COMPATIBLE ACTIONS AND COHOMOLOGY OF
CRYSTALLOGRAPHIC GROUPS
ALEJANDRO ADEM∗, JIANQUAN GE, JIANZHONG PAN, AND NANSEN PETROSYAN
Abstract. We compute the cohomology of crystallographic groups Γ = Zn ⋊Z/p with
holonomy of prime order by establishing the collapse at E2 of the spectral sequence
associated to their defining extension. As an application we compute the group of gerbes
associated to many six–dimensional toroidal orbifolds arising in string theory.
1. Introduction
Given a finite group G and an integral representation L for G (i.e. a homomorphism
G → GLn(Z), where L is the underlying ZG–module), we can define the semi–direct
product Γ = L⋊G. Calculating the cohomology of these groups is a problem of instrinsic
algebraic interest; indeed if the representation is faithful then these groups can be thought
of as crystallographic groups (see [7], page 74).
From the geometric point of view, the action on L gives rise to a G–action on the
n–torus X = Tn; this approach can be used to derive important examples of orbifolds,
known as toroidal orbifolds (see [2]). In the case when n = 6 these are of particular
interest in string theory (see [4], [11]).
Given the split group extension 0 → L → Γ → G → 1, the basic problem which
we address is that of providing conditions which imply the collapse (without extension
problems) of the associated Lyndon–Hochshild–Serre spectral sequence. The conditions
which we establish are representation–theoretic, namely depending solely on the structure
of the integral representation L. This can be a difficult problem (see [15] for further
background) and there are well–known examples where the spectral sequence does not
collapse.
Our approach is to systematically apply the methods used in [2]. The key idea is to
construct a free resolution F for the semidirect product L ⋊ G such that the Lyndon–
Hochschild–Serre spectral sequence of the group extension collapses at E2. This requires
Date: October 30, 2018.
Key words and phrases. spectral sequence, group cohomology.
∗The first author was partially supported by the NSF and by NSERC.
http://arxiv.org/abs/0704.1823v1
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 2
a chain–level argument, more specifically the construction of a compatible G–action on
a certain free resolution for the torsion–free abelian group L (see §2 for details). We
concentrate on the case of G = Z/p, a cyclic group of prime order p, as the representation
theory is well–understood. Our main algebraic result is the following
Theorem 1.1. Let G = Z/p, where p is any prime. If L is any finitely generated ZG–
lattice1, and Γ = L⋊G is the associated semi–direct product group, then for each k ≥ 0
Hk(Γ,Z) ∼=
i+j=k
H i(G,∧j(L∗))
where ∧j(L∗) denotes the j-th exterior power of the dual module L∗ = Hom(L,Z).
Expressed differently: these results imply a complete calculation for the integral coho-
mology of crystallographic groups Zn ⋊ Z/p where p is prime. These calculations can be
made explicit.
The theorem has an interesting geometric application:
Theorem 1.2. Let G = Z/p, where p is any prime. Suppose that G acts on a space X
homotopy equivalent to (S1)n with XG 6= ∅, then for each k ≥ 0
Hk(EG×G X,Z) ∼=
i+j=k
H i(G,Hj(X,Z)) ∼= Hk(Γ,Z).
where Γ = π1(X)⋊G.
On the other hand, the explicit computation for torsion–free crystallographic groups
with holonomy of prime order was carried out long ago by Charlap and Vásquez (see [8],
page 556). Combining the two results we obtain a complete calculation:
Theorem 1.3. Let Γ denote a crystallographic group with holonomy of prime order p,
expressed as an extension
1 → L → Γ → Z/p → 1
where L is a free abelian group of finite rank.
(1) If Γ is torsion–free, then L ∼= N ⊕ Z (it splits off a trivial direct summand) and
Hk(Γ,Z) ∼= H0(Z/p,∧k(L∗))⊕H1(Z/p,∧k−1(N∗))
for 0 ≤ k ≤ rk(L); Hk(Γ,Z) = 0 for k > rk(L).
(2) If Γ is not torsion–free, then H∗(Γ,Z) can be computed using Theorem 1.1.
1A ZG–lattice is a ZG–module which happens to be a free abelian group.
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 3
In this paper we also consider the situation for the cyclic group of order four; some
partial results are obtained but a general collapse has not been established. However,
based on these and other computations we conjecture that for G any cyclic group, the
spectral sequence associated to a semi–direct product of the form Zn ⋊ G must collapse
at E2.
In the last section we give an application of our methods to calculations for six–
dimensional toroidal orbifolds, showing that among the 18 inequivalent N = 1 supersym-
metric string theories on symmetric orbifolds of (2, 2)–type without discrete background,
only two of them cannot be analyzed using our methods i.e. we cannot show the existence
of compatible actions for the associated modules. If X = [T6/G] is an orbifold arising
this way, then our results provide a complete calculation for its associated group of gerbes
Gb(X ) ∼= H3(EG×G T
6,Z) (see [2] for more details).
2. Preliminary Results
The notion of a compatible action was first introduced in [9]. If such an action exists
it allows one to construct practical projective resolutions and from these to compute the
cohomology of the group. We will give the basic definition and the main theorem that
follows. More details can be found in [2].
