathbb{Z}&\mathbb{Z}^{\oplus
2}&\mathbb{Z}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p2}}&\mathbb{Z}_{2}&+&\mathbb{Z}&0&\mathbb{Z}\oplus\mathbb{Z}_{2}^{\oplus
3}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p3}}&\mathbb{Z}_{3}&+&\mathbb{Z}&0&\mathbb{Z}\oplus\mathbb{Z}_{3}^{\oplus
2}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p4}}&\mathbb{Z}_{4}&+&\mathbb{Z}&0&\mathbb{Z}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{4}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p6}}&\mathbb{Z}_{6}&+&\mathbb{Z}&0&\mathbb{Z}\oplus\mathbb{Z}_{6}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{pm}/{pg}}&D_{1}&-&\mathbb{Z}&\mathbb{Z}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}^{\oplus 2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{cm}}&D_{1}&-&\mathbb{Z}&\mathbb{Z}&\mathbb{Z}_{2}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{pmm}/{pmg}/{pgg}}&D_{2}&-&\mathbb{Z}&0&\mathbb{Z}_{2}^{\oplus
4}&\mathbb{Z}_{2}^{\oplus 4}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{cmm}}&D_{2}&-&\mathbb{Z}&0&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}^{\oplus 2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p3m1}}&D_{3}&-&\mathbb{Z}&0&\mathbb{Z}_{2}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p31m}}&D_{3}&-&\mathbb{Z}&0&\mathbb{Z}_{3}\oplus\mathbb{Z}_{2}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p4m}/{p4g}}&D_{4}&-&\mathbb{Z}&0&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}^{\oplus 3}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p6m}}&D_{6}&-&\mathbb{Z}&0&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}^{\oplus 2}\tsep{2pt}\bsep{2pt}\\\ \hline\cr\end{array}$
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 $\mathbb{Z}_{2}$, 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
$H^{1}_{P}(T^{2};\mathbb{Z}_{2})$ also admits a filtration

$\displaystyle
H^{1}_{P}\big{(}T^{2};\mathbb{Z}_{2}\big{)}=F^{0}H^{1}_{P}\big{(}T^{2};\mathbb{Z}_{2}\big{)}\supset
F^{1}H^{1}_{P}\big{(}T^{2};\mathbb{Z}_{2}\big{)}\supset
F^{2}H^{1}_{P}\big{(}T^{2};\mathbb{Z}_{2}\big{)}=0.$

Because the degree in question is $1$, the degeneration of the Leray–Serre
spectral sequence gives the identification

$\displaystyle
F^{1}H^{1}_{P}\big{(}T^{2};\mathbb{Z}_{2}\big{)}=\operatorname{Hom}\big{(}P,\mathbb{Z}_{2}\big{)}=H^{1}_{P}\big{(}\mathrm{pt};\mathbb{Z}_{2}\big{)},$

which is a direct summand of $H^{1}_{P}(T^{2};\mathbb{Z}_{2})$ and is also
computed immediately by using the knowledge of the equivariant cohomology of
the space consisting of one point, $\mathrm{pt}=\\{\mbox{one point}\\}$, in
Section 4.1.

######  Corollary 1.4.

Let $P$ be the point group of one of the $2$-dimensional space groups $S$,
acting on $T^{2}=\mathbb{R}^{2}/\Pi$ via $P\subset{\rm O}(2)$. Then the
$P$-equivariant cohomology $H^{1}_{P}(T^{2};\mathbb{Z}_{2})$ is as in Fig. 3.

$\begin{array}[]{|c|c|c||c|c|c|}\hline\cr\mbox{Space group
$S$}&P&\mbox{ori}&H^{1}_{P}(T^{2};\mathbb{Z}_{2})&F^{1}H^{1}_{P}(T^{2};\mathbb{Z}_{2})&E_{\infty}^{1,0}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p1}}&1&+&\mathbb{Z}_{2}^{\oplus 2}&0&\mathbb{Z}_{2}^{\oplus
2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p2}}&\mathbb{Z}_{2}&+&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}&\mathbb{Z}_{2}^{\oplus 2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p3}}&\mathbb{Z}_{3}&+&0&0&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p4}}&\mathbb{Z}_{4}&+&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p6}}&\mathbb{Z}_{6}&+&\mathbb{Z}_{2}&\mathbb{Z}_{2}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{pm}/{pg}}&D_{1}&-&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}&\mathbb{Z}_{2}^{\oplus 2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{cm}}&D_{1}&-&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{pmm}/{pmg}/{pgg}}&D_{2}&-&\mathbb{Z}_{2}^{\oplus
4}&\mathbb{Z}_{2}^{\oplus 2}&\mathbb{Z}_{2}^{\oplus 2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{cmm}}&D_{2}&-&\mathbb{Z}_{2}^{\oplus 3}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p3m1}}&D_{3}&-&\mathbb{Z}_{2}&\mathbb{Z}_{2}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p31m}}&D_{3}&-&\mathbb{Z}_{2}&\mathbb{Z}_{2}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p4m}/{p4g}}&D_{4}&-&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}^{\oplus 2}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p6m}}&D_{6}&-&\mathbb{Z}_{2}^{\oplus 2}&\mathbb{Z}_{2}^{\oplus
2}&0\tsep{2pt}\bsep{2pt}\\\ \hline\cr\end{array}$ Figure 3: The list of first
equivariant cohomology groups with coefficients $\mathbb{Z}_{2}$. The quotient
group $H^{1}_{P}(T^{2};\mathbb{Z}_{2})/F^{1}H^{1}_{P}(T^{2};\mathbb{Z}_{2})$
is denoted with $E_{\infty}^{1,0}$.

The grading of twists classified by
$F^{1}H^{1}_{P}(T^{2};\mathbb{Z}_{2})=\operatorname{Hom}(P,\mathbb{Z}_{2})$
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
$\mathbb{Z}_{2}$-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 $\phi\colon G\to\mathbb{Z}_{2}$. The
$K$-theory of Freed–Moore reduces to the $G$-equivariant $K$-theory if $\phi$
is trivial, and to the $KR$-theory if $G=\mathbb{Z}_{2}$ and $\phi$ 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 $\mathbb{Z}_{4}$-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 $H^{1}_{G}(X;\mathbb{Z}_{2})\times
H^{3}_{G}(X;\mathbb{Z}_{\phi})$ and the ungraded twists by
$H^{3}_{G}(X;\mathbb{Z}_{\phi})$. Here $\mathbb{Z}_{\phi}$ denotes a local
system for the Borel equivariant cohomology associated to the $G$-module
$\mathbb{Z}_{\phi}$ such that its underlying group is $\mathbb{Z}$ and $G$
acts via $\phi\colon G\to\mathbb{Z}_{2}$. The cohomology group
$H^{n}_{G}(X;\mathbb{Z}_{\phi})$ also admits a filtration

$\displaystyle H^{n}_{G}(X;\mathbb{Z}_{\phi})\supset
F^{1}H^{n}_{G}(X;\mathbb{Z}_{\phi})\supset
F^{2}H^{n}_{G}(X;\mathbb{Z}_{\phi})\supset\cdots\supset
F^{n+1}H^{n}_{G}(X;\mathbb{Z}_{\phi})=0.$

The associated graded quotient is computed by the Leray–Serre spectral
sequence, and the subgroups $F^{p}H^{3}_{G}(X;\mathbb{Z}_{\phi})\subset
H^{3}_{G}(X;\mathbb{Z}_{\phi})$ have geometric interpretations as well
(Proposition 5.1).

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 $\phi\colon P\to\mathbb{Z}_{2}$.

  * •

In the cases of p2, p4 and p6, the point group $P$ is the cyclic group
$\mathbb{Z}_{2m}=\langle C\,|\,C^{2m}\rangle$ of even order. We write
$\phi_{1}\colon\mathbb{Z}_{2m}\to\mathbb{Z}_{2}$ for the unique non-trivial
homomorphism given by $\phi_{1}(C)=-1$.

  * •

In the other case, the point group $P$ is the dihedral group $D_{n}=\langle
C,\sigma\,|\,C^{n},\sigma^{2},\sigma C\sigma C\rangle$ of degree $n$ and order
$2n$, and $D_{n}$ is embedded into ${\rm O}(2)$ so that $C$ is a rotation of
$\mathbb{R}^{2}$ and $\sigma$ is a reflection. We define $\phi_{0}\colon
D_{n}\to\mathbb{Z}_{2}$ to be the composition of the inclusion $D_{n}\to{\rm
O}(2)$ and $\det\colon{\rm O}(2)\to\mathbb{Z}_{2}$. Put differently,
$\phi_{0}(C)=1$ and $\phi_{0}(\sigma)=-1$. This provides the unique non-
trivial homomorphism $D_{n}\to\mathbb{Z}_{2}$ if $n$ is odd. In the case of
even $n$, we define two more non-trivial homomorphisms $\phi_{i}\colon
D_{n}\to\mathbb{Z}_{2}$ by

$\displaystyle\begin{cases}\phi_{1}(C)=-1,\\\
\phi_{1}(\sigma)=1,\end{cases}\qquad\begin{cases}\phi_{2}(C)=-1,\\\
\phi_{2}(\sigma)=-1.\end{cases}$

######  Theorem 1.5.

Let $P$ be the point group of one of the $2$-dimensional space groups $S$,
acting on $T^{2}=\mathbb{R}^{2}/\Pi$ via $P\subset{\rm O}(2)$, and $\phi\colon
P\to\mathbb{Z}_{2}$ a non-trivial homomorphism. Then,
$H^{3}_{P}(T^{2};\mathbb{Z}_{\phi})=F^{0}H^{3}_{P}(T^{2};\mathbb{Z}_{\phi})=F^{1}H^{3}_{P}(T^{2};\mathbb{Z}_{\phi})$.
This cohomology group and its subgroups
$F^{p}H^{3}_{P}(T^{2};\mathbb{Z}_{\phi})$ are as in Fig. 4.

$\begin{array}[]{|c|c|c||c|c|c||c|c|}\hline\cr\mbox{Space group
$S$}&P&\phi&H^{3}_{P}(T^{2};\mathbb{Z}_{\phi})&F^{2}&F^{3}&E^{1,2}_{\infty}&E^{2,1}_{\infty}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p2}}&\mathbb{Z}_{2}&\phi_{1}&\mathbb{Z}_{2}^{\oplus
4}&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}^{\oplus
2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p4}}&\mathbb{Z}_{4}&\phi_{1}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p6}}&\mathbb{Z}_{6}&\phi_{1}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{pm}/{pg}}&D_{1}&\phi_{0}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}&0&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{cm}}&D_{1}&\phi_{0}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&0&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{pmm}/{pmg}/{pgg}}&D_{2}&\phi_{0}&\mathbb{Z}_{2}^{\oplus
4}&\mathbb{Z}_{2}^{\oplus 4}&\mathbb{Z}_{2}^{\oplus
2}&0&\mathbb{Z}_{2}^{\oplus 2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{pmm}/{pmg}/{pgg}}&D_{2}&\phi_{1},\phi_{2}&\mathbb{Z}_{2}^{\oplus
6}&\mathbb{Z}_{2}^{\oplus 5}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}^{\oplus 3}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{cmm}}&D_{2}&\phi_{0}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}^{\oplus 2}&\mathbb{Z}_{2}^{\oplus
2}&0&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{cmm}}&D_{2}&\phi_{1},\phi_{2}&\mathbb{Z}_{2}^{\oplus
4}&\mathbb{Z}_{2}^{\oplus 3}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p3m1}}&D_{3}&\phi_{0}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&0&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p31m}}&D_{3}&\phi_{0}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}&0&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p4m}/{p4g}}&D_{4}&\phi_{0}&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}^{\oplus 3}&\mathbb{Z}_{2}^{\oplus
2}&0&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p4m}/{p4g}}&D_{4}&\phi_{1},\phi_{2}&\mathbb{Z}_{2}^{\oplus
4}&\mathbb{Z}_{2}^{\oplus 3}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p6m}}&D_{6}&\phi_{0}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}^{\oplus 2}&\mathbb{Z}_{2}^{\oplus
2}&0&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p6m}}&D_{6}&\phi_{1}&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}^{\oplus 2}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}&0\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p6m}}&D_{6}&\phi_{2}&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}^{\oplus 2}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}&0\tsep{2pt}\bsep{2pt}\\\ \hline\cr\end{array}$
Figure 4: The list of $H^{3}_{P}(T^{2};\mathbb{Z}_{\phi})$ and its subgroups
$F^{p}=F^{p}H^{3}_{P}(T^{2};\mathbb{Z}_{\phi})$. The $E_{\infty}$-term of the
Leray–Serre spectral sequence is related to these subgroups by
$E^{p,3-p}_{\infty}\cong F^{p}/F^{p+1}$.

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 $\phi\colon
P\to\mathbb{Z}_{2}$. Some of such modified actions differ from those given by
the inclusion $P\subset{\rm O}(2)$, 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 $H^{3}_{P}(T^{2};\mathbb{Z}_{\phi})$ and its
filtration is similar to the one computing $H^{3}_{P}(T^{2};\mathbb{Z})$. In
the computation, the cohomology $H^{n}_{P}(T^{2};\mathbb{Z}_{\phi})$ for
$n\leq 2$ 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 $T^{2}=\mathbb{R}^{2}/\Pi$ via $P\subset{\rm O}(2)$. For $n\leq 3$,
the $P$-equivariant cohomology $H^{n}_{P}(T^{2};\mathbb{Z}_{\phi})$ with
coefficients in the local system $\mathbb{Z}_{\phi}$ induced from a non-
trivial homomorphism $\phi\colon P\to\mathbb{Z}_{2}$ is as in Fig. 5.

$\begin{array}[]{|c|c|c||c|c|c|c|}\hline\cr\mbox{Space group
$S$}&P&\phi&H^{0}_{P}(T^{2})&H^{1}_{P}(T^{2})&H^{2}_{P}(T^{2})&H^{3}_{P}(T^{2})\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p2}}&\mathbb{Z}_{2}&\phi_{1}&0&\mathbb{Z}_{2}\oplus\mathbb{Z}^{\oplus
2}&0&\mathbb{Z}_{2}^{\oplus 4}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p4}}&\mathbb{Z}_{4}&\phi_{1}&0&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}^{\oplus
2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p6}}&\mathbb{Z}_{6}&\phi_{1}&0&\mathbb{Z}_{2}&\mathbb{Z}_{3}&\mathbb{Z}_{2}^{\oplus
2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{pm}/{pg}}&D_{1}&\phi_{0}&0&\mathbb{Z}_{2}\oplus\mathbb{Z}&\mathbb{Z}_{2}\oplus\mathbb{Z}&\mathbb{Z}_{2}^{\oplus
2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{cm}}&D_{1}&\phi_{0}&0&\mathbb{Z}_{2}\oplus\mathbb{Z}&\mathbb{Z}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{pmm}/{pmg}/{pgg}}&D_{2}&\phi_{0}&0&\mathbb{Z}_{2}&\mathbb{Z}_{2}^{\oplus
3}\oplus\mathbb{Z}&\mathbb{Z}_{2}^{\oplus 4}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{pmm}/{pmg}/{pgg}}&D_{2}&\phi_{1},\phi_{2}&0&\mathbb{Z}_{2}\oplus\mathbb{Z}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}^{\oplus 6}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{cmm}}&D_{2}&\phi_{0}&0&\mathbb{Z}_{2}&\mathbb{Z}_{2}^{\oplus
2}\oplus\mathbb{Z}&\mathbb{Z}_{2}^{\oplus 2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{cmm}}&D_{2}&\phi_{1},\phi_{2}&0&\mathbb{Z}_{2}\oplus\mathbb{Z}&\mathbb{Z}_{2}&\mathbb{Z}_{2}^{\oplus
4}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p3m1}}&D_{3}&\phi_{0}&0&\mathbb{Z}_{2}&\mathbb{Z}_{3}^{\oplus
2}\oplus\mathbb{Z}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p31m}}&D_{3}&\phi_{0}&0&\mathbb{Z}_{2}&\mathbb{Z}_{3}\oplus\mathbb{Z}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p4m}/{p4g}}&D_{4}&\phi_{0}&0&\mathbb{Z}_{2}&\mathbb{Z}_{4}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}&\mathbb{Z}_{2}^{\oplus
3}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p4m}/{p4g}}&D_{4}&\phi_{1},\phi_{2}&0&\mathbb{Z}_{2}&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}^{\oplus 4}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p6m}}&D_{6}&\phi_{0}&0&\mathbb{Z}_{2}&\mathbb{Z}_{6}\oplus\mathbb{Z}&\mathbb{Z}_{2}^{\oplus
2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p6m}}&D_{6}&\phi_{1}&0&\mathbb{Z}_{2}&\mathbb{Z}_{2}\oplus\mathbb{Z}_{3}&\mathbb{Z}_{2}^{\oplus
3}\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mbox{{p6m}}&D_{6}&\phi_{2}&0&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\mathbb{Z}_{2}^{\oplus
3}\tsep{2pt}\bsep{2pt}\\\ \hline\cr\end{array}$ 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 $\mathbb{Z}_{\phi}$
associated to a homomorphism $\phi\colon P\to\mathbb{Z}_{2}$. For the
cohomology with such a local system, the notion of reduced cohomology does not
make sense. 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 $\mathbb{Z}_{\phi}$. 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 $\phi_{2}$. 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 $\boldsymbol{K}$-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.)

###  2.1 Setting

Let us consider the following mathematical setting:

  * •

A lattice $\Pi\subset\Pi\otimes_{\mathbb{Z}}\mathbb{R}=\mathbb{R}^{d}$ of rank
$d$.