Let Γ = L ×ρ G = L ⋊ G be the semidirect product of a finite group G and a finite
dimensional Z-lattice L via a representation ρ : G → GL(L). G acts on the group L by the
homomorphism ρ, and this extends linearly to an action on the group algebra R[L], where
R denotes a commutative ring with unit. We write lg for ρ(g)l where l ∈ R[L], g ∈ G.
In the rest of this paper R will represent Z (the integers) or Z(p) (the ring of integers
localized at a fixed prime p).
Definition 2.1. Given a free resolution ǫ : F → R of R over R[L], we say that it admits
an action of G compatible with ρ if for all g ∈ G there is an augmentation-preserving
chain map τ(g) : F → F such that
(1) τ(g)[l · f ] = lg · [τ(g)f ] for all l ∈ R[L] and f ∈ F ,
(2) τ(g)τ(g′) = τ(gg′) for all g, g′ ∈ G,
(3) τ(1) = 1F .
The following two lemmas (see [2]) reduce the construction of compatible actions to the
case of faithful indecomposable representations.
Lemma 2.2. If ǫi : Fi → R is a projective R[Li]-resolution of R for i = 1, 2, then
ǫ1 ⊗ ǫ2 : F1 ⊗ F2 → R is a projective R[L1 × L2]-resolution of R. Furthermore, if G acts
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 4
compatibly on Fi by τi for i = 1, 2, then a compatible action of G on ǫ1⊗ ǫ2 : F1⊗F2 → R
is given by τ(g)(f1 ⊗ f2) = τ1(g)(f1)⊗ τ2(g)(f2).
Lemma 2.3. If L is a R[G1]-module, π : G2 → G1 a group homomorphism, and ǫ : F → R
is a R[L]-resolution of R such that G1 acts compatibly on it by τ
′, then G2 also acts
compatibly on it by τ(g)f = τ ′(π(g))f for any g ∈ G.
If a compatible action exists, we can give F a Γ-module structure as follows. An
element γ ∈ Γ can be expressed uniquely as γ = lg, with l ∈ L and g ∈ G. We set
γ · f = (lg) · f = l · τ(g)f . Note that given any G–module M , this inflates to a Γ–action
on M via the projection Γ → G.
We can always construct a special free resolution F of R over L, characterized by the
property that the cochain complex HomL(F,R) for computing the cohomology H
∗(L,R)
has all coboundary maps zero (more details will be provided in the next section). Using
this fact, the following was proved in [2]:
Theorem 2.4. (Adem-Pan) Let ǫ : F → R be a special free resolution of R over L and
suppose that there is a compatible action of G on F . Then for all integers k ≥ 0, we have
Hk(L⋊G,R) =
i+j=k
H i(G,Hj(L,R)).
This result can be interpreted as saying that the Lyndon-Hochschild-Serre spectral
sequence
2 = H
p(G,Hq(L,R)) ⇒ Hp+q(L⋊G,R)
collapses at E2 without extension problems. Note that this is not always the case; in fact
there are examples of semi–direct products of the form Zn ⋊ (Z/p)2 where the associated
spectral sequence has non–trivial differentials (see [15]). This will be discussed in §5.
3. Construction of Compatible Actions
Let R[L] denote the group ring of L, a free abelian group with basis {x1, . . . , xn}.
Then the elements x1 − 1, . . . , xn − 1 form a regular sequence in R[L], hence the Koszul
complex K∗ = K(x1−1, . . . , xn−1) is a free resolution of the trivial module R. It has the
additional property of being a differential graded algebra (or DGA). We briefly recall how
it looks. There are generators a1, . . . , an in degree one, and the graded basis for K∗ can
be identified with the usual basis for the exterior algebra they generate. The differential
is given by the following formula: if ai1...ip = ai1 . . . aip is a basis element in Kp, then
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 5
d(ai1...ip) =
(−1)j−1(xij − 1)ai1...îj ...ip.
Now the cohomology of the free abelian group L is precisely an exterior algebra on n one-
dimensional generators, which in fact can be identified with the dual elements a∗1, . . . , a
In particular we see that the cochain complex HomR[L](K∗, R) has zero differentials, and
hence K∗ is a special free resolution of R over R[L] (this resolution also appears in [7]
pp. 96–97).
We now consider how to construct a compatible G–action on K∗, given a G–module
structure on L.
Theorem 3.1. If G acts on the lattice L, let K∗ = K(x1 − 1, . . . xn − 1) denote the
special free resolution of R over R[L] defined using the Koszul complex associated to the
elements x1 − 1, . . . , xn − 1, where {x1, . . . , xn} form a basis for L. Suppose that there is
a homomorphism τ : G → Aut(K1) such that for every g ∈ G and a ∈ K1 it satisfies
dτ(g)(a) = d(a)g
where d : K1 → K0 is the usual Koszul differential, and d(a)
g ∈ K0 = R[L]. Then τ
extends to K∗ using its DGA structure and so defines a compatible G–action on K∗.
Proof. First we observe that τ(g) acts on K0 = R[L] via the original G–action, i.e.
τ(g)(x) = xg for any x ∈ K0. Next we define the action on the basis of K∗ as a graded
R[L]–module, namely:
τ(g)(ai1 . . . aip) = τ(g)(ai1) . . . τ(g)(aip).
If α ∈ R[L] and u ∈ K∗, we define τ(g)(αu) = α
gτ(g)(x). By linearity and the DGA
structure of K∗, this will define τ : G → Aut(K∗), with the desired properties. �
Generally speaking it can be quite difficult to construct a compatible action; however
there is an important special case where it is quite straightforward.
Theorem 3.2. Let φ : G → Σn denote a group homomorphism, where Σn denotes the
symmetric group on n elements. Let G act on Zn via2 this homomorphism. Then the
associated Koszul complex K∗ admits a compatible G–action.
2Such a module is called a permutation module.
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 6
Proof. By Lemma 2.3 we can assume that G is a subgroup of Σn, hence it will suffice to
prove this for Σn itself. If we take generators a1, . . . , an for the Koszul complex corre-
sponding to the elements x1, . . . , xn in the underlying module L, then we can define τ as
follows:
τ(σ)(ai) = aσ(i).
This obviously defines a permutation representation onK1, and compatibility follows from
the fact that for all ai, 1 ≤ i ≤ n and σ ∈ Σ we have
dτ(σ)(ai) = d(aσ(i)) = xσ(i) − 1 = (xi − 1)
This completes the proof. �
Aside from permutation representations, it is difficult to construct general examples
of compatible actions. However if G is a cyclic group then we can handle an important
additional type of module.
Proposition 3.3. Let the cyclic group G = 〈t|tn = 1〉 act on Zn−1 by:
ξ1 : t 7→
0 1 0 . . . 0 0
0 0 1 . . . 0 0
. . .
0 0 0 . . . 1 0
−1 −1 −1 . . . −1 −1
∈ GLn−1(Z).
If x1, . . . , xn is the canonical basis under which the action is represented by the matrix
above, then the free resolution K∗ = K∗(x1−1, . . . , xn−1) admits an action of G compatible
with ξ1, which can be defined by:
τ(t)(a1) = −x
n−1an−1 τ(t)(ak) = −x
n−1(an−1 − ak−1), 1 < k ≤ n− 1.
Proof. The proof is a straightforward calculation verifying that τ defines a compatible
action. First we verify that τn = 1. For this we observe that if A is the matrix in GLn−1(Z)
representing the generator t, then expressed in terms of the basis {a1, . . . , an} we have
that τ(t) = x−1n−1A. If we iterate this action and use the fact that τ(g)(αu) = α
gτ(g)(u)
then we obtain
τ(t)n = (x−1n−1)
tn−1(x−1n−1)
tn−2 . . . (x−1n−1)
t(x−1n−1)A
n = 1.
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 7
This follows from the fact that the characteristic polynomial of A is the cyclotomic poly-
nomial p(z) = 1+ z + · · ·+ zn−1, hence we have that p(A) = 0 on the underlying module
L and so in multiplicative notation we have that ut
· · · · ·ut ·u = 1 for any u ∈ L.
Next we verify compatibility:
τ(t)d(a1) = τ(t)(x1 − 1) = x
n−1 − 1
dτ(t)(a1) = d(−x
n−1an−1) = −x
n−1(xn−1 − 1) = x
n−1 − 1.
Similarly for all 1 < k ≤ n− 1 we have that:
τ(t)d(ak) = τ(t)(xk − 1) = xk−1x
n−1 − 1
= −x−1n−1(xn−1 − 1− xk−1 − 1) = d(−x
n−1(an−1 − ak−1)) = dτ(t)(ak).
For G = Z/n, the module which gives rise to the matrix in 3.3 is the augmentation
ideal IG, which has rank equal to n − 1. The following proposition is an application of
the results in this section.
Proposition 3.4. Let G = Z/n, and assume that L is a ZG–lattice such that
L ∼= M ⊕ IGt
where M is a permutation module. Then, for any coefficient ring R, the special free
resolution K∗ over R[L] admits a compatible G–action.
Proof. This follows from applying Lemma 2.2 to Theorem 3.2 and Proposition 3.3. �
For our cohomology calculations it will be practical to use the coefficient ring R = Z(p),
where p is a prime. In this situation, for G = Z/p (see [10]) there are only three distinct
isomorphism classes of indecomposable RG–lattices, namely R (the trivial module), IG
(the augmentation ideal) and RG, the group ring. Moreover, if L is any finitely generated
ZG–lattice, we can construct a ZG-homomorphism f : L′ → L such that
• L′ ∼= Zr ⊕ ZGs ⊕ IGt
• f is an isomorphism after tensoring with R.
We shall call L′ a representation of type (r, s, t).
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 8
4. Applications to Cohomology
We are now ready to prove our main result.
Theorem 4.1. Let G = Z/p, where p is any prime. If L is any finitely generated ZG–
lattice, and Γ = L⋊G is the associated semi–direct product group, then for each k ≥ 0
Hk(Γ,Z) ∼=
i+j=k
H i(G,∧j(L∗))
where ∧j(L∗) denotes the j-th exterior power of the dual module L∗ = Hom(L,Z).