  * •

A subgroup $S$ of the Euclidean group $\mathbb{R}^{d}\rtimes{\rm O}(d)$ of
$\mathbb{R}^{d}$ which is an extension of a finite group $P\subset{\rm O}(d)$
by $\Pi$:

$\displaystyle\begin{array}[]{c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c}1&\
\longrightarrow&\mathbb{R}^{d}&\ \longrightarrow&\mathbb{R}^{d}\rtimes{\rm
O}(d)&\ \longrightarrow&{\rm O}(d)&\ \longrightarrow&1\\\
&&\cup&&\cup&&\cup&&\\\ 1&\ \longrightarrow&\Pi&\ \longrightarrow&S&\
\overset{\pi}{\longrightarrow}&P&\ \longrightarrow&1.\end{array}$

  * •

A unitary representation $U\colon P\to U(V)$ 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 $\Pi$,
it is called symmorphic, otherwise nonsymmorphic.

Based on the mathematical setting above, we can introduce a quantum system on
$\mathbb{R}^{d}$ 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 $L^{2}(\mathbb{R}^{d},V)$, on which $g\in S$ acts by
$\psi(x)\mapsto(\rho(g)\psi)(x)=U(\pi(g))\psi(g^{-1}x)$.

  * •

The ‘Hamiltonian’ is a self-adjoint operator $H$ on $L^{2}(\mathbb{R}^{d},V)$
invariant under the $S$-action: $H\circ\rho(g)=\rho(g)\circ H$. A typical form
of $H$ is $H=\Delta+\Phi$, where $\Delta=\sum\partial^{2}/\partial x_{i}^{2}$
is the Laplacian and $\Phi\colon\mathbb{R}^{d}\to\mathrm{End}(V)$ is a
potential term.

###  2.2 Bloch transformation

Even if the Hamiltonian $H$ is invariant under the translation of $\Pi$, a
solution $\psi$ to the ‘time-independent Schrödinger equation’ $H\psi=E\psi$
with $E\in\mathbb{R}$ is not necessarily $S$-invariant. The so-called ‘Bloch
transformation’ allows us to deal with such a situation.

Let $\hat{\Pi}=\operatorname{Hom}(\Pi,U(1))$ denote the Pontryagin dual of the
lattice $\Pi$, which is often called the ‘Brillouin torus’ in condensed matter
physics. We define the space $L^{2}_{\Pi}(\hat{\Pi}\times\mathbb{R}^{d},V)$ by

$\displaystyle
L^{2}_{\Pi}\big{(}\hat{\Pi}\times\mathbb{R}^{d},V\big{)}=\big{\\{}\hat{\psi}\in
L^{2}\big{(}\hat{\Pi}\times\mathbb{R}^{d},V\big{)}\,|\,\hat{\psi}(\hat{k},x+m)=\hat{k}(m)\hat{\psi}(\hat{k},x)\
(m\in\Pi)\\}.$

We also define transformations $\hat{\mathcal{B}}$ and $\mathcal{B}$, inverse
to each other:

$\displaystyle\hat{\mathcal{B}}\colon\
L^{2}\big{(}\mathbb{R}^{d},V\big{)}\longrightarrow
L^{2}_{\Pi}\big{(}\hat{\Pi}\times\mathbb{R}^{d},V\big{)},\qquad$
$\displaystyle(\hat{\mathcal{B}}\psi)\big{(}\hat{k},x\big{)}=\sum_{n\in\Pi}\hat{k}(n)^{-1}\psi(x+n),$
$\displaystyle\mathcal{B}\colon\
L^{2}_{\Pi}\big{(}\hat{\Pi}\times\mathbb{R}^{d},V\big{)}\longrightarrow
L^{2}\big{(}\mathbb{R}^{d},V\big{)},\qquad$
$\displaystyle(\mathcal{B}\hat{\psi})(x)=\int_{\hat{k}\in\hat{\Pi}}\hat{\psi}\big{(}\hat{k},x\big{)}d\hat{k}.$

As is described in [8], the space
$L^{2}_{\Pi}(\hat{\Pi}\times\mathbb{R}^{d},V)$ can be identified with the
space $L^{2}(\hat{\Pi},\mathcal{E}\otimes V)$ of $L^{2}$-sections of a vector
bundle $\mathcal{E}\otimes V\to\hat{\Pi}$. The infinite-dimensional vector
bundle $\mathcal{E}\to\hat{\Pi}$ is given by

$\displaystyle\mathcal{E}=\bigcup_{\hat{k}\in\hat{\Pi}}L^{2}\big{(}\mathbb{R}^{d}/\Pi,\mathcal{L}|_{\\{\hat{k}\\}\times\mathbb{R}^{d}/\Pi}\big{)},$

where $\mathcal{L}\to\hat{\Pi}\times\mathbb{R}^{d}/\Pi$ is the Poincaré line
bundle, the quotient of the product line bundle
$\hat{\Pi}\times\mathbb{R}^{d}\times\mathbb{C}\to\hat{\Pi}\times\mathbb{R}^{d}$
by the following $\Pi$-action

$\displaystyle\Pi\times\big{(}\hat{\Pi}\times\mathbb{R}^{d}\times\mathbb{C}\big{)}\longrightarrow\hat{\Pi}\times\mathbb{R}^{d}\times\mathbb{C},\qquad\big{(}m,\hat{k},x,z\big{)}\mapsto\big{(}\hat{k},x+m,\hat{k}(m)z\big{)}.$

In summary, we get an identification of $L^{2}$-spaces

$\displaystyle L^{2}\big{(}\mathbb{R}^{d},V\big{)}\cong
L^{2}_{\Pi}\big{(}\hat{\Pi}\times\mathbb{R}^{d},V\big{)}\cong
L^{2}\big{(}\hat{\Pi},\mathcal{E}\otimes V\big{)}.$

The Hamiltonian $H$ on $L^{2}(\mathbb{R}^{d},V)$ then induces an operator
$\hat{H}$ on $L^{2}_{\Pi}(\hat{\Pi}\times\mathbb{R}^{d},V)\cong
L^{2}(\hat{\Pi},\mathcal{E}\otimes V)$ by
$\hat{H}\circ\mathcal{\hat{B}}=\mathcal{\hat{B}}\circ H$. If, for instance,
$H$ is of the form $H=\Delta+\Phi$, then $\hat{H}$ preserves the fiber of
$\mathcal{E}\otimes V$. 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 $\hat{H}(\hat{k})$
would be confined to a compact region in $\mathbb{R}$ as $\hat{k}\in\hat{\Pi}$
varies. Then the corresponding eigenfunctions form a finite rank subbundle
$E\subset\mathcal{E}\otimes V$, called the ‘Bloch bundle’. The $K$-class of
this vector bundle $E\to\hat{\Pi}$ is regarded as an invariant of the quantum
system under study.

###  2.3 Nonsymmorphic group and twisted $\boldsymbol{K}$-theory

We now take the symmetry into account. From the extension $1\to\Pi\to
S\overset{\pi}{\to}P\to 1$, we can associate a twisted $P$-equivariant vector
bundle on $\hat{\Pi}$ to the $S$-module $L^{2}(\mathbb{R}^{d},V)$. This is a
version of the so-called ‘Mackey machine’.

Recall that the Euclidean group $\mathbb{R}^{d}\rtimes{\rm O}(d)$ is the semi-
direct product of the orthogonal group ${\rm O}(d)$ and the group of
translations $\mathbb{R}^{d}$. Hence a collection of representatives
$\\{s_{p}\\}_{p\in P}$ of $p\in P\cong S/\Pi$ in $S$ is expressed as
$s_{p}=(a_{p},p)\in\mathbb{R}^{d}\rtimes{\rm O}(d)$ by means of a map $a\colon
P\to\mathbb{R}^{d}$. For $p_{1},p_{2}\in P$ we put

$\displaystyle\nu(p_{1},p_{2})=a_{p_{1}}+p_{1}a_{p_{2}}-a_{p_{1}p_{2}}.$

Since $\Pi\subset S$ is normal, the action of $P\subset{\rm O}(d)$ on
$\mathbb{R}^{d}$ preserves $\Pi\subset\mathbb{R}^{d}$. Then we have
$\nu(p_{1},p_{2})\in\Pi$, and $\nu\colon P\times P\to\Pi$ is a group
$2$-cocycle of $P$ with values in $\Pi$ regarded as a left $P$-module through
the action $m\mapsto pm$ of $p\in P$ on $m\in\Pi$. This group $2$-cocycle
measures the failure for $S$ to be symmorphic.

By means of the $S$-action $\rho$ on $L^{2}(\mathbb{R}^{d},V)$, we define an
‘action’ of $p\in P$ by

$\displaystyle\rho(p)\colon\
L^{2}\big{(}\mathbb{R}^{d},V\big{)}\longrightarrow
L^{2}\big{(}\mathbb{R}^{d},V\big{)},\qquad\rho(p)=\rho((a_{p},p)),$

whose explicit formula for $\psi\in L^{2}(\mathbb{R}^{d},V)$ is given by
$(\rho(p)\psi)(x)=U(p)\psi(p^{-1}x+a_{p^{-1}})$. The Bloch transformation then
induces the following ‘action’ of $P$,

$\displaystyle\hat{\rho}(p)\colon\
L^{2}_{\Pi}\big{(}\hat{\Pi}\times\mathbb{R}^{d},V\big{)}\to
L^{2}_{\Pi}\big{(}\hat{\Pi}\times\mathbb{R}^{d},V\big{)},\qquad\hat{\rho}(p)\circ\hat{\mathcal{B}}=\hat{\mathcal{B}}\circ\rho(p),$

whose explicit formula for $\hat{\psi}\in
L^{2}_{\Pi}(\hat{\Pi}\times\mathbb{R}^{d},V)$ is
$(\hat{\rho}(p)\hat{\psi})(\hat{k},x)=U(p)\hat{\psi}(p^{-1}\hat{k},p^{-1}x+a_{p^{-1}})$.
Here the left $P$-action on $\hat{\Pi}$ is defined by
$(p\hat{k})(m)=\hat{k}(p^{-1}m)$, where $p\in P$ acts on $m\in\Pi$ through the
inclusion $P\subset{\rm O}(d)$ and the left action of ${\rm O}(d)$ on
$\mathbb{R}^{d}$. Notice that $\rho$ and $\hat{\rho}$ 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:

$\displaystyle\big{(}\hat{\rho}(p_{1})\big{(}\hat{\rho}(p_{2})\hat{\psi}\big{)}\big{)}\big{(}\hat{k},\xi\big{)}=\big{(}p_{2}^{-1}p_{1}^{-1}\hat{k}\big{)}\big{(}\nu\big{(}p_{2}^{-1},p_{1}^{-1}\big{)}\big{)}\big{(}\hat{\rho}(p_{1}p_{2})\hat{\psi}\big{)}\big{(}\hat{k},\xi\big{)}.$

To interpret the ‘action’ $\hat{\rho}(p)$ in terms of the vector bundle
$\mathcal{E}\otimes V$ through
$L^{2}_{\Pi}(\hat{\Pi}\times\mathbb{R}^{d},V)\cong
L^{2}(\hat{\Pi},\mathcal{E}\otimes V)$, recall that the fiber of
$\mathcal{E}\otimes V$ at $\hat{k}\in\hat{\Pi}$ is

$\displaystyle\mathcal{E}|_{\hat{k}}\otimes
V=L^{2}\big{(}\mathbb{R}^{d}/\Pi,\mathcal{L}|_{\\{\hat{k}\\}\times\mathbb{R}^{d}/\Pi}\otimes
V\big{)},$

and $\hat{\psi}\in L^{2}_{\Pi}(\hat{\Pi}\times\mathbb{R}^{d},V)$ corresponds
to the following section $\Psi\in L^{2}(\hat{\Pi},\mathcal{E}\otimes V)$:

$\displaystyle\Psi\big{(}\hat{k}\big{)}\colon\
\mathbb{R}^{d}/\Pi\longrightarrow\mathcal{L}|_{\\{\hat{k}\\}\times\mathbb{R}^{d}/\Pi}\otimes
V,\qquad x\mapsto\big{[}\hat{k},x,\hat{\psi}(\hat{k},x)\big{]}.$

Define for $p\in P$ and $\hat{k}\in\hat{\Pi}$ a linear map

$\displaystyle\rho_{\mathcal{E}\otimes V}\big{(}p;\hat{k}\big{)}\colon\
\mathcal{E}|_{\hat{k}}\otimes V\longrightarrow\mathcal{E}|_{p\hat{k}}\otimes
V$

by the assignment of the sections

$\displaystyle\rho_{\mathcal{E}\otimes
V}\big{(}p;\hat{k}\big{)}\big{(}\big{[}x\mapsto\big{[}\hat{k},x,\hat{\psi}(\hat{k},x)\big{]}\big{]}\big{)}=\big{[}x\mapsto\big{[}p\hat{k},x,U(p)\hat{\psi}\big{(}\hat{k},p^{-1}x+a_{p^{-1}}\big{)}\big{]}\big{]}.$

These maps constitute a vector bundle map $\rho_{\mathcal{E}\otimes
V}(p)\colon\mathcal{E}\otimes V\to\mathcal{E}\otimes V$ covering the action
$\hat{k}\mapsto p\hat{k}$ on $\hat{\Pi}$

$\displaystyle\begin{CD}\mathcal{E}\otimes V@>{\rho_{\mathcal{E}\otimes
V}(p)}>{}>\mathcal{E}\otimes V\\\ @V{}V{}V@V{}V{}V\\\
<EMAIL_ADDRESS>
This is a $\tau$-twisted $P$-action, in the sense that the formula

$\displaystyle\rho_{\mathcal{E}\otimes
V}\big{(}p_{1};p_{2}\hat{k}\big{)}\rho_{\mathcal{E}\otimes
V}\big{(}p_{2};\hat{k}\big{)}\xi=\tau\big{(}p_{1},p_{2};\hat{k}\big{)}\rho_{\mathcal{E}\otimes
V}\big{(}p_{1}p_{2};\hat{k}\big{)}\xi$

holds for $p_{1},p_{2}\in P$, $\hat{k}\in\hat{\Pi}$ and
$\xi\in\mathcal{E}|_{\hat{k}}\otimes V$. Here $\tau\colon P\times
P\times\hat{\Pi}\to U(1)$ is defined by

$\displaystyle\tau(p_{1},p_{2};\hat{k})=\hat{k}\big{(}\nu\big{(}p_{2}^{-1},p_{1}^{-1}\big{)}\big{)},$

and is regarded as a group $2$-cocycle of $P$ with its coefficients in the
group $C(\hat{\Pi},U(1))$ of $U(1)$-valued functions on $\hat{\Pi}$ thought of
as a right $P$-module through the pull-back under the left action
$\hat{k}\mapsto p\hat{k}$ of $p\in P$ on $\hat{k}\in\hat{\Pi}$. The map
$\rho_{\mathcal{E}\otimes V}(p)$ on the vector bundle induces the
transformation on the sections

$\displaystyle\rho_{\mathcal{E}\otimes V}(p)\colon\
L^{2}\big{(}\hat{\Pi},\mathcal{E}\otimes V\big{)}\longrightarrow
L^{2}\big{(}\hat{\Pi},\mathcal{E}\otimes V\big{)}$

by $(\rho_{\mathcal{E}\otimes V}(p)\Psi)(\hat{k})=\rho_{\mathcal{E}\otimes
V}(p;p^{-1}\hat{k})\Psi(p^{-1}\hat{k})$. One can verify that: if $\Psi\in
L^{2}(\hat{\Pi},\mathcal{E}\otimes V)$ corresponds to $\hat{\psi}\in
L^{2}_{\Pi}(\hat{\Pi}\times\mathbb{R}^{d},V)$, then $\rho_{\mathcal{E}\otimes
V}(p)\Psi$ corresponds to $\hat{\rho}(p)\hat{\psi}$. Hence the ‘action’
$\hat{\rho}(p)$ on $L^{2}_{\Pi}(\hat{\Pi}\times\mathbb{R}^{d},V)\cong
L^{2}(\hat{\Pi},\mathcal{E}\otimes V)$ agrees with the one induced from the
$\tau$-twisted $P$-action on $\mathcal{E}\otimes V$.

Now, under the assumption that $\hat{H}$ describes an insulator, the Bloch
bundle $E\subset\mathcal{E}\otimes V$ inherits a $\tau$-twisted $P$-action
from $\mathcal{E}\otimes V$. This is a consequence of the invariance of the
Hamiltonian under the space group action. Therefore the Bloch bundle, being a
$\tau$-twisted $P$-equivariant vector bundle of finite rank, defines a class
in the $\tau$-twisted $P$-equivariant $K$-theory $K^{\tau+0}_{P}(\hat{\Pi})$,
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 $\tau$ to symmorphic space groups. However, in the
symmorphic case, the cocycle $\nu$ and hence $\tau$ 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 $\omega\colon P\times P\to U(1)$. In this case, the resulting
Bloch bundle defines a class in the twisted equivariant $K$-theory
$K^{\tau+\omega+0}_{P}(\hat{\Pi})$.

######  Remark 2.1.

The phase factor in the composition rule of $\hat{\rho}$,

$\displaystyle\tau_{R}\big{(}\hat{k};p_{1},p_{2}\big{)}=\big{(}p_{2}^{-1}p_{1}^{-1}\hat{k}\big{)}\big{(}\nu\big{(}p_{2}^{-1},p_{1}^{-1}\big{)}\big{)}=\hat{k}\big{(}p_{1}p_{2}\nu\big{(}p_{2}^{-1},p_{1}^{-1}\big{)}\big{)}$

defines a group $2$-cocycle of $P$ with coefficients in $C(\hat{\Pi},U(1))$,
when regarded as a left $P$-module by the right action
$\hat{k}\mapsto\hat{k}p=p^{-1}\hat{k}$ of $p\in P$ on $\hat{k}\in\hat{\Pi}$.
The $2$-cocycles $\tau$ and $\tau_{R}$ are related by
$\tau_{R}(\hat{k};p_{1},p_{2})=\tau(p_{1},p_{2};(p_{1}p_{2})^{-1}\hat{k})$.
This extends to a cochain bijection of group cochains with coefficients in the
left/right $P$-modules $C(\hat{\Pi},U(1))$. Thus, $\tau$ and $\tau_{R}$ have
cohomologically the same information. We also remark that $\tau$ and
$\tau_{R}$ are respectively cohomologous to the following $2$-cocycles:

$\displaystyle\tau^{\prime}\big{(}p_{1},p_{2};\hat{k}\big{)}=\big{(}p_{1}p_{2}\hat{k}\big{)}(\nu(p_{1},p_{2}))^{-1},\qquad\tau^{\prime}_{R}\big{(}\hat{k};p_{1},p_{2}\big{)}=\hat{k}(\nu(p_{1},p_{2}))^{-1}.$

######  Remark 2.2.

Given a homomorphism $c\colon P\to\mathbb{Z}_{2}$, we can impose that the
Hamiltonian $H$ and the symmetry $\rho(g)$ with $g\in S$ are graded
commutative, $H\circ\rho(g)=c(\pi(g))\rho(g)\circ H$. Then the quantum system
with symmetry in question leads to an element of the twisted equivariant
$K$-theory $K^{\tau+c+0}_{P}(\hat{\Pi})$, where the (ungraded) twist $\tau$ is
now graded by $c\in H^{1}_{P}(\hat{\Pi};\mathbb{Z}_{2})$. 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 $\tau$ 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]).