Proof. First, let us prove the analogous result for the cohomology with coefficients in
R = Z(p). We make the assumption that L is a module of type (r, s, t). We need to verify:
Hk(Γ, R) ∼=
i+j=k
H i(G,∧j(L∗R))
where L∗R = L
∗ ⊗ R. In fact, we see that in the associated Lyndon-Hochschild-Serre
spectral sequence for the extension 0 → L → Γ → G → 1 with
2 (R) = H
i(G,Hj(L,R)) ⇒ Hk(Γ, R)
there are no differentials and no extension problems. This follows from applying Theorem
2.4 and the fact that the module L gives rise to a special resolution with a compatible
action by Proposition 3.4.
Now let us consider the case when L is not of type (r, s, t). As observed previously, we
can construct a ZG–lattice L′ and a map f : L′ → L such that L′ is of type (r, s, t) and
f is an isomorphism after tensoring with R. Under these conditions f will induce a map
between the spectral sequences with R–coefficients for the extensions corresponding to L
and L′. However by our hypotheses, ∧k(L′∗R) and ∧
k(L∗R) are isomorphic as RG–modules
for all k ≥ 0, with the isomorphism induced by f . Hence the corresponding E2–terms are
isomorphic, and so the spectral sequences both collapse and the result follows.
It now remains to prove the result with coefficients in the integers Z. Note that by
the universal coefficient theorem, we have H∗(Γ,Z(p)) ∼= H
∗(Γ,Z) ⊗ Z(p) hence the only
relevant discrepancy between H∗(Γ,Z(p)) and H
∗(Γ,Z) might arise from the presence of
torsion prime to p in the integral cohomology of Γ. However, a quick inspection of the
spectral sequence of the extension 0 → L → Γ → G → 1 with Z(q) coefficients shows that
there is no torsion prime to p in the cohomology, as L is free abelian and G is a p–group.
This completes our proof. �
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 9
In [2], Corollary 3.3, it was observed that the spectral sequence for the extension L⋊G
satisfies a collapse at E2 wthout extension problems if the same is true for all the restricted
extensions L ⋊ Gp, where the Gp ⊂ G are the p–Sylow subgroups of G. We obtain the
following
Corollary 4.2. Let G denote a finite group of square–free order, and L any finitely
generated ZG–lattice. Then for all k ≥ 0 we have
Hk(L⋊G,Z) ∼=
i+j=k
H i(G,∧j(L∗)).
We now consider a more geometric situation. Suppose that the group G = Z/p acts on
a space X which has the homotopy type of a product of circles.
Theorem 4.3. Let G = Z/p, where p is any prime. Suppose G acts on a space X
homotopy equivalent to (S1)n with XG 6= ∅, then for each k ≥ 0
Hk(EG×G X,Z) ∼=
i+j=k
H i(G,Hj(X,Z)) ∼= Hk(Γ,Z)
where Γ = π1(X)⋊G.
Proof. The space EG×GX fits into a fibration X →֒ EG×GX → BG which has a section
due to the fact that XG 6= ∅. Let Γ denote the fundamental group of EG×GX . The long
exact sequence for the homotopy groups of the fibration gives rise to a split extension
1 → π1(X) → Γ → G → 1. Since π1(X) ∼= L, a ZG–lattice, this shows that Γ ∼= L ⋊G,
where the G action is induced on L via the action on the fiber. Note that EG ×G X is
an Eilenberg-MacLane space of type K(Γ, 1). Hence, H∗(EG ×G X,Z) ∼= H
∗(Γ,Z) and
the result follows from Theorem 4.1.
Note that a special case of this result was proved in [1], namely for actions where
π1(X)⊗ Z(p) is isomorphic to a direct sum of indecomposables of rank p− 1. The terms
H i(Z/p,∧j(L∗)) can be computed if L is known up to isomorphism. In fact, all we need
is to know L up to Z/p cohomology, as this will determine its indecomposable factors (at
least up to Z(p)–equivalence).
As we mentioned in the introduction, our results complete the calculation for the co-
homology of crystallographic groups with prime order holonomy when combined with
previous work on the torsion–free case (Bieberbach groups). The terms appearing in the
formulas in Theorem 1.3 can be explicitly computed, as was observed in [8].
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 10
5. Extensions to Other Groups
In this section we explore to what degree our results can be extended to other groups.
In the case of the cyclic group of order four, the indecomposable integral representations
are easy to describe, so it is a useful test case.
Let G = Z/4, from [3] we can give a complete list of all (nine) indecomposable pair-
wise nonequivalent integral representations by the following adopted table, where a is a
generator for Z/4:
ρ1 : a → 1; ρ2 : a → −1; ρ3 : a →
; ρ4 : a →
ρ5 : a →
0 0 −1
1 0 −1
0 1 −1
; ρ6 : a →
0 1 0
−1 0 1
0 0 1
ρ7 : a →
0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0
; ρ8 : a →
0 0 −1 1
1 0 −1 1
0 1 −1 0
0 0 0 1
ρ9 : a →
0 1 0 0
−1 0 0 1
0 0 −1 1
0 0 0 1
Theorem 5.1. Let G = Z/4 and L a finitely generated ZG–lattice. If L is a direct sum
of indecomposables of type ρi for i ≤ 7, and i 6= 6, then there is a compatible action and
Hk(L⋊G,Z) =
i+j=k
H i(G,Hj(L,Z)).