###  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, $\Pi$ its lattice, and $P$
its point group.

  1. (A)

By the inclusion $P\subset{\rm O}(d)$ and the standard left action of ${\rm
O}(d)$ on $\mathbb{R}^{d}$, the point group $P$ acts on $\mathbb{R}^{d}$,
preserving $\Pi\subset\mathbb{R}^{d}$. Hence the left action of $P$ on
$\mathbb{R}^{d}$ descends to give a left action of $P$ on the torus
$\mathbb{R}^{d}/\Pi$.

  2. (B)

By the action (A), the point group $P$ acts on the Pontryagin dual
$\hat{\Pi}=\operatorname{Hom}(\Pi,U(1))$ of $\Pi$ from the left: For $p\in P$
and $\hat{k}\in\hat{\Pi}$, we define $p\hat{k}\in\hat{\Pi}$ by
$(p\hat{k})(m)=\hat{k}(p^{-1}m)$ for all $m\in\Pi$.

  3. (C)

By the inclusion $S\subset\mathbb{R}^{d}\rtimes{\rm O}(d)$ and the standard
left action of $\mathbb{R}^{d}\rtimes{\rm O}(d)$ on $\mathbb{R}^{d}$, the
space group $S$ acts on $\mathbb{R}^{d}$. The subgroup $\Pi\subset S$
preserves $\Pi\subset\mathbb{R}^{d}$, so that the point group $P\cong S/\Pi$
acts on $\mathbb{R}^{d}/\Pi$.

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 $0\in\mathbb{R}^{d}/\Pi$
and $0\in\hat{\Pi}$, respectively, where we regard $\hat{\Pi}$ as
$\operatorname{Hom}(\Pi,\mathbb{R}/\mathbb{Z})$ via
$\mathbb{R}/\mathbb{Z}\cong U(1)$ and $0\in\hat{\Pi}$ stands for the trivial
homomorphism. On the other hand, if $(a_{p},p)\in\mathbb{R}^{d}\rtimes{\rm
O}(d)$ is a lift of $p\in P\subset{\rm O}(d)$, then the action of $p\in P$ on
$k\in\mathbb{R}^{d}/\Pi$ in (C) admits the description $k\mapsto pk+a_{p}$. If
$S$ is symmorphic, then we can choose $a_{p}$ to be in $\Pi$. In this case,
the actions (A) and (C) are equivalent. However, if $S$ is nonsymmorphic, then
$a_{p}$ cannot be in $\Pi$. Thus, in this case, the action of $p\in P$ does
not fix any point on $\mathbb{R}^{d}/\Pi$, so that the actions (A) and (C) are
not equivalent. For example, in the case of pg, the action of
$P=\mathbb{Z}_{2}$ 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 $\mathbb{R}^{d}/\Pi$
with $\hat{\Pi}=\operatorname{Hom}(\Pi,\mathbb{R}/\mathbb{Z})$, 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 $\Pi\cong\mathbb{Z}^{d}$. This choice induces the
following identifications of tori inverse to each other:

$\displaystyle\hat{\Pi}\longrightarrow\mathbb{R}^{d}/\Pi,\qquad$
$\displaystyle\hat{k}\mapsto\sum_{j}\hat{k}(v_{j})v_{j},$
$\displaystyle\mathbb{R}^{d}/\Pi\longrightarrow\hat{\Pi},\qquad$
$\displaystyle\sum_{j}k_{j}v_{j}\mapsto\bigg{[}\sum_{j}m_{j}v_{j}\mapsto\sum_{j}m_{j}k_{j}\bigg{]}.$

With this identification of tori, the left $P$-action on $\hat{\Pi}$ in (B)
induces a right $P$-action on $\mathbb{R}^{d}/\Pi$. Considering the action of
$p^{-1}$ instead of $p$, we finally get a left action of $P$ on
$\mathbb{R}^{d}/\Pi$, induced from (B) and the identification
$\hat{\Pi}\cong\mathbb{R}^{d}/\Pi$. In general, this left action of $P$ on
$\mathbb{R}^{d}/\Pi$ 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, $P_{S}$ its point group, $\Pi_{S}$
its lattice, and $\hat{\Pi}_{S}$ the Pontryagin dual of $\Pi_{S}$.

  * $(a)$

We choose a basis $\\{v_{j}\\}$ of $\Pi_{S}$ to identify $\hat{\Pi}_{S}$ with
$\mathbb{R}^{2}/\Pi_{S}$, and let the action in $(B)$ induce an action of
$P_{S}$ on $\mathbb{R}^{2}/\Pi_{S}$. 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 $P_{S}$ on
$\mathbb{R}^{2}/\Pi_{S}$ induced from $(B)$ is equivalent to the action of
$P_{S}$ on $\mathbb{R}^{2}/\Pi_{S}$ in $(A)$.

  * (c)

If $S$ is p3m1 $($respectively p31m$)$, then the action of $P_{S}$ on
$\mathbb{R}^{2}/\Pi_{S}$ induced from $(B)$ is equivalent to the action of
$P_{S^{\prime}}$ on $\mathbb{R}^{2}/\Pi_{S^{\prime}}$ in $(A)$, where
$S^{\prime}$ 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
$P_{S}=P_{S^{\prime}}$ and $\Pi_{S}=\Pi_{S^{\prime}}$ in the third item in the
lemma above.

###### Proof.

Considering the action $(A)$, we define $\psi(p)_{\ell j}\in\mathbb{Z}$ by
$pv_{j}=\sum_{\ell}\psi(p)_{\ell j}v_{\ell}$, and a homomorphism $\psi\colon
P_{S}\to{\rm GL}(2,\mathbb{Z})$ by $\psi(p)=(\psi(p)_{\ell j})$. Since $P_{S}$
is the point group of a $2$-dimensional space group $S$, the homomorphism
$\psi$ is injective and its image $\psi(P_{S})$ is a finite subgroup of ${\rm
GL}(2,\mathbb{Z})$. Let $S^{\prime}$ be another $2$-dimensional space group
with its point group $P^{\prime}$. Choosing a basis of its lattice
$\Pi^{\prime}$, we similarly get from the action (A) a homomorphism
$\psi^{\prime}\colon P_{S^{\prime}}\to{\rm GL}(2,\mathbb{Z})$. If the images
$\psi(P_{S})$ and $\psi^{\prime}(P_{S^{\prime}})$ are conjugate to each other
in ${\rm GL}(2,\mathbb{Z})$, then the actions of $P_{S}$ and $P_{S^{\prime}}$
in (A) are equivalent. The action of $P_{S}$ on $\mathbb{R}^{2}/\Pi_{S}$
induced from (B) also yields an associated homomorphism $P_{S}\to{\rm
GL}(2,\mathbb{Z})$. This homomorphism turns out to be the transpose inverse
${}^{t}\psi^{-1}\colon P_{S}\to{\rm GL}(2,\mathbb{Z})$, which is again
injective and defines a finite subgroup ${}^{t}\psi^{-1}(P_{S})\subset{\rm
GL}(2,\mathbb{Z})$. If we alter the basis $\\{v_{j}\\}$, then
${}^{t}\psi^{-1}$ changes by a conjugation of a matrix in ${\rm
GL}(2,\mathbb{Z})$. Thus, up to conjugations, the image
${}^{t}\psi^{-1}(P_{S})\subset{\rm GL}(2,\mathbb{Z})$ is independent of the
choice of $\\{v_{j}\\}$, showing (a). Now we can directly verify (b) and (c),
by computing the homomorphism $\psi$ based on the explicit basis in Appendix,
and comparing the images $\psi(P_{S})$ and ${}^{t}\psi^{-1}(P_{S})$ in ${\rm
GL}(2,\mathbb{Z})$. ∎

Another way of identifying $\hat{\Pi}$ with $\mathbb{R}^{d}/\Pi$ is to choose
a bilinear form $\langle\ ,\ \rangle\colon\Pi\times\Pi\to\mathbb{Z}$. We
assume that this form is non-degenerate in the sense that the matrix $(\langle
v_{i},v_{j}\rangle)$ is invertible with respect to any basis $\\{v_{i}\\}$ of
$\Pi$. A non-degenerate bilinear form induces an identification of the tori as
follows

$\displaystyle\mathbb{R}^{d}/\Pi\longrightarrow\hat{\Pi}=\operatorname{Hom}(\Pi,\mathbb{R}/\mathbb{Z}),\qquad
k\mapsto[m\mapsto\langle m,k\rangle].$

If the bilinear form is $P$-invariant in the sense that $\langle
pm,pm^{\prime}\rangle=\langle m,m^{\prime}\rangle$ for all
$m,m^{\prime}\in\Pi$ and $p\in P$, then the action (A) on $\mathbb{R}^{d}/\Pi$
agrees with the action (B) on $\hat{\Pi}$ under the induced identification
$\mathbb{R}^{d}/\Pi\cong\hat{\Pi}$. For the $2$-dimensional space groups such
that $\Pi$ can be the standard lattice $\mathbb{Z}^{2}\subset\mathbb{R}^{2}$,
the standard inner product on $\mathbb{R}^{2}$ restricts to give a
$P$-invariant non-degenerate bilinear form. If we choose an orthonormal basis
$\\{v_{j}\\}$, then the identifications $\mathbb{R}^{d}/\Pi\cong\hat{\Pi}$
given by $\langle\ ,\ \rangle$ and by $\\{v_{j}\\}$ are $P$-equivariantly the
same.

In Section 4, we will work with the torus $\mathbb{R}^{2}/\Pi$ with the action
(A), and the relation to $\hat{\Pi}$ 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
$H^{3}_{G}(X;\mathbb{Z})$ 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].

###  3.1 Spectral sequences

The Borel equivariant cohomology $H^{n}_{G}(X;\mathbb{Z})$ is defined to be
the (singular) cohomology of the quotient space $EG\times_{G}X$ of $EG\times
X$ under the diagonal $G$-action $(\xi,x)\mapsto(\xi g,g^{-1}x)$, where $EG$
is the total space of the universal $G$-bundle $EG\to BG$. Associated to the
fibration $X\to EG\times_{G}X\to BG$ is the Leray–Serre spectral sequence

$\displaystyle E_{r}^{p,q}\Longrightarrow H^{p+q}_{G}(X;\mathbb{Z})$

converging to the graded quotient of a filtration

$\displaystyle H^{n}_{G}(X;\mathbb{Z})=F^{0}H^{n}_{G}(X;\mathbb{Z})\supset
F^{1}H^{n}_{G}(X;\mathbb{Z})\supset\cdots\supset
F^{n+1}H^{n}_{G}(X;\mathbb{Z})=0,$

that is,
$E_{\infty}^{p,q}=F^{p}H^{p+q}_{G}(X;\mathbb{Z})/F^{p+1}H^{p+q}_{G}(X;\mathbb{Z})$.
The $E_{2}$-term is given by the group cohomology of $G$

$\displaystyle E_{2}^{p,q}=H^{p}_{\mathrm{group}}(G;H^{q}(X;\mathbb{Z})),$

where the coefficient $H^{q}(X;\mathbb{Z})$ 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
$C^{p}_{\mathrm{group}}(G;M)=C(G^{p},M)=\\{\tau\colon G^{p}\to M\\}$, and the
coboundary $\partial\colon C^{p}_{\mathrm{group}}(G;M)\to
C^{p+1}_{\mathrm{group}}(G;M)$ is given by

$\displaystyle(\partial\tau)(g_{1},\ldots,g_{p+1})=\tau(g_{2},\ldots,g_{p+1})+\sum_{i=1}^{p}(-1)^{i}\tau(g_{1},\ldots,g_{i}g_{i+1},\ldots,g_{p+1})$
$\displaystyle\hphantom{(\partial\tau)(g_{1},\ldots,g_{p+1})=}{}+(-1)^{p+1}\tau(g_{1},\ldots,g_{p})g_{p+1}.$

As an application of the spectral sequence, we can obtain an identification
$H^{n}_{G}(\mathrm{pt};\mathbb{Z})\cong H^{n}_{\mathrm{group}}(G;\mathbb{Z})$.
(We also have $H^{n}_{\mathrm{group}}(G;\mathbb{Z})\cong
H^{n-1}_{\mathrm{group}}(G;U(1))$ for $n\geq 2$ 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 $H^{n}_{G}(X;\mathbb{Z})$
is isomorphic to the cohomology $H^{n}(G^{\bullet}\times X;\mathbb{Z})$ of a
simplicial space $G^{\bullet}\times X$ with its coefficients in the constant
sheaf $\mathbb{Z}$. This is a consequence of a more general theorem about
simplicial space (see [6] for example) together with the fact that the
geometric realization $\lvert G^{\bullet}\times X\rvert$ of $G^{\bullet}\times
X$ is identified with $EG\times_{G}X$.

The simplicial space $G^{\bullet}\times X$ is associated to the left
$G$-action on $X$, and consists of a sequence of spaces $\\{G^{p}\times
X\\}_{p\geq 0}$ together with the face map $\partial_{i}\colon G^{p}\times
X\to G^{p-1}\times X$, $i=0,\ldots,p$, and the degeneracy map $s_{i}\colon
G^{p}\times X\to G^{p+1}\times X$, $i=0,\ldots,p$, given by

$\displaystyle\partial_{i}(g_{1},\ldots,g_{p},x)=\begin{cases}(g_{2},\ldots,g_{p},x),&i=0,\\\
(g_{1},\ldots,g_{i}g_{i+1},\ldots,g_{p},x),&i=1,\ldots,p-1,\\\
(g_{1},\ldots,g_{p-1},g_{p}x),&i=p,\end{cases}$ $\displaystyle
s_{i}(g_{1},\ldots,g_{p},x)=(g_{1},\ldots,g_{i-1},1,g_{i},\ldots,g_{p},x).$

The cohomology $H^{n}(G^{\bullet}\times X;\mathbb{Z})$ is then defined to be
the total cohomology of the double complex $(C^{i}(G^{j}\times
X;\mathbb{Z}),\delta,\partial)$, where $(C^{i}(G^{j}\times
X;\mathbb{Z}),\delta)$ is the complex computing the cohomology of $G^{j}\times
X$ with coefficients in $\mathbb{Z}$ and $\partial\colon C^{i}(G^{j}\times
X;\mathbb{Z})\to C^{i}(G^{j+1}\times X;\mathbb{Z})$ is
$\partial=\sum\limits_{i=0}^{j+1}(-1)^{i}\partial_{i}^{*}$. The double complex
admits a natural filtration $\\{\oplus_{j\geq p}C^{i}(G^{j}\times
X;\mathbb{Z})\\}_{p\geq 0}$. The associated spectral sequence agrees with the
Leray–Serre spectral sequence $E_{r}^{p,q}$, since $G$ is finite.

Now, let us consider the standard exponential exact sequence of sheaves on the
simplicial space
$0\to\mathbb{Z}\to\underline{\mathbb{R}}\to\underline{U(1)}\to 0$, where
$\underline{\mathbb{R}}$ consists of the sheaf of $\mathbb{R}$-valued
functions on $G^{p}\times X$ and $\underline{U(1)}$ consists of the sheaf of
$U(1)$-valued functions on $G^{p}\times X$. As in [9, Lemma 4.4], we can
readily show that $H^{n}(G^{\bullet}\times X;\underline{\mathbb{R}})=0$ for
$n>0$. This vanishing together with the associated long exact sequence leads
to the following isomorphism for $n\geq 1$

$\displaystyle H^{n}(G^{\bullet}\times X;\underline{U(1)})\cong
H^{n+1}(G^{\bullet}\times X;\mathbb{Z}).$

The cohomology $H^{n}(G^{\bullet}\times X;\underline{U(1)})$ can be defined
exactly in the same way as in the case of $H^{n}(G^{\bullet}\times
X;\mathbb{Z})$ by using a double complex. Therefore we have a spectral
sequence

$\displaystyle{}^{\prime}E^{p,q}_{r}\Longrightarrow H^{p+q}(G^{\bullet}\times
X;\underline{U(1)})$

converging to the graded quotient of a filtration

$\displaystyle{}^{\prime}F^{0}H^{n}(G^{\bullet}\times
X;\underline{U(1)})=H^{n}(G^{\bullet}\times
X;\underline{U(1)})\supset{}^{\prime}F^{1}H^{n}(G^{\bullet}\times
X;\underline{U(1)})\supset\cdots,$

whose $E_{2}$-term is

$\displaystyle{}^{\prime}E_{2}^{p,q}=H^{p}_{\mathrm{group}}(G;H^{q}(X;\underline{U(1)})),$

where $H^{q}(X;\underline{U(1)})$ is regarded as a right $G$-module by pull-
back. It is clear that $H^{0}(X;\underline{U(1)})\cong C(X,U(1))$ and
$H^{n}(X;\underline{U(1)})\cong H^{n+1}(X;\mathbb{Z})$ for $n\geq 1$. Since
${}^{\prime}E_{2}^{p,q}$ involves the group cohomology with coefficients in
$C(X,U(1))$, its computation seems to be more complicated than that of
$E_{2}^{p,q}$. 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
$H^{n+1}_{G}(X;\mathbb{Z})\cong H^{n}(G^{\bullet}\times X;\underline{U(1)})$
for $n\geq 1$ are related as follows

$\displaystyle\begin{array}[]{c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c}H^{n}(G^{\bullet}\times
X;\underline{U(1)})&\ =&{}^{\prime}F^{0}H^{n}&\
\supset&{}^{\prime}F^{1}H^{n}&\
\supset&\cdots&\supset&{}^{\prime}F^{p}H^{n}&\supset&\cdots\\\
\parallel&&&&\downarrow&&&&\downarrow&&\\\ H^{n+1}_{G}(X;\mathbb{Z})&\
=&F^{0}H^{n+1}&\ \supset&F^{1}H^{n+1}&\ \supset&\cdots&\supset&F^{p}H^{n+1}&\
\supset&\cdots.\end{array}$

The spectral sequences are related by a map ${}^{\prime}E^{p,q}_{r}\to
E^{p,q+1}_{r}$. In particular, the $E_{2}$-terms ${}^{\prime}E_{2}^{p,0}$ and
$E_{2}^{p,1}$ are related by a map $C(X,U(1))\to H^{1}(X;\mathbb{Z})$ fitting
into the exact sequence

$\displaystyle 0\to H^{0}(X;\mathbb{Z})\to C(X,\mathbb{R})\to C(X,U(1))\to
H^{1}(X;\mathbb{Z})\to 0.$

As is mentioned, because of the isomorphism $H^{q}(X;\underline{U(1)})\cong
H^{q+1}(X;\mathbb{Z})$, we have ${}^{\prime}E_{2}^{p,q}\cong E_{2}^{p,q+1}$
for $q\geq 1$. 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 $G\times X$.