Proof. For the indecomposables ρ1, ρ2, ρ3, ρ4, compatible actions are known to exist on
the associated resolutions by the results in [2]. The same is true3 for ρ5 and ρ7 by 3.3 and
3.2. Hence if L is any integral representation expressed as a direct sum of ρi, i ≤ 7 and
i 6= 6, the result follows from 2.2 and 2.4. �
3In fact ρ5 corresponds to the dual module IG
∗, but for cyclic groups there is an isomorphism IG ∼=
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 11
In the case of ρ6, ρ8 and ρ9, a compatible action is not known to exist. However, via an
explicit computation done in [14], we can establish the collapse of the spectral sequence
for the extension Γ6 associated to ρ6, yielding
H i(Γ6,Z) =
Z if i = 0, 1
Z/4⊕ Z if i = 2
Z/2⊕ Z if i = 3
Z/4⊕ Z/2 if i ≥ 4
which verifies the statement analogous to 5.1 for the cohomology of Γ6.
Indeed, for all the examples of semidirect products we have considered so far, there is
a collapse at E2 in the Lyndon-Hochschild-Serre spectral sequence of the group extension
0 → L → G⋊ L → G → 1 and therefore we can make the following:
Conjecture 5.2. Suppose that G is a finite cyclic group and L a finitely generated ZG–
lattice; then for any k ≥ 0 we have
Hk(G⋊ L,Z) =
i+j=k
H i(G,Hj(L,Z)).
In [15] examples were given of semi-direct products of the form L ⋊ (Z/p)2 where the
associated mod p Lyndon-Hochschild-Serre spectral sequence has non–zero differentials.
This relies on the fact that for G = Z/p × Z/p, there exist ZG–modules M which are
not realizable as the cohomology of a G–space. These are the counterexamples to the
Steenrod Problem given by G.Carlsson (see [5]), where M can be identified with L∗. No
such counterexamples exist for finite cyclic groups Z/N , which means that disproving our
conjecture will require a different approach.
We should also mention that by using results due to Nakaoka (see [12], pages 19 and
50) we know that the spectral sequence for a wreath product Zn ⋊ G, where G ⊂ Σn
(the symmetric group) acting on Zn via permutations will always collapse at E2, without
extension problems. This can be interpreted as the fact that a strong collapse theorem
holds for all permutation modules and all finite groups G. A simple proof of this result
can be obtained by applying Theorem 2.4 to Proposition 3.2.
As suggested by [15], the results here can be considered part of a very general problem,
which is both interesting and quite challenging:
Problem: Given a finite group G, find suitable conditions on a ZG–lattice L so that the
spectral sequence for L⋊G collapses at E2.
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 12
6. Application to Computations for Toroidal Orbifolds
Interesting examples arise from calculations for six–dimensional orbifolds, where the
usual spectral sequence techniques become rather complicated. Here our methods provide
an important new ingredient that allows us to compute rigorously beyond the known
range. An important class of examples in physics arises from actions of a cyclic group G =
Z/N on T6. In our scheme, these come from six-dimensional integral representations of
Z/N . However, the constraints from physics impose certain restrictions on them (see [11],
[16]). If θ ∈ GL6(Z) is an element of order N , then it can be diagonalized over the complex
numbers. The associated eigenvalues, denoted α1, α2, α3, should satisfy α1α2α3 = 1, and
in addition all of the αi 6= 1. The first condition implies that the orbifold T
6 → T6/G
is a Calabi–Yau orbifold, and so admits a crepant resolution. These more restricted
representations have been classified4 in [11], where it is shown that there are precisely 18
inequivalent lattices of this type.
It turns out that calculations are focused on computing the equivariant cohomology
H∗(EG ×G T
6,Z) (see [2] and [4] for more details). As was observed in [2], we can
compute the group of gerbes associated to the orbifold X = [T6/G] via the isomorphism
Gb(X ) ∼= H3(EG×G T
6,Z) ∼= H3(Z6 ⋊G,Z),
whence our methods can be used to obtain some fairly complete results in this setting.
Before proceeding we recall that as in Corollary 4.2 the collapse of the spectral sequence
for an extension L ⋊ G will follow from the existence of compatible Sylp(G) actions on
the Koszul complex K∗ for every prime p dividing |G|. If these exist we shall say that K∗
admits a local compatible action.