######  Definition 3.1.

A central extension $(L,\tau)$ of the groupoid $X/\\!/G$ consists of the
following data:

  * •

a Hermitian line bundle $L\to G\times X$, which we write $L_{g}\to X$ for the
restriction to $\\{g\\}\times X$ for each $g\in G$,

  * •

unitary isomorphisms of Hermitian line bundles $\tau_{g,h}\colon
h^{*}L_{g}\otimes L_{h}\to L_{gh}$ on $X$ for each $g,h\in G$, which we write
$\tau_{g,h}(x)\colon L_{g}|_{hx}\otimes L_{h}|_{x}\to L_{gh}|_{x}$ for the
restriction to $x\in X$. We assume the following diagram is commutative

$\displaystyle\begin{CD}L_{g}|_{hkx}\otimes L_{h}|_{kx}\otimes
L_{k}|_{x}@>{1\otimes\tau_{h,k}(x)}>{}>L_{g}|_{hkx}\otimes L_{hk}|_{x}\\\
@V{\tau_{g,h}(kx)\otimes 1}V{}V@V{}V{\tau_{g,hk}(x)}V\\\ L_{gh}|_{kx}\otimes
L_{k}|_{x}@>{\tau_{gh,k}(x)}>{}>L_{ghk}|_{x}.\end{CD}$

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
$(K,\beta_{g})\colon(L_{g},\tau_{g,h})\to(L^{\prime}_{g},\tau^{\prime}_{g,h})$
of central extensions of $X/\\!/G$ consists of the following data:

  * •

a Hermitian line bundle $K\to X$,

  * •

unitary isomorphisms of Hermitian line bundles $\beta_{g}\colon L_{g}\otimes
K\to g^{*}K\otimes L^{\prime}_{g}$ on $X$ for each $g\in G$, which we write
$\beta_{g}(x)\colon L_{g}|_{x}\otimes K|_{x}\to K|_{gx}\otimes
L^{\prime}_{g}|_{x}$ for the restriction to $x\in X$. We assume the following
diagram is commutative

$\displaystyle\begin{CD}L_{g}|_{hx}\otimes L_{h}|_{x}\otimes
K|_{x}@>{1\otimes\beta_{h}(x)}>{}>L_{g}|_{hx}\otimes K|_{hx}\otimes
L^{\prime}_{h}|_{x}\\\ @V{\tau_{g,h}(x)\otimes
1}V{}V@V{}V{\beta_{g}(hx)\otimes 1}V\\\ L_{gh}|_{x}\otimes
K|_{x}K|_{ghx}\otimes L^{\prime}_{g}|_{hx}\otimes L^{\prime}_{h}|_{x}\\\
\Big{\|}@V{}V{1\otimes\tau^{\prime}_{g,h}(x)}V\\\ L_{gh}|_{x}\otimes
K|_{x}@>{\beta_{gh}(x)}>{}>K|_{ghx}\otimes L^{\prime}_{gh}|_{x}.\end{CD}$

The isomorphisms $(K,\beta_{g})$ and $(K^{\prime},\beta^{\prime}_{g})$ from
$(L_{g},\tau_{g,h})$ to $(L^{\prime}_{g},\tau^{\prime}_{g,h})$ are identified
if there is a unitary isomorphism $f\colon K\to K^{\prime}$ making the
following diagram commutative

$\displaystyle\begin{CD}L_{g}|_{x}\otimes
K|_{x}@>{\beta_{g}(x)}>{}>K|_{gx}\otimes L^{\prime}_{g}|x\\\ @V{1\otimes
f(x)}V{}V@V{}V{f(gx)\otimes 1}V\\\ L_{g}|_{x}\otimes
K^{\prime}|_{x}@>{\beta^{\prime}_{g}(x)}>{}>K^{\prime}|_{gx}\otimes
L^{\prime}_{g}|x.\end{CD}$

######  Definition 3.3.

An ungraded $G$-equivariant twist of $X$, or a twist for short, is a central
extension of a groupoid $\tilde{X}$ which has a local equivalence to
$X/\\!/G$.

A point in this definition is that a twist needs an extra groupoid
$\tilde{X}$. A central extension of $X/\\!/G$ is a special type of a twist
such that $\tilde{X}=X/\\!/G$. 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 $H^{3}_{G}(X;\mathbb{Z})$.

A key to the classification is the isomorphism $H^{3}_{G}(X;\mathbb{Z})\cong
H^{2}(G^{\bullet}\times X;\underline{U(1)})$. A close look at the proof of the
classification leads to:

######  Lemma 3.5.

The following holds true:

  * $(i)$

${}^{\prime}F^{1}H^{2}(G^{\bullet}\times X;\underline{U(1)})$ classifies
twists represented by central extensions of the groupoid $X/\\!/G$.

  * $(ii)$

${}^{\prime}F^{2}H^{2}(G^{\bullet}\times X;\underline{U(1)})$ 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 $K=X\times\mathbb{C}$. 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 $(L,\tau)$ with a
$\mathbb{Z}_{2}$-graded Hermitian line bundle. Since $L$ is of rank $1$, its
$\mathbb{Z}_{2}$-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 $H^{1}_{G}(X;\mathbb{Z}_{2})\times
H^{3}_{G}(X;\mathbb{Z})$.

###  3.3 Comparison of two spectral sequences

The relation between the spectral sequences $E_{r}^{p,q}$ and
${}^{\prime}E_{r}^{p,q}$ 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

$\displaystyle 0\to H^{0}(X;\mathbb{Z})\to C(X,\mathbb{R})\to C(X,U(1))\to
H^{1}(X;\mathbb{Z})\to 0.$

Let us fold this into a short exact sequence

$\displaystyle 0\to C(X,\mathbb{R})/H^{0}(X;\mathbb{Z})\to C(X,U(1))\to
H^{1}(X;\mathbb{Z})\to 0.$

In general, this does not split as an exact sequence of $G$-modules. (Such an
example is provided by the circle $S^{1}\subset\mathbb{R}^{2}$ with the action
of $D_{2}\subset{\rm O}(2)$.) Notice that if $X$ is path connected, then
$H^{0}(X;\mathbb{Z})=\mathbb{Z}$.

######  Lemma 3.8.

If a finite group $G$ acts on a compact and path connected space $X$ fixing a
point $\mathrm{pt}\in X$, then the following exact sequence of $G$-modules
splits

$\displaystyle 0\to C(X,\mathbb{R})/\mathbb{Z}\to C(X,U(1))\to
H^{1}(X;\mathbb{Z})\to 0.$

###### Proof.

For notational convenience, we use the identification
$U(1)\cong\mathbb{R}/\mathbb{Z}$ in this proof. Let
$C(X,\mathrm{pt},\mathbb{R})\subset C(X,\mathbb{R})$ be the subgroup
consisting of functions taking $0$ at $\mathrm{pt}$. The inclusion
$\iota\colon\mathrm{pt}\to X$ induces an isomorphism of $G$-modules

$\displaystyle C(X,\mathbb{R})\longrightarrow
C(X,\mathrm{pt},\mathbb{R})\oplus\mathbb{R},\qquad
f\mapsto(f-\iota^{*}f,\iota^{*}f).$

Similarly, we have an isomorphism $C(X,\mathbb{R}/\mathbb{Z})\cong
C(X,\mathrm{pt},\mathbb{R}/\mathbb{Z})\oplus\mathbb{R}/\mathbb{Z}$ of
$G$-modules. Thus the exact sequence of $G$-modules in question is equivalent
to

$\displaystyle 0\to C(X,\mathrm{pt},\mathbb{R})\to
C(X,\mathrm{pt},\mathbb{R}/\mathbb{Z})\overset{\delta}{\to}H^{1}(X;\mathbb{Z})\to
0.$

Since $X$ is supposed to be compact, $H^{1}(X;\mathbb{Z})$ is a free abelian
group of finite rank. Let us choose a basis
$H^{1}(X;\mathbb{Z})\cong\bigoplus_{i}\mathbb{Z}a_{i}$, and also
$\varphi_{i}\colon X\to\mathbb{R}/\mathbb{Z}$ such that
$\delta\varphi_{i}=a_{i}$ and $\varphi_{i}(\mathrm{pt})=0$. Modifying the
splitting $a_{i}\mapsto\varphi_{i}$ 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 $g\in G$ by $g^{*}a_{i}=\sum_{j}A_{ij}(g)a_{j}$.
It holds that $A(gh)=A(g)A(h)$. Because of the exact sequence, there are
functions $f^{i}_{g}\in C(X,\mathrm{pt},\mathbb{R})$ such that the following
holds in $C(X,\mathrm{pt},\mathbb{R}/\mathbb{Z})$:

$\displaystyle g^{*}\varphi_{i}=\sum_{j}A_{ij}(g)\varphi_{j}+\big{(}f_{g}^{i}\
\mathrm{mod}\ \mathbb{Z}\big{)}.$

This can be expressed as $g^{*}\Phi=A(g)\Phi+F_{g}$ by using the vectors
$\Phi=(\varphi_{i})$ and $F_{g}=(F^{i}_{g})$. It then holds that
$F_{gh}=A(g)F_{h}+h^{*}F_{g}$ in $C(X,\mathrm{pt},\mathbb{R})$. Since $A(g)$
is invertible, this is equivalent to

$\displaystyle
A(gh)^{-1}F_{gh}=A(h)^{-1}F_{h}+A(h)^{-1}h^{*}\big{(}A(g)^{-1}F_{g}\big{)}.$

Write $\lvert G\rvert$ for the order of $G$, and put
$\overline{F}=\frac{1}{\lvert G\rvert}\sum\limits_{g\in G}A(g)^{-1}F_{g}$.
Taking the average over $g\in G$ in the formula above, we get

$\displaystyle\overline{F}=A(h)^{-1}F_{h}+A(h)^{-1}h^{*}\overline{F},$

which is equivalent to $F_{g}=A(g)\overline{F}-g^{*}\overline{F}$. Now
$g^{*}(\Phi+\overline{F})=A(g)(\Phi+\overline{F})$. Thus, under the expression
$\overline{F}=(\overline{f}^{i})$ by using $\overline{f}^{i}\in
C(X,\mathrm{pt},\mathbb{R})$, the assignment
$a_{i}\mapsto\varphi_{i}+(\overline{f}^{i}\ \mathrm{mod}\ \mathbb{Z})$ defines
a splitting $H^{1}(X;\mathbb{Z})\to C(X,\mathrm{pt},\mathbb{R}/\mathbb{Z})$
compatible with the $G$-module structures. ∎

######  Lemma 3.9.

Let $G$ be a finite group acting on a compact and path connected space $X$
fixing a point $\mathrm{pt}\in X$. Then, for $n\geq 1$, there is an
isomorphism

$\displaystyle H^{n}_{\mathrm{group}}(G;C(X,U(1)))\cong
H^{n}_{\mathrm{group}}(G;U(1))\oplus
H^{n}_{\mathrm{group}}\big{(}G;H^{1}(X;\mathbb{Z})\big{)},$

where $U(1)$ is the trivial $G$-module, and $H^{1}(X;\mathbb{Z})$ is regarded
as a $G$-module through the action of $G$ on $X$.

###### Proof.

Lemma 3.8 implies

$\displaystyle H^{n}_{\mathrm{group}}(G;C(X,U(1)))\cong
H^{n}_{\mathrm{group}}(G;C(X,\mathbb{R})/\mathbb{Z})\oplus
H^{n}_{\mathrm{group}}\big{(}G;H^{1}(X;\mathbb{Z})\big{)}$

for all $n\geq 1$. By the $G$-module isomorphism
$C(X,\mathbb{R})/\mathbb{Z}\cong
C(X,\mathrm{pt},\mathbb{R})\oplus\mathbb{R}/\mathbb{Z}$ utilized in Lemma 3.8,
we have

$\displaystyle H^{n}_{\mathrm{group}}(G;C(X,\mathbb{R})/\mathbb{Z})\cong
H^{n}_{\mathrm{group}}(G;C(X,\mathrm{pt},\mathbb{R}))\oplus
H^{n}_{\mathrm{group}}(G;\mathbb{R}/\mathbb{Z}).$

Since $C(X,\mathrm{pt},\mathbb{R})$ is a vector space over $\mathbb{R}$, we
can prove the vanishing
$H^{n}_{\mathrm{group}}(G;C(X,\mathrm{pt},\mathbb{R}))=0$ for $n\geq 1$ 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 $\mathrm{pt}\in X$. Then for $r\geq 2$ we have

$\displaystyle{}^{\prime}E^{p,0}_{r}\cong E^{p,1}_{r}\oplus
E^{p+1,0}_{r},\quad p\geq 1,\qquad{}^{\prime}E^{p,q}_{r}\cong
E^{p,q+1}_{r},\quad p\geq 0,q\geq 1.$

###### Proof.

Recall that the exponential exact sequence induces the connecting homomorphism
$\delta\colon H^{q}(X;\underline{U(1)})\to H^{q+1}(X;\mathbb{Z})$ and this
induces a natural homomorphism $\delta\colon{}^{\prime}E^{p,q}_{r}\to
E^{p,q+1}_{r}$ compatible with the differentials ${}^{\prime}d_{r}$ and
$d_{r}$. In the case of $r=2$, the homomorphism
$\delta\colon{}^{\prime}E_{2}^{p,q}\to E^{p,q+1}_{2}$ is bijective for $q\geq
1$ and $p\geq 0$, and we have ${}^{\prime}E^{p,0}_{2}\cong E^{p,1}_{2}\oplus
E^{p+1,0}_{2}$ for $p\geq 1$ as a consequence of Lemma 3.9. Notice that, under
this isomorphism, $\delta\colon{}^{\prime}E^{p,0}_{2}\to E^{p,1}_{2}$ for
$p\geq 1$ restricts to the identity on the direct summand
$E^{p,1}_{2}\subset{}^{\prime}E^{p,0}_{2}$. Note also that
$E^{p,0}_{2}=E^{p,0}_{\infty}$ for any $p$, because

$\displaystyle
E^{p,0}_{2}=H^{p}_{\mathrm{group}}(G;\mathbb{Z})=H^{p}(BG;\mathbb{Z})=H^{p}_{G}(\mathrm{pt};\mathbb{Z})$

is a direct summand of $H^{p}_{G}(X;\mathbb{Z})\cong
H^{0}_{G}(\mathrm{pt};\mathbb{Z})\oplus\tilde{H}^{p}_{G}(X;\mathbb{Z})$, where
$\tilde{H}^{p}_{G}(X;\mathbb{Z})$ is the reduced cohomology. Thus, for $p\geq
1$, the map $\delta\colon{}^{\prime}E^{p,0}_{2}\to E^{p,1}_{2}$ is the
projection onto $E^{p,1}_{2}$ and the image of the differential
${}^{\prime}d_{2}\colon{}^{\prime}E^{p-2,1}_{2}\to{}^{\prime}E_{2}^{p,0}$ is
in the direct summand $E^{p,0}_{2}$. This leads to

$\displaystyle{}^{\prime}E^{p,0}_{3}\cong E^{p,0}_{3}\oplus E^{p,1}_{3},\quad
p\geq 1,\qquad{}^{\prime}E^{p,q}_{3}\cong E^{p,q+1}_{3},\quad p\geq 0,q\geq
1.$

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 $n\geq 1$ and
$p=0,\ldots,n$, there is a natural isomorphism

$\displaystyle{}^{\prime}F^{p}H^{n}(G^{\bullet}\times X;\underline{U(1)})\cong
F^{p}H^{n+1}_{G}(X;\mathbb{Z}).$

In addition, we have a decomposition

$\displaystyle{}^{\prime}F^{n}H^{n}(G^{\bullet}\times
X\colon\underline{U(1)})\cong F^{n}H^{n+1}_{G}(X;\mathbb{Z})\cong
E^{n,1}_{\infty}\oplus F^{n+1}H^{n+1}_{G}(X;\mathbb{Z}),$

in which $F^{n+1}H^{n+1}_{G}(X;\mathbb{Z})\cong
H^{n+1}_{G}(\mathrm{pt};\mathbb{Z})\cong H^{n}_{\mathrm{group}}(G;U(1))$.

###### Proof.

Put ${}^{\prime}F^{p}H^{n}={}^{\prime}F^{p}H^{n}(G^{\bullet}\times
X;\underline{U(1)})$ and $F^{p}H^{n+1}=F^{p}H^{n+1}_{G}(X;\mathbb{Z})$ for
short. The exponential exact sequence induces a homomorphism of short exact
sequences

$\displaystyle\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>{}^{\prime}F^{p+1}H^{n}@>{}>{}>{}^{\prime}F^{p}H^{n}@>{}>{}>{}^{\prime}E^{p,n-p}_{\infty}@>{}>{}>0\\\
@V{\delta}V{}V@V{\delta}V{}V@V{\delta}V{}V\\\
0@>{}>{}>F^{p+1}H^{n+1}@>{}>{}>F^{p}H^{n+1}<EMAIL_ADDRESS>
In the case of $p=n$, the diagram above becomes

$\displaystyle\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>0@>{}>{}>{}^{\prime}F^{n}H^{n}@>{\cong}>{}>{}^{\prime}E^{n,0}_{\infty}@>{}>{}>0\\\
@V{\delta}V{}V@V{\delta}V{}V@V{\delta}V{}V\\\
0@>{}>{}>F^{n+1}H^{n+1}@>{}>{}>F^{n}H^{n+1}<EMAIL_ADDRESS>
Notice that $F^{n+1}H^{n+1}\cong E^{n+1,0}_{\infty}\cong E^{n+1,0}_{2}$ since
$E^{n+1,0}_{2}\cong H^{n+1}_{G}(\mathrm{pt};\mathbb{Z})$ must survive into
$H^{n+1}_{G}(X;\mathbb{Z})\cong
H^{n+1}_{G}(\mathrm{pt};\mathbb{Z})\oplus\tilde{H}^{n+1}_{G}(X;\mathbb{Z})$.
Hence $F^{n}H^{n+1}\cong E^{n+1,0}_{\infty}\oplus E^{n,1}_{\infty}$. If $n\geq
1$, then this isomorphism is compatible with the isomorphism
${}^{\prime}E^{n,0}_{\infty}\cong E^{n+1,0}_{\infty}\oplus E^{n,1}_{\infty}$
in Proposition 3.10 through $\delta$, so that

$\displaystyle{}^{\prime}F^{n}H^{n}\overset{\delta}{\cong}F^{n}H^{n+1}\cong
F^{n+1}H^{n+1}\oplus E^{n,1}_{\infty}.$

For $p=n-1,n-2,\ldots,1,0$, we know
$\delta\colon{}^{\prime}E^{p,n-p}_{\infty}\to E^{p,n-p+1}_{\infty}$ is
bijective by Proposition 3.10. Therefore ${}^{\prime}F^{p}H^{n}\cong
F^{p}H^{n+1}$ inductively. ∎

Combining the above corollary with Lemma 3.5, we get the interpretations of
$F^{p}H^{3}_{G}(X;\mathbb{Z})$ by twists presented in Introduction:

######  Corollary 3.12.

Let $G$ and $X$ be as in Proposition 3.10.

  * $(i)$

$F^{1}H^{3}_{G}(X;\mathbb{Z})$ classifies twists which can be represented by
central extensions of the groupoid $X/\\!/G$.

  * $(ii)$

$F^{2}H^{3}_{G}(X;\mathbb{Z})$ classifies twists which can be represented by
$2$-cocycles of $G$ with coefficients in the $G$-module $C(X,U(1))$.

  * $(iii)$

$F^{3}H^{3}_{G}(X;\mathbb{Z})=H^{2}_{\mathrm{group}}(G;U(1))$ 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 ${}^{\prime}F^{1}H^{n}(G^{\bullet}\times
X;\underline{U(1)})=F^{1}H^{n+1}(X;\mathbb{Z})$ in Corollary 3.11 holds true
for $n\geq 0$ without the assumption that $G$ fixes a point on $X$. This is
because ${}^{\prime}E^{0,n}$ and $E^{0,n+1}$ are subgroups of
$H^{n}(X;\underline{U(1)})\cong H^{n+1}(X;\mathbb{Z})$ and it holds that

$\displaystyle{}^{\prime}F^{1}H^{n}(G^{\bullet}\times X;\underline{U(1)})$
$\displaystyle=F^{1}H^{n+1}(X;\mathbb{Z})$
$\displaystyle=\operatorname{Ker}\big{[}f\colon H^{n}(G^{\bullet}\times
X;\underline{U(1)})\cong H^{n+1}_{G}(X;\mathbb{Z})\to
H^{n+1}(X;\mathbb{Z})\big{]},$

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.

###  4.1 Some generality

The cohomology $H^{n}(T^{2};\mathbb{Z})$ 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
$H^{3}(T^{2};\mathbb{Z})=0$ implies $E^{0,3}_{\infty}=0$, so that

$\displaystyle
H^{3}_{P}\big{(}T^{2};\mathbb{Z}\big{)}=F^{0}H^{3}_{P}\big{(}T^{2};\mathbb{Z}\big{)}=F^{1}H^{3}_{P}\big{(}T^{2};\mathbb{Z}\big{)}.$

Note that each point group $P$ fixes a point on $T^{2}$, so that

$\displaystyle
F^{3}H^{3}_{P}\big{(}T^{2};\mathbb{Z}\big{)}=H^{3}_{P}(\mathrm{pt};\mathbb{Z})=H^{3}_{\mathrm{group}}(P;\mathbb{Z})=H^{2}_{\mathrm{group}}(P;U(1)).$

Then the main task for the proof of Theorem 1.1 is to compute
$H^{3}_{P}(T^{2};\mathbb{Z})$ and $F^{2}H^{3}_{P}(T^{2};\mathbb{Z})$, since in
the case where $P$ is the cyclic group $\mathbb{Z}_{n}$ or the dihedral group
$D_{n}$, the cohomology $H^{m}_{P}(\mathrm{pt};\mathbb{Z})$ is summarized as
follows:

$\begin{array}[]{|c|c|c|c|c|}\hline\cr
P&H^{0}_{P}(\mathrm{pt};\mathbb{Z})&H^{1}_{P}(\mathrm{pt};\mathbb{Z})&H^{2}_{P}(\mathrm{pt};\mathbb{Z})&H^{3}_{P}(\mathrm{pt};\mathbb{Z})\tsep{2pt}\bsep{2pt}\\\
\hline\cr\mathbb{Z}_{n}&\mathbb{Z}&0&\mathbb{Z}_{n}&0\\\ \hline\cr
D_{n}&\mathbb{Z}&0&\begin{cases}\mathbb{Z}_{2}&(\mbox{$n$: odd})\\\
\mathbb{Z}_{2}^{\oplus 2}&(\mbox{$n$:
even})\end{cases}&\begin{cases}0&(\mbox{$n$: odd})\\\
\mathbb{Z}_{2}&(\mbox{$n$: even})\end{cases}\tsep{10pt}\bsep{10pt}\\\
\hline\cr\end{array}$

The degree $0$ part
$H^{0}_{P}(\mathrm{pt};\mathbb{Z})=H^{0}(BP;\mathbb{Z})=\mathbb{Z}$ is clear.
Since $P$ is finite, the degree $1$ part
$H^{1}_{P}(\mathrm{pt};\mathbb{Z})\cong\operatorname{Hom}(P,\mathbb{Z})$ is
trivial. The degree $2$ part
$H^{2}_{P}(\mathrm{pt};\mathbb{Z})\cong\operatorname{Hom}(P,U(1))$ can be seen
by the classification of irreducible representations. Finally, the degree $3$
part $H^{3}_{P}(\mathrm{pt};\mathbb{Z})\cong H^{2}_{\mathrm{group}}(P;U(1))$
for $P=\mathbb{Z}_{n},D_{n}$ 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 $G/H\times e^{d}$, where $H\subset G$ 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:

######  Lemma 4.1.

Suppose that a finite group $G$ acts on a path connected space $Y$ fixing at
least one point $\mathrm{pt}\in Y$. 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 $n\geq 0$.

$\displaystyle
H^{n}_{\mathrm{group}}\big{(}G;H^{1}(Y;\mathbb{Z})\big{)}\cong\tilde{H}^{n+1}_{G}(Y;\mathbb{Z}),$

where $\tilde{H}^{n+1}_{G}(Y;\mathbb{Z})$ stands for the reduced cohomology.

Notice that a $G$-CW complex is naturally a CW complex.

###### Proof.

Consider the Leray–Serre spectral sequence

$\displaystyle
E_{2}^{p,q}=H^{p}_{\mathrm{group}}(G;H^{q}(Y;\mathbb{Z}))\Longrightarrow
H^{*}_{G}(Y;\mathbb{Z}).$

Note that $H^{q}(Y;\mathbb{Z})=0$ for $q\neq 0,1$. The $E_{2}$-term
$E_{2}^{p,0}=H^{p}_{\mathrm{group}}(G;\mathbb{Z})$ must survive into the
direct summand $H^{p}_{G}(\mathrm{pt};\mathbb{Z})$ in
$H^{p}_{G}(Y;\mathbb{Z})\cong
H^{p}_{G}(\mathrm{pt};\mathbb{Z})\oplus\tilde{H}^{p}_{G}(Y;\mathbb{Z})$.
Therefore it must hold that

$\displaystyle\tilde{H}^{p}_{G}(Y;\mathbb{Z})\cong
E^{p-1,1}_{\infty}=E^{p-1,1}_{2}\cong
H^{p-1}_{\mathrm{group}}\big{(}G;H^{1}(Y;\mathbb{Z})\big{)},$

which completes the proof. ∎

We also prepare a simple lemma about group cohomology: Let $G$ be a finite
group, $c\colon G\to\mathbb{Z}_{2}=\\{\pm 1\\}$ a surjective homomorphism, and
$\tilde{\mathbb{Z}}=\mathbb{Z}_{c}$ the $G$-module such that its underlying
group is $\mathbb{Z}$ and $G$ acts (from the right) by $m\mapsto mc(g)$. A
typical example is a finite subgroup $P\subset{\rm O}(2)$ such that
$P\not\subset{\rm SO}(2)$ with $c$ the composition of the inclusion $P\to{\rm
O}(2)$ and the determinant ${\rm O}(2)\to\mathbb{Z}_{2}$.

######  Lemma 4.2.

Let $G$, $c$ and $\tilde{\mathbb{Z}}$ be as above. Then,
$H^{0}_{\mathrm{group}}(G;\tilde{\mathbb{Z}})=0$ and
$H^{1}_{\mathrm{group}}(G;\tilde{\mathbb{Z}})\cong\mathbb{Z}_{2}$.

###### Proof.

For any $n\in C^{0}_{\mathrm{group}}(G;\tilde{\mathbb{Z}})=\mathbb{Z}$, its
coboundary $\partial n\colon G\to\mathbb{Z}$ is $(\partial n)(g)=n(1-c(g))$.
Thus, the assumption that $c$ is surjective implies the vanishing of the $0$th
cohomology. The inclusion $\operatorname{Ker}(c)\subset G$ induces an
injection on $1$-cocycles

$\displaystyle Z^{1}_{\mathrm{group}}\big{(}G;\tilde{\mathbb{Z}}\big{)}\to
Z^{1}_{\mathrm{group}}\big{(}\operatorname{Ker}(c);\tilde{\mathbb{Z}}\big{)}=\operatorname{Hom}(\operatorname{Ker}(c),\mathbb{Z})=0.$

Thus, given a group $1$-cocycle $\phi\in
Z^{1}_{\mathrm{group}}(G;\tilde{\mathbb{Z}})$, it holds that $\phi(g)=0$ for
all $g\in\operatorname{Ker}(c)$. If $g,h\not\in\operatorname{Ker}(c)$, then
the cocycle condition $(\partial\phi)(g,h)=0$ implies $\phi(g)=\phi(h)$.
Therefore $\phi\colon G\to\mathbb{Z}$ is always of the form
$\phi(g)=n(1-c(g))/2$ for some $n\in\mathbb{Z}$. This provides the
identification $Z^{1}_{\mathrm{group}}(G;\tilde{\mathbb{Z}})\cong\mathbb{Z}$
as well as $B^{1}_{\mathrm{group}}(G;\tilde{\mathbb{Z}})\cong 2\mathbb{Z}$,
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 $\mathrm{pt}\in T^{2}$,

  * •

the $G$-action does not preserve the orientation of $T^{2}$.

Then the following holds true about the Leray–Serre spectral sequence:

  * $(a)$

$F^{2}H^{3}_{G}(T^{2};\mathbb{Z})\cong E_{2}^{2,1}\oplus E_{2}^{3,0}$,

  * $(b)$

$H^{n}_{G}(T^{2};\mathbb{Z})\cong\bigoplus_{p+q=n}E_{2}^{p,q}$ for $n\leq 2$.

###### Proof.

In the Leray–Serre spectral sequence
$E_{2}^{p,q}=H^{p}_{\mathrm{group}}(G;H^{q}(T^{2};\mathbb{Z}))$, the
coefficient in the group cohomology $H^{0}(T^{2})\cong\mathbb{Z}$ is
identified with the trivial $G$-module, and $H^{2}(T^{2})\cong\mathbb{Z}$ with
the $G$-module in Lemma 4.2. Then the relevant $E_{2}$-terms can be summarized
as follows:

$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr q=3&0&0&0&0&0\\\ \hline\cr
q=2&0&\mathbb{Z}_{2}&&&\\\ \hline\cr
q=1&E_{2}^{0,1}&E_{2}^{1,1}&E_{2}^{2,1}&&\tsep{2pt}\bsep{2pt}\\\ \hline\cr
q=0&E_{2}^{0,0}&E_{2}^{1,0}&E_{2}^{2,0}&E_{2}^{3,0}&E_{2}^{4,0}\tsep{2pt}\bsep{2pt}\\\
\hline\cr E_{2}^{p,q}\tsep{2pt}\bsep{2pt}&p=0&p=1&p=2&p=3&p=4\\\
\hline\cr\end{array}$

Since $G$ fixes $\mathrm{pt}\in T^{2}$, we have the decomposition
$H^{n}_{G}(T^{2};\mathbb{Z})\cong
H^{n}_{G}(\mathrm{pt};\mathbb{Z})\oplus\tilde{H}^{n}_{G}(T^{2};\mathbb{Z})$,
where $\tilde{H}^{n}_{G}(T^{2};\mathbb{Z})$ is the reduced cohomology.
Therefore the $E_{2}$-term
$E_{2}^{n,0}=H^{n}_{\mathrm{group}}(G;\mathbb{Z})\cong
H^{n}_{G}(\mathrm{pt};\mathbb{Z})$ must survive into the direct summand
$H^{n}_{G}(\mathrm{pt};\mathbb{Z})$ in $H^{n}_{G}(T^{2};\mathbb{Z})$. This
implies that $E_{2}^{n,0}=E_{\infty}^{n,0}$ is always a direct summand of the
subgroups $F^{p}H^{n}_{G}(T^{2};\mathbb{Z})\subset
H^{n}_{G}(T^{2};\mathbb{Z})$ and that $d_{2}\colon E_{2}^{p-2,1}\to
E_{2}^{p,0}$ is trivial. As a result, we get $E_{2}^{2,1}=E_{\infty}^{2,1}$
and the isomorphism (a). Also $E_{2}^{p,q}=E_{\infty}^{p,q}$ for $p+q\leq 2$,
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]).

######  Lemma 4.4.

Suppose a finite group $G$ acts on the torus $T^{2}=S^{1}\times S^{1}$ and

  * •

there is a fixed point $\mathrm{pt}=(x_{0},y_{0})\in T^{2}$ under the
$G$-action,

  * •

$G$ preserves the subspace $S^{1}\vee
S^{1}=S^{1}\times\\{y_{0}\\}\cup\\{x_{0}\\}\times S^{1}\subset T^{2}$.

Then there is a $G$-equivariant homotopy equivalence

$\displaystyle\Sigma T^{2}\simeq\Sigma\big{(}S^{1}\vee
S^{1}\big{)}\vee\Sigma\big{(}T^{2}/S^{1}\vee S^{1}\big{)},$

where $\Sigma$ 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.

######  Lemma 4.5.

Under the assumption in Lemma 4.4, we have the following isomorphism of groups
for all $n\in\mathbb{Z}$

$\displaystyle H^{n}_{G}\big{(}T^{2};\mathbb{Z}\big{)}\cong
H^{n}_{G}(\mathrm{pt};\mathbb{Z})\oplus\tilde{H}^{n}_{G}\big{(}S^{1}\vee
S^{1};\mathbb{Z}\big{)}\oplus\tilde{H}^{n}_{G}\big{(}T^{2}/S^{1}\vee
S^{1};\mathbb{Z}\big{)}.$

Further, the Leray–Serre spectral sequence for $H^{n}_{G}(T^{2};\mathbb{Z})$
degenerates at $E_{2}$ and the relevant extension problems are trivial, so
that

  * $(a)$

$F^{2}H^{3}_{G}(T^{2};\mathbb{Z})\cong E_{2}^{2,1}\oplus E_{2}^{3,0}$,

  * $(b)$

$H^{n}_{G}(T^{2};\mathbb{Z})\cong\bigoplus_{p+q=n}E_{2}^{p,q}$ for all
$n\in\mathbb{Z}$.