Theorem 6.1. Among the 18 inequivalent integral representations associated to the six–
dimensional orbifolds T6/Z/N described above, only two of them are not known to admit
(local) compatible actions. Hence for those 16 examples there is an isomorphism
Hk(EZ/N ×Z/N T
6,Z) ∼=
i+j=k
H i(Z/N,Hj(T6,Z))
Proof. Consider the defining matrix of an indecomposable action of Z/N on Zn with
determinant one, expressed in canonical form as
4In the language of physics, they show that there exist 18 inequivalent N = 1 supersymmetric string
theories on symmetric orbifolds of (2, 2)–type without discrete background.
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 13
0 . . . 0 v1
1 0 . . . 0 v2
0 1 0 . . . 0 v3
. . .
0 . . . 1 vn
where v1 = ±1.
In [11] it was determined that the matrices that specify the indecomposable modules
appearing as summands for the N = 1 supersymmetric Z/N -orbifolds can be given as
follows, where the vectors represent the values (v1, v2, . . . , vn):
Indecomposable matrices relevant for N = 1 supersymmetry
n = 1 n = 2 n = 3 n = 4
Z/2(1) : (−1) Z/3(2) : (−1,−1) Z/4(3) : (−1,−1,−1) Z/6(4) : (−1, 0,−1, 0)
Z/4(2) : (−1, 0) Z/6(3) : (−1, 0, 0) Z/8(4) : (−1, 0, 0, 0)
Z/6(2) : (−1, 1) Z/12(4) : (−1, 0, 1, 0)
n = 5 n = 6
Z/6(5) : (−1,−1,−1,−1,−1) Z/7(6) : (−1,−1,−1,−1,−1,−1)
Z/8(5) : (−1,−1, 0, 0,−1) Z/8(6) : (−1, 0,−1, 0,−1, 0)
Z/12(6) : (−1,−1, 0, 1, 0,−1)
We will show that all of these, except possibly Z/8(5) and Z/12(6), admit local compat-
ible actions. The examples of rank two or less were dealt with in [2]; for N = 2, 3, 6, 7 the
result follows directly from 4.1 and 4.2. The case Z/4(3) was covered in 5.1. We will deal
explicitly with the cases Z/8(4), Z/12(4) and Z/8(6).
(1) The group Z/8 acts on Z4 with generator represented by the matrix:
0 0 0 −1
1 0 0 0
0 1 0 0
0 0 1 0
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 14
We define a compatible action by the following formulas:
τ(t)(a1) = −x
4 a4, τ(t)(a2) = a1, τ(t)(a3) = a2, τ(t)(a4) = a3.
(2) The group Z/12 acts on Z4 with generator represented by the matrix:
0 0 0 −1
1 0 0 0
0 1 0 1
0 0 1 0
For this example it suffices to construct a compatible action for Z/4 (with generator rep-
resented by the matrix T 3) as we already know that a compatible action exists restricted
to Z/3. Now we have that Z/4 acts on Z4 with
T 3 =
0 −1 0 −1
0 0 −1 0
0 1 0 0
1 0 1 0
which is a matrix whose square is −I. This implies that the module is a sum of two copies
of the faithful rank two indecomposable (see §5), for which a compatible action is known
to exist (as explained in Theorem 5.1), and so this case is taken care of.
(3) The group Z/8 acts on Z6 with generator represented by the matrix:
0 0 0 0 0 −1
1 0 0 0 0 0
0 1 0 0 0 −1
0 0 1 0 0 0
0 0 0 1 0 −1
0 0 0 0 1 0
The formulas for a compatible action are given by
τ(t)(a1) = −x
6 a6, τ(t)(a2) = a1, τ(t)(a3) = x
6 (a2 − a6)
τ(t)(a4) = a3, τ(t)(a5) = x
6 (a4 − a6), τ(t)(a6) = a5.
We have shown that (local) compatible actions exist for all representations constructed
using indecomposables other than Z/8(5) and Z/12(6). However, these indecomposables
can only appear once in the list due to dimensional constraints, namely in the form
Z/8(5) ⊕ Z/2(1) and Z/12(6) itself. Thus our proof is complete. �
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 15
References
[1] Adem, A., Z/pZ actions on (Sn)k, Trans. Amer. Math. Soc. 300 (1987), no. 2, 791–809.
[2] Adem, A. and Pan, J., Toroidal Orbifolds, Gerbes and Group Cohomology, Trans. Amer. Math.
Soc. 358 (2006), 3969-3983.
[3] Berman, S. and Gudikov, P. Indecomposable representations of finite groups over the ring of
p-adic integers, Izv. Akad. Nauk. SSSR 28 (1964), 875–910.
[4] de Boer, J., Dijkgraaf, R., Hori, K., Keurentjes, A., Morgan, J., Morrison, D. and Sethi, S.,
Triples, Fluxes, and Strings, Adv. Theor. Math. Phys. 4 (2000), no. 5, 995–1186.
[5] Carlsson, G., A counterexample to a conjecture of Steenrod, Inv. Math. 64 (1981), no. 1, 171–174.
[6] Cartan, H. and Eilenberg, S., Homological Algebra, Oxford University Press, Oxford, 1956.
[7] Charlap, L., Bieberbach Groups and Flat Manifolds, Universitext, Springer–Verlag, Berlin,
1986.