###### Proof.

The stable splitting in Lemma 4.4 immediately gives the first isomorphism. For
the trivial $G$-space $\mathrm{pt}$, the Leray–Serre spectral sequence clearly
degenerates at $E_{2}$, and we have

$\displaystyle H^{n}_{G}(\mathrm{pt};\mathbb{Z})\cong
H^{n}_{\mathrm{group}}\big{(}G;H^{0}(\mathrm{pt};\mathbb{Z})\big{)}.$

Since $H^{0}(\mathrm{pt};\mathbb{Z})\cong H^{0}(T^{2};\mathbb{Z})$ as
$G$-modules, we get the following identification of the $E_{2}$-term
$E_{2}^{n,0}$ of the Leray–Serre spectral sequence for
$H^{n}_{G}(T^{2};\mathbb{Z})$

$\displaystyle
E_{2}^{n,0}=H^{n}_{\mathrm{group}}\big{(}G;H^{0}\big{(}T^{2};\mathbb{Z}\big{)}\big{)}\cong
H^{n}_{G}(\mathrm{pt};\mathbb{Z}).$

For the $G$-space $S^{1}\vee S^{1}$, 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 $H^{1}(S^{1}\vee
S^{1};\mathbb{Z})\cong H^{1}(T^{2};\mathbb{Z})$ as $G$-modules, the
$E_{2}$-term $E_{2}^{n-1,1}$ of the Leray–Serre spectral sequence for
$H^{n}_{G}(T^{2};\mathbb{Z})$ is

$\displaystyle
E_{2}^{n-1,1}=H^{n-1}_{\mathrm{group}}\big{(}G;H^{1}\big{(}T^{2};\mathbb{Z}\big{)}\big{)}\cong\tilde{H}^{n}_{G}\big{(}S^{1}\vee
S^{1};\mathbb{Z}\big{)}.$

Exactly in the same way, we have

$\displaystyle
E_{2}^{n-2,2}=H^{n-2}_{\mathrm{group}}\big{(}G;H^{2}\big{(}T^{2};\mathbb{Z}\big{)}\big{)}\cong\tilde{H}^{n}_{G}\big{(}T^{2}/S^{1}\vee
S^{1};\mathbb{Z}\big{)},$

since $H^{2}(T^{2}/S^{1}\vee S^{1};\mathbb{Z})\cong H^{2}(T^{2};\mathbb{Z})$
as $G$-modules. The first isomorphism now gives $H^{n}_{G}(T^{2};\mathbb{Z})$
$\cong E_{2}^{n,0}\oplus E_{2}^{n-1,1}\oplus E_{2}^{n-2,2}$, which also
implies the triviality of the spectral sequence. ∎

###  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. 1.

In the cases of p2 and pm/pg, the point group $\mathbb{Z}_{2}=D_{1}$ acts on
the torus $T^{2}=S^{1}\times S^{1}$ preserving the direct product structure,
so that we can think of $T^{2}$ as a stack of certain equivariant circle
bundles ($\mathbb{Z}_{2}$-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. 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 $S^{1}\vee S^{1}$, which we compute by
using the Mayer–Vietoris exact sequence.

  3. 3.

In the cases of p3, p6, p3m1, p31m and p6m, the computation can be divided
into two parts. One part is to compute $H^{3}_{P}(T^{2};\mathbb{Z})$. 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 $H^{1}(T^{2};\mathbb{Z})$. For
this aim, we take an invariant subspace $Y\subset T^{2}$ 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
$H^{1}(Y;\mathbb{Z})$ through Lemma 4.1. The coefficients
$H^{1}(Y;\mathbb{Z})$ and $H^{1}(T^{2};\mathbb{Z})$ are related by a short
exact sequence. The associated long exact sequence then computes the group
cohomology with coefficients in $H^{1}(T^{2};\mathbb{Z})$. 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 $\Pi\subset\mathbb{R}^{2}$ is the standard one
$\Pi=\mathbb{Z}\oplus\mathbb{Z}$ and the point group $P=\mathbb{Z}_{2}=\\{\pm
1\\}$ acts on $\Pi$ and $\mathbb{R}^{2}$ by $(x,y)\mapsto(-x,-y)$.

######  Theorem 4.6 (p2).

The $\mathbb{Z}_{2}$-equivariant cohomology of $T^{2}$ is given as follows

$\begin{array}[]{|c|c|c|c|c|}\hline\cr&n=0&n=1&n=2&n=3\\\ \hline\cr
H^{n}_{\mathbb{Z}_{2}}(T^{2};\mathbb{Z})&\mathbb{Z}&0&\mathbb{Z}\oplus\mathbb{Z}_{2}^{\oplus
3}&0\tsep{2pt}\bsep{2pt}\\\ \hline\cr\end{array}$

###### Proof.

We use the Gysin exact sequence for ‘Real’ circle bundles in [10]: We write
$H^{n}_{\mathbb{Z}_{2}}(X)=H^{n}_{\mathbb{Z}_{2}}(X;\mathbb{Z})$ for the
equivariant cohomology and $H^{n}_{\pm}(X)\cong
H^{n}_{\mathbb{Z}}(X;\mathbb{Z}(1))$ 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
$\tilde{S}^{1}$, where $\tilde{S}^{1}=U(1)$ is the circle with the involution
$z\mapsto z^{-1}$. We can think of $\tilde{S}^{1}\times\tilde{S}^{1}$ as the
trivial ‘Real’ circle bundle on $\tilde{S}^{1}$. Similarly, $\tilde{S}^{1}$ is
the trivial ‘Real’ circle bundle on $\mathrm{pt}$. The Gysin exact sequences
for these ‘Real’ circle bundles are split, and we find

$\displaystyle H^{n}_{\mathbb{Z}_{2}}\big{(}T^{2}\big{)}\cong
H^{n}_{\mathbb{Z}_{2}}\big{(}\tilde{S}^{1}\big{)}\oplus
H^{n-1}_{\pm}\big{(}\tilde{S}^{1}\big{)}\cong
H^{n}_{\mathbb{Z}_{2}}(\mathrm{pt})\oplus H^{n-1}_{\pm}(\mathrm{pt})\oplus
H^{n-1}_{\pm}(\mathrm{pt})\oplus H^{n-2}_{\mathbb{Z}_{2}}(\mathrm{pt}).$

As given in [10], the cohomology $H^{n}_{\pm}(\mathrm{pt})$ is isomorphic to
$\mathbb{Z}_{2}$ if $n>0$ is odd, and is trivial otherwise. We already know
$H^{n}_{\mathbb{Z}_{2}}(\mathrm{pt})$, and get $H^{n}_{\mathbb{Z}_{2}}(T^{2})$
easily. ∎

###  4.4 p4m/p4g

The lattice $\Pi=\mathbb{Z}^{2}\subset\mathbb{R}^{2}$ is standard. The point
group is $P=D_{4}=\langle
C_{4},\sigma_{x}\,|\,C_{4}^{4},\sigma_{x}^{2},\sigma_{x}C_{4}\sigma_{x}C_{4}\rangle$.
The $D_{4}$-action on $\Pi$ and $\mathbb{R}^{2}$ is given by the following
matrix presentation:

$\displaystyle C_{4}=\left(\begin{matrix}0&-1\\\
1&0\end{matrix}\right),\qquad\sigma_{x}=\left(\begin{matrix}-1&0\\\
0&1\end{matrix}\right).$

In the rest of this subsection, we will use the following notations to
indicate elements in $D_{4}$:

$\displaystyle 1,\qquad$ $\displaystyle C_{4},\qquad$ $\displaystyle
C_{2}=C_{4}^{2},\qquad$ $\displaystyle C_{4}^{-1}=C_{4}^{3},$
$\displaystyle\sigma_{x},\qquad$
$\displaystyle\sigma_{d}=\sigma_{x}C_{4},\qquad$
$\displaystyle\sigma_{y}=C_{2}\sigma_{x},\qquad$
$\displaystyle\sigma_{d}^{\prime}=C_{4}\sigma_{x}.$

The closure of a fundamental domain is
$\\{s(1,0)+t(0,1)\in\mathbb{R}^{2}\,|\,0\leq s,t\leq 1\\}$. Then we find that
the $D_{4}$-action on $T^{2}=\mathbb{R}^{2}/\Pi$ satisfies the assumptions in
Lemma 4.4, in which $\mathrm{pt}=(0,0)$ and

$\displaystyle S^{1}\vee S^{1}\cong Y=((\mathbb{R}\oplus 0)/(\mathbb{Z}\oplus
0))\vee((0\oplus\mathbb{R})/(0\oplus\mathbb{Z})).$

To apply Lemma 4.5, we compute the cohomology of $Y$:

######  Lemma 4.7.

The equivariant cohomology of $Y$ is as follows:

$\begin{array}[]{|c|c|c|c|c|}\hline\cr&n=0&n=1&n=2&n=3\\\ \hline\cr
H^{n}_{D_{4}}(Y;\mathbb{Z})&\mathbb{Z}&0&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}^{\oplus 2}\tsep{2pt}\bsep{2pt}\\\ \hline\cr\end{array}$

###### Proof.

We use the Mayer–Vietoris exact sequence: Cover $Y$ by invariant subspaces $U$
and $V$ with the following $D_{4}$-equivariant homotopy equivalences

$\displaystyle U\simeq\mathrm{pt},\qquad V\simeq D_{4}/D_{2}^{(v)},\qquad
U\cap V\simeq D_{4}/\mathbb{Z}_{2}^{(v)},$

where $D_{2}^{(v)}=\\{1,C_{2},\sigma_{x},\sigma_{y}\\}\cong D_{2}$ and
$\mathbb{Z}_{2}^{(v)}=\\{1,\sigma_{y}\\}\cong\mathbb{Z}_{2}$. We can summarize
the equivariant cohomology of these spaces in low degrees as follows:

$\begin{array}[]{|c|c|c|c|}\hline\cr
n=3&&\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}&0\\\ \hline\cr
n=2&&\mathbb{Z}_{2}^{\oplus 2}\oplus\mathbb{Z}_{2}^{\oplus
2}\tsep{2pt}\bsep{2pt}&\mathbb{Z}_{2}\\\ \hline\cr n=1&&0&0\\\ \hline\cr
n=0&&\mathbb{Z}\oplus\mathbb{Z}&\mathbb{Z}\\\
\hline\cr&H^{n}_{D_{4}}(Y)&H^{n}_{D_{4}}(U)\oplus
H^{n}_{D_{4}}(V)&H^{n}_{D_{4}}(U\cap V)\tsep{2pt}\bsep{2pt}\\\
\hline\cr\end{array}$

In the Mayer–Vietoris exact sequence

$\displaystyle\cdots\to H^{n}_{D_{4}}(Y)\to H^{n}_{D_{4}}(U)\oplus
H^{n}_{D_{4}}(V)\overset{\Delta}{\to}H^{n}_{D_{4}}(U\cap V)\to
H^{n+1}_{D_{4}}(Y)\to\cdots,$

the map $\Delta\colon H^{n}_{D_{4}}(U)\oplus H^{n}_{D_{4}}(V)\to
H^{n}_{D_{4}}(U\cap V)$ is expressed as
$\Delta(u,v)=j_{U}^{*}(u)-j_{V}^{*}(v)$, where $j_{U}\colon U\cap V\to U$ and
$j_{V}\colon U\cap V\to V$ are the inclusions. Under the natural
identifications

$\displaystyle\begin{split}&H^{2}_{D_{4}}(V)\cong
H^{2}_{D_{2}^{(v)}}(\mathrm{pt})\cong\operatorname{Hom}\big{(}D_{2}^{(v)},U(1)\big{)},\\\
&H^{2}_{D_{4}}(U\cap V)\cong
H^{2}_{\mathbb{Z}_{2}^{(v)}}(\mathrm{pt})\cong\operatorname{Hom}\big{(}\mathbb{Z}_{2}^{(v)},U(1)\big{)},\end{split}$

the map $j_{U}^{*}$ agrees with the homomorphism induced from the inclusion
$\mathbb{Z}_{2}^{(v)}\to D_{2}^{(v)}$. This implies that $j_{U}^{*}$ is
surjective, and so is $\Delta$ in degree $2$. Clearly, $\Delta\colon
H^{0}_{D_{4}}(U)\oplus H^{0}_{D_{4}}(V)\to H^{0}_{D_{4}}(U\cap V)$ is
identified with the homomorphism $\mathbb{Z}\oplus\mathbb{Z}\to\mathbb{Z}$
given by $(m,n)\mapsto m-n$. 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 $D_{4}$-equivariant cohomology of $T^{2}$ in low degrees is as follows:

$\begin{array}[]{|c|c|c|c|c|}\hline\cr&n=0&n=1&n=2&n=3\\\ \hline\cr
H^{n}_{D_{4}}(T^{2};\mathbb{Z})&\mathbb{Z}&0&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}^{\oplus 3}\tsep{2pt}\bsep{2pt}\\\ \hline\cr\end{array}$

We also have $F^{2}H^{3}_{D_{4}}(T^{2};\mathbb{Z})\cong\mathbb{Z}_{2}^{\oplus
2}$.

###### Proof.

In the Leray–Serre spectral sequence
$E_{2}^{p,q}=H^{p}_{\mathrm{group}}(D_{4};H^{q}(T^{2};\mathbb{Z}))$, 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 $\mathbb{Z}$, $H^{1}(Y)$ and the
$D_{4}$-module $\tilde{\mathbb{Z}}$ in Lemma 4.2, respectively. Using Lemmas
4.1 and 4.2, we can summarize the $E_{2}$-terms as follows:

$\begin{array}[]{|c|c|c|c|c|}\hline\cr q=3&0&0&0&0\\\ \hline\cr
q=2&0&\mathbb{Z}_{2}&&\\\ \hline\cr q=1&0&\mathbb{Z}_{2}&\mathbb{Z}_{2}&\\\
\hline\cr q=0&\mathbb{Z}&0&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\ \hline\cr
E_{2}^{p,q}&p=0&p=1&p=2&p=3\tsep{2pt}\bsep{2pt}\\\ \hline\cr\end{array}$

Now the proof is completed by Lemma 4.5. ∎

###  4.5 p6m

We let $\Pi=\mathbb{Z}a\oplus\mathbb{Z}b\subset\mathbb{R}^{2}$ be the lattice
spanned by $a=\left(\begin{matrix}1\\\ 0\end{matrix}\right)$ and
$b=\left(\begin{matrix}1/2\\\ \sqrt{3}/2\end{matrix}\right)$. The point group
$P$ is $D_{6}=\langle
C,\sigma_{1}\,|\,C^{6},\sigma_{1}^{2},\sigma_{1}C\sigma_{1}C\rangle=\\{1,C,C^{2},C^{3},C^{4},C^{5},\sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4},\sigma_{5},\sigma_{6}\\}$,
where $\sigma_{\ell}=C^{\ell-1}\sigma_{1}$. This group acts on $\Pi$ and
$\mathbb{R}^{2}$ through the inclusion $D_{6}\subset{\rm O}(2)$ defined by

$\displaystyle C=\left(\begin{matrix}1/2&-\sqrt{3}/2\\\
\sqrt{3}/2&1/2\end{matrix}\right),\qquad\sigma_{1}=\left(\begin{matrix}1&0\\\
0&-1\end{matrix}\right).$

If we use the identifications $a=1$ and $b=\tau=\exp 2\pi i/6$ under
$\mathbb{R}^{2}=\mathbb{C}$, then the actions of $C\in D_{6}$ and $\sigma_{1}$
are given by the multiplication by $\tau$ and the complex conjugation,
respectively. The closure of a fundamental domain is $\\{sa+tb\,|\,0\leq
s,t\leq 1\\}$ or equivalently $\\{s+t\tau\,|\,0\leq s,t\leq 1\\}$. We
decompose this region to define a $D_{6}$-CW decomposition of $T^{2}$ as
follows:

$\begin{array}[]{|c|c|c|}\hline\cr\mbox{0-cell}&\mbox{1-cell}&\mbox{2-cell}\\\
\hline\cr\tilde{e}^{0}_{0}=\mathrm{pt}&\tilde{e}^{1}_{01}=(D_{6}/\\{1,\sigma_{1}\\})\times
e^{1}&\tilde{e}^{2}=D_{6}\times e^{2}\tsep{2pt}\\\
\tilde{e}^{0}_{1}=D^{6}/\\{1,C^{3},\sigma_{1},\sigma_{4}\\}&\tilde{e}^{1}_{02}=(D_{6}/\\{1,\sigma_{2}\\})\times
e^{1}&\\\
\tilde{e}^{0}_{2}=D^{6}/\\{1,C^{2},C^{4},\sigma_{2},\sigma_{4},\sigma_{6}\\}&\tilde{e}^{1}_{12}=(D_{6}/\\{1,\sigma_{4}\\})\times
e^{1}&\bsep{2pt}\\\ \hline\cr\end{array}$

  * •

($0$-cell) The $0$-cell $\tilde{e}^{0}_{0}=(D_{6}/D_{6})\times
e^{0}=\mathrm{pt}$ is the unique fixed point on $T^{2}$. The other $0$-cells
are defined as follows:

$\displaystyle\tilde{e}^{0}_{1}=\left\\{\frac{1}{2},\frac{\tau}{2},\frac{1+\tau}{2}\right\\}\cong\big{(}D_{6}/\big{\\{}1,C^{3},\sigma_{1},\sigma_{4}\big{\\}}\big{)}\times
e^{0},$
$\displaystyle\tilde{e}^{0}_{2}=\left\\{\frac{1+\tau}{3},\frac{2(1+\tau)}{3}\right\\}\cong\big{(}D_{6}/\big{\\{}1,C^{2},C^{4},\sigma_{2},\sigma_{4},\sigma_{6}\big{\\}}\big{)}\times
e^{0}.$
$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$

  * •

($1$-cell) For $0\leq i<j\leq 2$, the $1$-cell $\tilde{e}^{1}_{ij}$ consists
of the six segments connecting $\tilde{e}^{0}_{i}$ and $\tilde{e}^{0}_{j}$.
They are of the forms $\tilde{e}^{1}_{01}=(D_{6}/\\{1,\sigma_{1}\\})\times
e^{1}$, $\tilde{e}^{1}_{02}=(D_{6}/\\{1,\sigma_{2}\\})\times e^{1}$ and
$\tilde{e}^{1}_{12}=(D_{6}/\\{1,\sigma_{4}\\})\times e^{1}$.

$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$$\textstyle{\circ}$

  * •

($2$-cell) The $2$-cell $\tilde{e}^{2}=D_{6}\times e^{2}$ consists of the
twelve small triangular regions surrounded by the $1$-cells.

Let $Y\subset T^{2}$ be the invariant subspace
$Y=\tilde{e}^{0}_{0}\cup\tilde{e}^{0}_{1}\cup\tilde{e}^{1}_{01}$.

######  Lemma 4.9.

The equivariant cohomology of $Y$ is given as follows:

$\begin{array}[]{|c|c|c|c|c|}\hline\cr&n=0&n=1&n=2&n=3\\\ \hline\cr
H^{n}_{D_{6}}(Y;\mathbb{Z})&\mathbb{Z}&0&\mathbb{Z}_{2}^{\oplus
3}&\mathbb{Z}_{2}^{\oplus 2}\tsep{2pt}\bsep{2pt}\\\ \hline\cr\end{array}$

###### Proof.

We can find $D_{6}$-invariant subspaces $U$ and $V$ in $Y$ which have the
following equivariant homotopy equivalences

$\displaystyle U\simeq\tilde{e}^{0}_{0}=\mathrm{pt},\qquad
V\simeq\tilde{e}^{0}_{1}=D_{6}/D_{2},\qquad U\cap
V\simeq\tilde{e}^{1}_{01}\simeq D_{6}/\mathbb{Z}_{2}^{(1)},$

where $D_{2}=\\{1,C^{3},\sigma_{1},\sigma_{4}\\}$ and
$\mathbb{Z}_{2}^{(1)}=\\{1,\sigma_{1}\\}$. The equivariant cohomology groups
of these spaces can be summarized as follows:

$\begin{array}[]{|c|c|c|c|}\hline\cr
n=3&&\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}&0\\\ \hline\cr
n=2&&\mathbb{Z}_{2}^{\oplus 2}\oplus\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\ \hline\cr n=1&&0&0\\\ \hline\cr
n=0&&\mathbb{Z}\oplus\mathbb{Z}&\mathbb{Z}\\\
\hline\cr&H^{n}_{D_{6}}(Y)&H^{n}_{D_{6}}(U)\oplus
H^{n}_{D_{6}}(V)&H^{n}_{D_{6}}(U\cap V)\tsep{2pt}\bsep{2pt}\\\
\hline\cr\end{array}$

In the Mayer–Vietoris exact sequence

$\displaystyle\cdots\to H^{n}_{D_{6}}(Y)\to H^{n}_{D_{6}}(U)\oplus
H^{n}_{D_{6}}(V)\overset{\Delta}{\to}H^{n}_{D_{6}}(U\cap V)\to
H^{n+1}_{D_{6}}(Y)\to\cdots,$

the homomorphism $\Delta$ is expressed as
$\Delta(u,v)=j_{U}^{*}(u)-j_{V}^{*}(v)$ with $j_{U}\colon U\cap V\to U$ and
$j_{V}\colon U\cap V\to V$ the inclusions. This immediately determines
$H^{0}_{D_{6}}(Y)\cong\mathbb{Z}$ and $H^{1}_{D_{6}}(Y)=0$. To complete the
proof, we recall the identifications

$\displaystyle
H^{2}_{D_{6}}(U)\cong\operatorname{Hom}(D_{6},U(1))\cong\mathbb{Z}_{2}^{\oplus
2},\qquad
H^{2}_{D_{6}}(V)\cong\operatorname{Hom}(D_{2},U(1))\cong\mathbb{Z}_{2}^{\oplus
2},$ $\displaystyle H^{2}_{D_{6}}(U\cap
V)\cong\operatorname{Hom}\big{(}\mathbb{Z}_{2}^{(1)},U(1)\big{)}\cong\mathbb{Z}_{2},$

under which $j_{U}^{*}$ and $j_{V}^{*}$ are induced from the inclusions
$D_{2}\to D_{6}$ and $\mathbb{Z}_{2}^{(1)}\to D_{6}$. As a basis of
$H^{2}_{D_{6}}(U)$ we can choose the following $1$-dimensional representations
$\rho_{i}\colon D_{6}\to U(1)$ of $D_{6}$

$\displaystyle\rho_{1}\colon\ \begin{cases}C\mapsto 1,\\\
\sigma_{1}\mapsto-1,\end{cases}\qquad\rho_{2}\colon\
\begin{cases}C\mapsto-1,\\\ \sigma_{1}\mapsto 1.\end{cases}$

Similarly, we can choose the following $1$-dimensional representations
$\rho^{\prime}_{i}$ of $D_{2}=\\{1,\sigma_{1},C^{3},\sigma_{4}\\}$ as a basis
of $H^{2}_{D_{6}}(V)$

$\displaystyle\rho^{\prime}_{1}\colon\ \begin{cases}C^{3}\mapsto 1,\\\
\sigma_{1}\mapsto-1,\end{cases}\qquad\rho^{\prime}_{2}\colon\
\begin{cases}C^{3}\mapsto-1,\\\ \sigma_{1}\mapsto 1.\end{cases}$

Now, we can see
$H^{2}_{D_{6}}(Y)\cong\operatorname{Ker}\Delta\cong\mathbb{Z}_{2}^{\oplus 3}$,
and it has the following basis

$\displaystyle\\{(\rho_{1},\rho^{\prime}_{1}),(\rho_{2},\rho_{2}^{\prime}),(0,\rho^{\prime}_{2})\\}\subset\operatorname{Hom}(D_{6},U(1))\oplus\operatorname{Hom}(D_{4},U(1)).$

We can also see that $\Delta$ is surjective, and
$H^{3}_{D_{6}}(Y)\cong\mathbb{Z}_{2}^{\oplus 2}$. ∎

Let $X_{1}$ be the $1$-skeleton of the $D_{6}$-CW complex $T^{2}$.

######  Lemma 4.10.

$H^{3}_{D_{6}}(X_{1};\mathbb{Z})\cong\mathbb{Z}_{2}^{\oplus 2}$.

###### Proof.

We cover
$X_{1}=\tilde{e}^{0}_{0}\cup\tilde{e}^{0}_{1}\cup\tilde{e}^{0}_{2}\cup\tilde{e}^{1}_{01}\cup\tilde{e}^{1}_{02}\cup\tilde{e}^{1}_{12}$
by invariant subspaces $U^{\prime}$ and $V^{\prime}$ which admit the following
equivariant homotopy equivalences

$\displaystyle U^{\prime}\simeq Y,\qquad
V^{\prime}\simeq\tilde{e}^{0}_{2}=D_{6}/D_{3},\qquad U^{\prime}\cap
V^{\prime}\simeq\tilde{e}^{1}_{02}\sqcup\tilde{e}^{1}_{12}\simeq
D_{6}/\mathbb{Z}_{2}^{(2)}\sqcup D_{6}/\mathbb{Z}_{2}^{(4)},$

where $D_{3}=\\{1,C^{2},C^{4},\sigma_{2},\sigma_{4},\sigma_{6}\\}$,
$\mathbb{Z}_{2}^{(2)}=\\{1,\sigma_{2}\\}$ and
$\mathbb{Z}_{2}^{(4)}=\\{1,\sigma_{4}\\}$. The equivariant cohomology groups
of these spaces are summarized as follows:

$\begin{array}[]{|c|c|c|c|}\hline\cr n=3&&\mathbb{Z}_{2}^{\oplus 2}\oplus
0&0\tsep{2pt}\bsep{2pt}\\\ \hline\cr n=2&&\mathbb{Z}_{2}^{\oplus
3}\oplus\mathbb{Z}_{2}&\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\
\hline\cr n=1&&0&0\\\ \hline\cr
n=0&&\mathbb{Z}\oplus\mathbb{Z}&\mathbb{Z}\oplus\mathbb{Z}\\\
\hline\cr&H^{n}_{D_{6}}(X_{1})&H^{n}_{D_{6}}(U^{\prime})\oplus
H^{n}_{D_{6}}(V^{\prime})&H^{n}_{D_{6}}(U^{\prime}\cap
V^{\prime})\tsep{2pt}\bsep{2pt}\\\ \hline\cr\end{array}$

The homomorphism $\Delta$ in the Mayer–Vietoris exact sequence

$\displaystyle\cdots\to H^{2}_{D_{6}}(X_{1})\to
H^{2}_{D_{6}}(U^{\prime})\oplus
H^{2}_{D_{6}}(V^{\prime})\overset{\Delta}{\to}H^{2}_{D_{6}}(U^{\prime}\cap
V^{\prime})\to H^{3}_{D_{6}}(X_{1})\to\cdots$

is expressed as $\Delta(u,v)=j_{U^{\prime}}^{*}(u)-j_{V^{\prime}}^{*}(v)$ by
using the inclusions $j_{U^{\prime}}\colon U^{\prime}\cap V^{\prime}\to
U^{\prime}$ and $j_{V^{\prime}}\colon U^{\prime}\cap V^{\prime}\to
V^{\prime}$. An inspection proves that $j_{U^{\prime}}^{*}$ agrees with the
composition of the following two homomorphisms:

  * (i)

the inclusion that follows from the calculation of $H^{2}_{D_{6}}(Y)$ in Lemma
4.9

$\displaystyle H^{2}_{D_{6}}(U^{\prime})\cong
H^{2}_{D_{6}}(Y)\longrightarrow\operatorname{Hom}(D_{6},U(1))\oplus\operatorname{Hom}(D_{2},U(1)).$

  * (ii)

the direct sum $i_{2}^{*}\oplus i_{4}^{*}$ of the homomorphisms

$\displaystyle i^{*}_{2}\colon\
\operatorname{Hom}(D_{6},U(1))\to\operatorname{Hom}\big{(}\mathbb{Z}_{2}^{(2)},U(1)\big{)},\qquad
i^{*}_{4}\colon\
\operatorname{Hom}(D_{2},U(1))\to\operatorname{Hom}\big{(}\mathbb{Z}_{2}^{(4)},U(1)\big{)},$

induced from the inclusions $i_{2}\colon\mathbb{Z}_{2}^{(2)}\to D_{6}$ and
$\mathbb{Z}_{2}^{(4)}\to D_{2}$.

Then, using the basis presented in the calculation of $H^{2}_{D_{6}}(Y)$, we
find

$\displaystyle
j_{U^{\prime}}^{*}(\rho_{1},\rho_{1}^{\prime})=(\rho,\rho),\qquad
j_{U^{\prime}}^{*}(\rho_{2},\rho_{2}^{\prime})=(\rho,\rho),\qquad
j_{U^{\prime}}^{*}(0,\rho_{2}^{\prime})=(0,\rho),$

where $\rho\colon\mathbb{Z}_{2}\to\mathbb{Z}_{2}$ is the identity map
generating $\operatorname{Hom}(\mathbb{Z}_{2},U(1))\cong\mathbb{Z}_{2}$. Hence
$j_{U^{\prime}}^{*}$ as well as $\Delta$ are surjective, and
$H^{3}_{D_{6}}(X_{1})\cong H^{3}_{D_{6}}(Y)\cong\mathbb{Z}_{2}^{\oplus 2}$. ∎

######  Theorem 4.11 (p6m).

$H^{3}_{D_{6}}(T^{2};\mathbb{Z})\cong\mathbb{Z}_{2}^{\oplus 2}$.

###### Proof.

The relevant part of the exact sequence for the pair $(T^{2},X_{1})$ is

$\displaystyle H^{3}_{D_{6}}\big{(}T^{2},X_{1}\big{)}\to
H^{3}_{D_{6}}\big{(}T^{2}\big{)}\to H^{3}_{D_{6}}(X_{1})\to
H^{4}_{D_{6}}\big{(}T^{2},X_{1}\big{)}.$

By means of the excision axiom, we have $H^{n}_{D_{6}}(T^{2},X_{1})\cong
H^{n-2}(\mathrm{pt})$. Therefore we get $H^{3}_{D_{6}}(T^{2})\cong
H^{3}_{D_{6}}(X_{1})\cong\mathbb{Z}_{2}^{\oplus 2}$. ∎

Let $\hat{\mathbb{Z}}=\mathbb{Z}_{\phi_{1}}$ be the $D_{6}$-module such that
its underlying group is $\mathbb{Z}$ and $D_{6}$ acts via the homomorphism
$\phi_{1}\colon D_{6}\to\mathbb{Z}_{2}$ given by $\phi_{1}(C)=-1$ and
$\phi_{1}(\sigma_{1})=1$.

######  Lemma 4.12.

There is an exact sequence of $D_{6}$-modules

$\displaystyle 0\to H^{1}\big{(}T^{2};\mathbb{Z}\big{)}\to
H^{1}(Y;\mathbb{Z})\overset{\pi}{\to}\hat{\mathbb{Z}}\to 0$

admitting a module homomorphism $s\colon\hat{\mathbb{Z}}\to
H^{1}(Y;\mathbb{Z})$ such that $\pi\circ s=3$.

###### Proof.

Let $\eta_{1},\eta_{2}\in H_{1}(T^{2})$ be the homology classes of the loops
going along the vectors $1$ and $\tau$ respectively in the fundamental domain,
which form a basis of $H_{1}(T^{2})\cong\mathbb{Z}^{2}$. Also, let
$\gamma_{1},\gamma_{2},\gamma_{3}\in H_{1}(Y)$ be the homology classes of
loops along $1$, $\tau$ and $\tau-1$, which form a basis of
$H_{1}(Y)\cong\mathbb{Z}^{3}$. The inclusion map $i\colon Y\to T^{2}$ relates
these bases by $i_{*}\gamma_{1}=\eta_{1}$, $i_{*}\gamma_{2}=\eta_{2}$ and
$i_{*}\gamma_{3}=\eta_{2}-\eta_{1}$. The actions of $C\in D_{6}$ and
$\sigma_{1}$ on these bases are

$\displaystyle\begin{cases}C_{*}\eta_{1}=\eta_{2},\\\
C_{*}\eta_{2}=\eta_{2}-\eta_{1},\end{cases}\qquad\begin{cases}{\sigma_{1}}_{*}\eta_{1}=\eta_{1},\\\
{\sigma_{1}}_{*}\eta_{2}=\eta_{1}-\eta_{2},\end{cases}\qquad\begin{cases}C_{*}\gamma_{1}=\gamma_{2},\\\
C_{*}\gamma_{2}=\gamma_{3},\\\
C_{*}\gamma_{3}=-\gamma_{1},\end{cases}\qquad\begin{cases}{\sigma_{1}}_{*}\gamma_{1}=\gamma_{1},\\\
{\sigma_{1}}_{*}\gamma_{2}=-\gamma_{3},\\\
{\sigma_{1}}_{*}\gamma_{3}=-\gamma_{2}.\end{cases}\\!\\!\\!$

Let $\\{h_{1},h_{2}\\}\subset H^{1}(T^{2})$ and
$\\{g_{1},g_{2},g_{3}\\}\subset H^{1}(Y)$ 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.

$\displaystyle\begin{cases}C^{*}h_{1}=-h_{2},\\\
C^{*}h_{2}=h_{1}+h_{2},\end{cases}\qquad\begin{cases}\sigma_{1}^{*}h_{1}=h_{1}+h_{2},\\\
\sigma_{1}^{*}h_{2}=-h_{2},\end{cases}\qquad\begin{cases}C^{*}g_{1}=-g_{3},\\\
C^{*}g_{2}=g_{1},\\\
C^{*}g_{3}=g_{2},\end{cases}\qquad\begin{cases}\sigma_{1}^{*}g_{1}=g_{1},\\\
\sigma_{1}^{*}g_{2}=-g_{3},\\\ \sigma_{1}^{*}g_{3}=-g_{2}.\end{cases}$

These expressions allow us to prove that the cokernel of the homomorphism
$i^{*}\colon H^{1}(T^{2})\to H^{2}(Y)$ is isomorphic to $\hat{\mathbb{Z}}$,
yielding the exact sequence. The homomorphism $s\colon\hat{\mathbb{Z}}\to
H^{1}(Y)$ is given by $s(1)=g_{1}-g_{2}+g_{3}$. ∎

######  Lemma 4.13.

$H^{n}_{\mathrm{group}}(D_{6};H^{1}(T^{2};\mathbb{Z}))=0$ for $n=0,1,2$.

###### Proof.

We use the long exact sequence in group cohomology induced from the exact
sequence $0\to H^{1}(T^{2})\to H^{1}(Y)\overset{\pi}{\to}\hat{\mathbb{Z}}\to
0$ in coefficients. By Lemmas 4.1, 4.9 and 5.6 to be given in Section 5, we
get the following:

$\begin{array}[]{|c|c|c|c|}\hline\cr n=2&&\mathbb{Z}_{2}&\mathbb{Z}_{2}\\\
\hline\cr n=1&&\mathbb{Z}_{2}&\mathbb{Z}_{2}\\\ \hline\cr n=0&&0&0\\\
\hline\cr&H^{n}_{\mathrm{group}}(D_{6};H^{1}(T^{2}))&H^{n}_{\mathrm{group}}(D_{6};H^{1}(Y))&H^{n}_{\mathrm{group}}(D_{6};\hat{\mathbb{Z}})\tsep{2pt}\bsep{2pt}\\\
\hline\cr\end{array}$

It is clear that $H^{0}_{\mathrm{group}}(D_{6};H^{1}(T^{2}))=0$. The
homomorphism in group cohomology induced from $\pi\colon
H^{1}(Y)\to\hat{\mathbb{Z}}$ is surjective in degree $1$ and $2$, because
$\pi\circ s=3$. This leads to the remaining vanishing. ∎

######  Theorem 4.14 (p6m).

The following holds true:

  * $(a)$

$F^{2}H^{3}_{D_{6}}(T^{2};\mathbb{Z})\cong\mathbb{Z}_{2}$,

  * $(b)$

the $D_{6}$-equivariant cohomology of $T^{2}$ in low degrees is as follows:

$\displaystyle
H^{0}_{D_{6}}\big{(}T^{2};\mathbb{Z}\big{)}\cong\mathbb{Z},\qquad
H^{1}_{D_{6}}\big{(}T^{2};\mathbb{Z}\big{)}=0,\qquad
H^{2}_{D_{6}}\big{(}T^{2};\mathbb{Z}\big{)}\cong\mathbb{Z}_{2}^{\oplus 2}.$

###### Proof.

In the $E_{2}$-term of the Leray–Serre spectral sequence
$E_{2}^{p,q}=H^{p}_{\mathrm{group}}(D_{6};H^{q}(T^{2};\mathbb{Z}))$, the
coefficient $H^{0}(T^{2})$ is identified with the trivial $D_{6}$-module
$\mathbb{Z}$, and $H^{2}(T^{2})$ with $\tilde{\mathbb{Z}}$ as in Lemma 4.2.
The group cohomology with coefficients in $H^{1}(T^{2})$ is already computed,
and that in $\tilde{\mathbb{Z}}$ is also computed in Lemma 4.2. The
$E_{2}$-terms are summarized as follows:

$\begin{array}[]{|c|c|c|c|c|}\hline\cr q=3&0&0&0&0\\\ \hline\cr
q=2&0&\mathbb{Z}_{2}&&\\\ \hline\cr q=1&0&0&0&\\\ \hline\cr
q=0&\mathbb{Z}&0&\mathbb{Z}_{2}^{\oplus
2}&\mathbb{Z}_{2}\tsep{2pt}\bsep{2pt}\\\ \hline\cr
E_{2}^{p,q}&p=0&p=1&p=2&p=3\tsep{2pt}\bsep{2pt}\\\ \hline\cr\end{array}$

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

$\displaystyle F^{2}H^{3}_{P}\big{(}T^{2};\mathbb{Z}\big{)}\cong
E^{2,1}_{2}\oplus E^{3,0}_{2}$

by Lemma 4.3, in which the direct summands are

$\displaystyle
E^{2,1}_{2}=H^{2}_{\mathrm{group}}\big{(}P;H^{1}\big{(}T^{2};\mathbb{Z}\big{)}\big{)},\qquad
E^{3,0}_{2}=H^{3}_{\mathrm{group}}(P;\mathbb{Z})\cong
H^{2}_{\mathrm{group}}(P;U(1)).$

Thus, it suffices to prove that the group cocycles $\tau$ induced from the
nonsymmorphic $2$-dimensional space groups as in Section 2 generate
$E^{2,1}_{2}$.