[8] Charlap, L. and Vasquez, A., Compact Flat Riemannian Manifolds II: the Cohomology of Zp–
manifolds, Amer. J. Math. 87 (1965), 551–563. Trans, Amer. Math. Soc.
[9] Brady, T., Free resolutions for semi-direct products, Tohoku Math. J. (2) 45 (1993), no. 4, 535–
[10] Curtis, C.W. and Reiner, I., Representation Theory of Finite Groups and Associative
Algebras Wiley-Interscience (1987).
[11] Erler, J. and Klemm, A., Comment on the generation number in orbifold compactifications,
Comm. Math. Phys. 153 (1993), 579–604.
[12] Evens, L., Cohomology of groups, Oxford Mathematical Monographs, Oxford University Press
(1991).
[13] Joyce, D., Deforming Calabi-Yau orbifolds, Asian J. Math. 3 (1999), no. 4, 853–867.
[14] Petrosyan, N. Jumps in cohomology of groups, periodicity, and semi–direct products, Ph.D. Dis-
sertation, University of Wisconsin–Madison (2006).
[15] Totaro, B., Cohomology of Semidirect Product Groups, J. of Algebra 182 (1996), 469–475.
[16] Vafa, C. and Witten, E., On orbifolds with discrete torsion, J. Geom. Phys. 15 (1995), no. 3,
189–214.
Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2,
Canada
E-mail address : adem@math.ubc.ca
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
E-mail address : gejq04@mails.tsinghua.edu.cn
Institute of Mathematics, Academia Sinica, Beijing 100080, China
E-mail address : pjz@amss.ac.cn
Department of Mathematics, Indiana University, Bloomington IN 47405, USA
E-mail address : nanpetro@indiana.edu
	1. Introduction
	2. Preliminary Results
	3. Construction of Compatible Actions
	4. Applications to Cohomology
	5. Extensions to Other Groups
	6. Application to Computations for Toroidal Orbifolds
	References

  We compute the cohomology of crystallographic groups with holonomy of prime
order. As an application we compute the group of gerbes associated to many
six--dimensional toroidal orbifolds arising in string theory.

Introduction
Given a finite group G and an integral representation L for G (i.e. a homomorphism
G → GLn(Z), where L is the underlying ZG–module), we can define the semi–direct
product Γ = L⋊G. Calculating the cohomology of these groups is a problem of instrinsic
algebraic interest; indeed if the representation is faithful then these groups can be thought
of as crystallographic groups (see [7], page 74).
From the geometric point of view, the action on L gives rise to a G–action on the
n–torus X = Tn; this approach can be used to derive important examples of orbifolds,
known as toroidal orbifolds (see [2]). In the case when n = 6 these are of particular
interest in string theory (see [4], [11]).
Given the split group extension 0 → L → Γ → G → 1, the basic problem which
we address is that of providing conditions which imply the collapse (without extension
problems) of the associated Lyndon–Hochshild–Serre spectral sequence. The conditions
which we establish are representation–theoretic, namely depending solely on the structure
of the integral representation L. This can be a difficult problem (see [15] for further
background) and there are well–known examples where the spectral sequence does not
collapse.
Our approach is to systematically apply the methods used in [2]. The key idea is to
construct a free resolution F for the semidirect product L ⋊ G such that the Lyndon–
Hochschild–Serre spectral sequence of the group extension collapses at E2. This requires
Date: October 30, 2018.
Key words and phrases. spectral sequence, group cohomology.
∗The first author was partially supported by the NSF and by NSERC.
http://arxiv.org/abs/0704.1823v1
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 2
a chain–level argument, more specifically the construction of a compatible G–action on
a certain free resolution for the torsion–free abelian group L (see §2 for details). We
concentrate on the case of G = Z/p, a cyclic group of prime order p, as the representation
theory is well–understood. Our main algebraic result is the following
Theorem 1.1. Let G = Z/p, where p is any prime. If L is any finitely generated ZG–
lattice1, and Γ = L⋊G is the associated semi–direct product group, then for each k ≥ 0
Hk(Γ,Z) ∼=
i+j=k
H i(G,∧j(L∗))
where ∧j(L∗) denotes the j-th exterior power of the dual module L∗ = Hom(L,Z).
Expressed differently: these results imply a complete calculation for the integral coho-
mology of crystallographic groups Zn ⋊ Z/p where p is prime. These calculations can be
made explicit.
The theorem has an interesting geometric application:
Theorem 1.2. Let G = Z/p, where p is any prime. Suppose that G acts on a space X
homotopy equivalent to (S1)n with XG 6= ∅, then for each k ≥ 0
Hk(EG×G X,Z) ∼=
i+j=k
H i(G,Hj(X,Z)) ∼= Hk(Γ,Z).
where Γ = π1(X)⋊G.
On the other hand, the explicit computation for torsion–free crystallographic groups
with holonomy of prime order was carried out long ago by Charlap and Vásquez (see [8],
page 556). Combining the two results we obtain a complete calculation:
Theorem 1.3. Let Γ denote a crystallographic group with holonomy of prime order p,
expressed as an extension
1 → L → Γ → Z/p → 1
where L is a free abelian group of finite rank.