Recall from Section 2 that the group $2$-cocycle $\nu\in
Z^{2}_{\mathrm{group}}(P;\Pi)$ measures the failure for a space group $S$ to
be a semi-direct product of its point group $P$ and the lattice $\Pi$, where
$\Pi$ is regarded as a left $P$-module naturally. In other words, $S$ is
nonsymmorphic if and only if $[\nu]\in H^{2}_{\mathrm{group}}(P;\Pi)$ is non-
trivial.

######  Lemma 4.15.

Let $\Pi$ and $P$ be the lattice and the point group of a $d$-dimensional
space group $S$. Then there is an isomorphism of groups

$\displaystyle H^{2}_{\mathrm{group}}(P;\Pi)\cong
H^{2}_{\mathrm{group}}\big{(}P;H^{1}\big{(}\hat{\Pi};\mathbb{Z}\big{)}\big{)}.$

In particular, this factors through the homomorphisms

$\displaystyle H^{2}_{\mathrm{group}}(P;\Pi)\longrightarrow
H^{2}_{\mathrm{group}}\big{(}P;C\big{(}\hat{\Pi},U(1)\big{)}\big{)}$

given by the assignment of the cocycles $\nu\mapsto\tau$ in Section 2 and

$\displaystyle
H^{2}_{\mathrm{group}}\big{(}P;C\big{(}\hat{\Pi},U(1)\big{)}\big{)}\longrightarrow
H^{2}_{\mathrm{group}}\big{(}P;H^{1}\big{(}\hat{\Pi};\mathbb{Z}\big{)}\big{)}$

induced from the natural surjection $\delta\colon C(\hat{\Pi},U(1))\to
H^{1}(\hat{\Pi};\mathbb{Z})$.

###### Proof.

Instead of the left $P$-action on the Pontryagin dual
$\hat{\Pi}=\operatorname{Hom}(\Pi,U(1))$ defined in Section 2, we consider the
natural right action $\hat{k}(m)\mapsto\hat{k}(pm)$ of $p\in P$ on
$\hat{k}\in\hat{\Pi}$, from which the left action originates. This choice of
the actions does not affect the group cohomology. The right $P$-action on
$\hat{\Pi}$ induces by pull-back a left $P$-action on the cohomology
$H^{1}(\hat{\Pi};\mathbb{Z})$. Thus, the isomorphism of the group cohomologies
will be established once we see $H^{1}(\hat{\Pi};\mathbb{Z})\cong\Pi$ as left
$P$-modules. In general, for each element $m\in\Pi\subset\mathbb{R}^{d}=V$,
the path $[0,1]\to V$, ($t\mapsto tm$) defines a loop in $V/\Pi$. This induces
an isomorphism of left $P$-modules $\Pi\cong H_{1}(V/\Pi;\mathbb{Z})$. By the
universal coefficient theorem, the dual
$\Pi^{*}=\operatorname{Hom}(\Pi,\mathbb{Z})$ of $\Pi$ is identified with the
first homology group of $V/\Pi$ as a right $P$-module:

$\displaystyle\Pi^{*}=\operatorname{Hom}(\Pi,\mathbb{Z})\cong\operatorname{Hom}(H_{1}(V/\Pi;\mathbb{Z}),\mathbb{Z})\cong
H^{1}(V/\Pi;\mathbb{Z}).$

Considering the dual space $V^{*}=\operatorname{Hom}(V,\mathbb{R})$ and its
lattice $\Pi^{*}$ instead, we similarly get an isomorphism of left $P$-modules

$\displaystyle\Pi\cong H^{1}(V^{*}/\Pi^{*};\mathbb{Z}).$

Since there is a natural isomorphism of tori
$V^{*}/\Pi^{*}\to\hat{\Pi}=\operatorname{Hom}(\Pi,U(1))$ 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 $H^{2}_{\mathrm{group}}(\mathbb{Z}_{2};\Pi)\cong\mathbb{Z}_{2}$
through $\nu$, and the element corresponds by Lemma 4.15 to the non-trivial
element of
$H^{2}_{\mathrm{group}}(\mathbb{Z}_{2};H^{1}(T^{2};\mathbb{Z}))\cong\mathbb{Z}_{2}$
represented by the group $2$-cocycle $\tau$ induced from pg. The same holds
true in the case of p4m/p4g. In the case of pmm/pmg/pgg, we have
$H^{2}_{\mathrm{group}}(D_{2};\Pi)\cong
H^{2}_{\mathrm{group}}(D_{2};H^{1}(T^{2};\mathbb{Z}))\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$.
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
$\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$, 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
$H^{2}_{\mathrm{group}}(D_{2};H^{1}(T^{2};\mathbb{Z}))\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$
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 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.

###  5.1 The Leray–Serre spectral sequence

For a finite group $G$ and a homomorphism $\phi\colon G\to\mathbb{Z}_{2}$, we
define $\mathbb{Z}_{\phi}$ to be the $G$-module $\mathbb{Z}_{\phi}$ such that
its underlying group is $\mathbb{Z}$ and $G$ acts via $\phi$. Suppose that $G$
acts on a space $X$. Associated to the fibration $X\to EG\times_{G}X\to BG$ is
the long exact sequence of homotopy groups:

$\displaystyle\cdots\to\pi_{n}(X)\to\pi_{n}(EG\times_{G}X)\to\pi_{n}(BG)\to\cdots.$

Since $\pi_{1}(BG)\cong G$, we get a homomorphism
$\pi_{1}(EG\times_{G}X){\to}\mathbb{Z}_{2}$ by composing
$\pi_{1}(EG\times_{G}X){\to}G$ with $\phi\colon G\to\mathbb{Z}_{2}$. The
homomorphism defines a local system on the Borel construction $EG\times_{G}X$,
which we denote with the same notation $\mathbb{Z}_{\phi}$. By using this
local system, we define the $G$-equivariant cohomology with local coefficients

$\displaystyle
H^{n}_{G}(X;\mathbb{Z}_{\phi})=H^{n}(EG\times_{G}X;\mathbb{Z}_{\phi}).$

Of course, if $\phi$ is trivial, then the cohomology above recovers the usual
equivariant cohomology with integer coefficients $\mathbb{Z}$.

For $H^{n}_{G}(X;\mathbb{Z}_{\phi})$, we also have the Leray–Serre spectral
sequence $E_{r}^{p,q}$ converging to the graded quotient of a filtration

$\displaystyle
H^{n}_{G}(X;\mathbb{Z}_{\phi})=F^{0}H^{n}_{G}(X;\mathbb{Z}_{\phi})\supset
F^{1}H^{n}_{G}(X;\mathbb{Z}_{\phi})\supset
F^{2}H^{n}_{G}(X;\mathbb{Z}_{\phi})\supset\cdots.$

Its $E_{2}$-term is again given by the group cohomology

$\displaystyle
E_{2}^{p,q}=H^{p}_{\mathrm{group}}(G;H^{q}(X;\mathbb{Z})\otimes\mathbb{Z}_{\phi}),$

where $H^{q}(X;\mathbb{Z})\otimes\mathbb{Z}_{\phi}$ is the tensor product of
the $G$-modules $H^{q}(X;\mathbb{Z})$ and $\mathbb{Z}_{\phi}$, namely, its
underlying group is

$\displaystyle
H^{q}(X;\mathbb{Z})\otimes\mathbb{Z}_{\phi}=H^{q}(X;\mathbb{Z})\otimes\mathbb{Z}\cong
H^{q}(X;\mathbb{Z}),$

and $g\in G$ acts on $x\in H^{q}(X;\mathbb{Z})$ by $x\mapsto\phi(g)\cdot
g^{*}x$.

As in the usual equivariant cohomology, $H^{n}_{G}(X;\mathbb{Z}_{\phi})$ can
be identified with a sheaf cohomology of the simplicial space
$G^{\bullet}\times X$. (See for instance the appendix of [10] in the case of
$G=\mathbb{Z}_{2}$.) 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
$H^{1}_{G}(X;\mathbb{Z}_{2})\times H^{3}_{G}(X;\mathbb{Z}_{\phi})$ and the
ungraded twists by $H^{3}_{G}(X;\mathbb{Z}_{\phi})$.

The results in Section 3.3 can be generalized to the equivariant cohomology
with coefficients in $\mathbb{Z}_{\phi}$, 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 $\phi\colon G\to\mathbb{Z}_{2}$ a
homomorphism.

  * $(i)$

$F^{1}H^{3}_{G}(X;\mathbb{Z}_{\phi})$ classifies $($ungraded$)$ twists which
can be represented by $\phi$-twisted central extensions of the groupoid
$X/\\!/G$.

  * $(ii)$

$F^{2}H^{3}_{G}(X;\mathbb{Z}_{\phi})$ classifies $($ungraded$)$ twists which
can be represented by $2$-cocycles of $G$ with coefficients in the $G$-module
$C(X,U(1))_{\phi}$, where $C(X,U(1))_{\phi}=C(X,U(1))$ is the group of
$U(1)$-valued functions on $X$ and $g\in G$ acts on $f\colon X\to U(1)$ by
$f\mapsto g^{*}f^{\phi(g)}$.

  * $(iii)$

$F^{3}H^{3}_{G}(X;\mathbb{Z}_{\phi})=H^{2}_{\mathrm{group}}(G;U(1)_{\phi})$
classifies $($ungraded$)$ twists which can be represented by $2$-cocycles of
$G$ with coefficients in the $G$-module $U(1)_{\phi}$, where
$U(1)=U(1)_{\phi}$ as a group and $g\in G$ acts on $u\in U(1)$ by $u\mapsto
u^{\phi(g)}$.

###  5.2 Tools

As long as we are concerned with the local system $\mathbb{Z}_{\phi}$
associated to a homomorphism $\phi\colon G\to\mathbb{Z}_{2}$, the reduced
cohomology $\tilde{H}^{n}_{G}(X;\mathbb{Z}_{\phi})$ makes sense for a
$G$-space $X$ with a fixed point $\mathrm{pt}\in X$, and we have the direct
sum decomposition $H^{n}_{G}(X;\mathbb{Z}_{\phi})\cong
H^{n}_{G}(\mathrm{pt};\mathbb{Z}_{\phi})\oplus\tilde{H}^{n}_{G}(X;\mathbb{Z}_{\phi})$.
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 $\mathrm{pt}\in Y$. Suppose further that $Y$ is a CW complex
consisting of only cells of dimension less than or equal to $1$. Then, for any
homomorphism $\phi\colon G\to\mathbb{Z}_{2}$ and $n\geq 0$, the following
holds true:

$\displaystyle
H^{n}_{\mathrm{group}}\big{(}G;H^{1}(Y;\mathbb{Z})\otimes\mathbb{Z}_{\phi}\big{)}\cong\tilde{H}_{G}^{n+1}(Y;\mathbb{Z}_{\phi}).$

Similarly, we can generalize Lemma 4.5 as follows:

######  Lemma 5.3.

Suppose a finite group $G$ acts on the torus $T^{2}=S^{1}\times S^{1}$ and

  * •

there is a fixed point $\mathrm{pt}=(x_{0},y_{0})\in T^{2}$ under the
$G$-action,

  * •

$G$ preserves the subspace $S^{1}\vee
S^{1}=S^{1}\times\\{y_{0}\\}\cup\\{x_{0}\\}\times S^{1}\subset T^{2}$.

Then there is the following isomorphism of groups for any homomorphism
$\phi\colon G\to\mathbb{Z}_{2}$ and all $n\in\mathbb{Z}$

$\displaystyle H^{n}_{G}\big{(}T^{2};\mathbb{Z}_{\phi}\big{)}\cong
H^{n}_{G}(\mathrm{pt};\mathbb{Z}_{\phi})\oplus\tilde{H}^{n}_{G}\big{(}S^{1}\vee
S^{1};\mathbb{Z}_{\phi}\big{)}\oplus\tilde{H}^{n}_{G}\big{(}T^{2}/S^{1}\vee
S^{1};\mathbb{Z}_{\phi}\big{)}.$

Further, the Leray–Serre spectral sequence for
$H^{n}_{G}(T^{2};\mathbb{Z}_{\phi})$ degenerates at $E_{2}$ and the relevant
extension problems are trivial, so that

  * $(a)$

$F^{2}H^{3}_{G}(T^{2};\mathbb{Z}_{\phi})\cong E_{2}^{2,1}\oplus E_{2}^{3,0}$,

  * $(b)$

$H^{n}_{G}(T^{2};\mathbb{Z}_{\phi})\cong\bigoplus_{p+q=n}E_{2}^{p,q}$ for all
$n\in\mathbb{Z}$.

Besides the generalizations above, we will use the following universal
coefficient theorem in the sequel:

######  Lemma 5.4.

For any $\phi\colon G\to\mathbb{Z}_{2}$, there is a split exact sequence of
groups

$\displaystyle 0\to H^{n}_{G}(X;\mathbb{Z}_{\phi})\otimes\mathbb{Z}_{2}\to
H^{n}_{G}(X;\mathbb{Z}_{2})\to\operatorname{Tor}\big{(}H^{n+1}_{G}(X;\mathbb{Z}_{\phi}),\mathbb{Z}_{2}\big{)}\to
0.$

###### Proof.

For any homomorphism $\phi\colon G\to\mathbb{Z}_{2}$, let
$(\mathbb{Z}_{2})_{\phi}$ be the $G$-module such that its underlying group is
$\mathbb{Z}_{2}$ and its $G$-action is given by $\phi\colon
G\to\mathbb{Z}_{2}$. This $G$-module $(\mathbb{Z}_{2})_{\phi}$ agrees with the
trivial $G$-module $\mathbb{Z}_{2}$, even if $\phi$ 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 $\underline{\mathbb{R}}_{\phi}\to X$ be the $G$-equivariant real line
bundle on $X$ whose underlying bundle is $X\times\mathbb{R}$ and the action of
$g\in G$ on $(x,r)\in X\times\mathbb{R}$ is $(x,r)\mapsto(gx,\phi(g)r)$. The
Thom isomorphism theorem then provides

$\displaystyle H^{n}_{G}(X;A_{\phi})\cong H^{n+1}_{G}(D,S;A),$

where $A$ is $\mathbb{Z}_{2}$ or $\mathbb{Z}$, and
$D\subset\underline{\mathbb{R}}_{\phi}$ and
$S\subset\underline{\mathbb{R}}_{\phi}$ 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 $H^{n}_{G}(X;\mathbb{Z}_{\phi})$, we
usually need the cohomology of the point
$H^{n}_{G}(\mathrm{pt};\mathbb{Z}_{\phi})$. This cohomology is identified with
the group cohomology $H^{n}_{\mathrm{group}}(G;\mathbb{Z}_{\phi})$ 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:

######  Lemma 5.5.

For any $\phi\colon G\to\mathbb{Z}_{2}$, there are natural exact sequences

$\displaystyle\cdots\to H^{n-1}_{G}(\mathrm{pt};\mathbb{Z}_{\phi})\to
H^{n}_{G}(\mathrm{pt};\mathbb{Z})\overset{i^{*}}{\to}H^{n}_{\operatorname{Ker}\phi}(\mathrm{pt};\mathbb{Z})\to
H^{n}_{G}(\mathrm{pt};\mathbb{Z}_{\phi})\to\cdots,$ $\displaystyle\cdots\to
H^{n-1}_{G}(\mathrm{pt};\mathbb{Z})\to
H^{n}_{G}(\mathrm{pt};\mathbb{Z}_{\phi})\overset{i^{*}}{\to}H^{n}_{\operatorname{Ker}\phi}(\mathrm{pt};\mathbb{Z})\to
H^{n}_{G}(\mathrm{pt};\mathbb{Z})\to\cdots,$

where $i^{*}$ is induced from the inclusion $i\colon\operatorname{Ker}\phi\to
G$.

###### Proof.

Let $G$ act on $\mathbb{R}_{\phi}=\mathbb{R}$ via $\phi\colon
G\to\mathbb{Z}_{2}$. We can regard $\mathbb{R}_{\phi}$ as a $G$-equivariant
real line bundle on $\mathrm{pt}$. We have the Thom isomorphisms