(1) If Γ is torsion–free, then L ∼= N ⊕ Z (it splits off a trivial direct summand) and
Hk(Γ,Z) ∼= H0(Z/p,∧k(L∗))⊕H1(Z/p,∧k−1(N∗))
for 0 ≤ k ≤ rk(L); Hk(Γ,Z) = 0 for k > rk(L).
(2) If Γ is not torsion–free, then H∗(Γ,Z) can be computed using Theorem 1.1.
1A ZG–lattice is a ZG–module which happens to be a free abelian group.
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 3
In this paper we also consider the situation for the cyclic group of order four; some
partial results are obtained but a general collapse has not been established. However,
based on these and other computations we conjecture that for G any cyclic group, the
spectral sequence associated to a semi–direct product of the form Zn ⋊ G must collapse
at E2.
In the last section we give an application of our methods to calculations for six–
dimensional toroidal orbifolds, showing that among the 18 inequivalent N = 1 supersym-
metric string theories on symmetric orbifolds of (2, 2)–type without discrete background,
only two of them cannot be analyzed using our methods i.e. we cannot show the existence
of compatible actions for the associated modules. If X = [T6/G] is an orbifold arising
this way, then our results provide a complete calculation for its associated group of gerbes
Gb(X ) ∼= H3(EG×G T
6,Z) (see [2] for more details).
2. Preliminary Results
The notion of a compatible action was first introduced in [9]. If such an action exists
it allows one to construct practical projective resolutions and from these to compute the
cohomology of the group. We will give the basic definition and the main theorem that
follows. More details can be found in [2].
Let Γ = L ×ρ G = L ⋊ G be the semidirect product of a finite group G and a finite
dimensional Z-lattice L via a representation ρ : G → GL(L). G acts on the group L by the
homomorphism ρ, and this extends linearly to an action on the group algebra R[L], where
R denotes a commutative ring with unit. We write lg for ρ(g)l where l ∈ R[L], g ∈ G.
In the rest of this paper R will represent Z (the integers) or Z(p) (the ring of integers
localized at a fixed prime p).
Definition 2.1. Given a free resolution ǫ : F → R of R over R[L], we say that it admits
an action of G compatible with ρ if for all g ∈ G there is an augmentation-preserving
chain map τ(g) : F → F such that
(1) τ(g)[l · f ] = lg · [τ(g)f ] for all l ∈ R[L] and f ∈ F ,
(2) τ(g)τ(g′) = τ(gg′) for all g, g′ ∈ G,
(3) τ(1) = 1F .
The following two lemmas (see [2]) reduce the construction of compatible actions to the
case of faithful indecomposable representations.
Lemma 2.2. If ǫi : Fi → R is a projective R[Li]-resolution of R for i = 1, 2, then
ǫ1 ⊗ ǫ2 : F1 ⊗ F2 → R is a projective R[L1 × L2]-resolution of R. Furthermore, if G acts
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 4
compatibly on Fi by τi for i = 1, 2, then a compatible action of G on ǫ1⊗ ǫ2 : F1⊗F2 → R
is given by τ(g)(f1 ⊗ f2) = τ1(g)(f1)⊗ τ2(g)(f2).
Lemma 2.3. If L is a R[G1]-module, π : G2 → G1 a group homomorphism, and ǫ : F → R
is a R[L]-resolution of R such that G1 acts compatibly on it by τ
′, then G2 also acts
compatibly on it by τ(g)f = τ ′(π(g))f for any g ∈ G.
If a compatible action exists, we can give F a Γ-module structure as follows. An
element γ ∈ Γ can be expressed uniquely as γ = lg, with l ∈ L and g ∈ G. We set
γ · f = (lg) · f = l · τ(g)f . Note that given any G–module M , this inflates to a Γ–action
on M via the projection Γ → G.
We can always construct a special free resolution F of R over L, characterized by the
property that the cochain complex HomL(F,R) for computing the cohomology H
∗(L,R)
has all coboundary maps zero (more details will be provided in the next section). Using
this fact, the following was proved in [2]:
Theorem 2.4. (Adem-Pan) Let ǫ : F → R be a special free resolution of R over L and
suppose that there is a compatible action of G on F . Then for all integers k ≥ 0, we have
Hk(L⋊G,R) =
i+j=k
H i(G,Hj(L,R)).
This result can be interpreted as saying that the Lyndon-Hochschild-Serre spectral
sequence
2 = H
p(G,Hq(L,R)) ⇒ Hp+q(L⋊G,R)
collapses at E2 without extension problems. Note that this is not always the case; in fact
there are examples of semi–direct products of the form Zn ⋊ (Z/p)2 where the associated
spectral sequence has non–trivial differentials (see [15]). This will be discussed in §5.
3. Construction of Compatible Actions
Let R[L] denote the group ring of L, a free abelian group with basis {x1, . . . , xn}.
Then the elements x1 − 1, . . . , xn − 1 form a regular sequence in R[L], hence the Koszul
complex K∗ = K(x1−1, . . . , xn−1) is a free resolution of the trivial module R. It has the
additional property of being a differential graded algebra (or DGA). We briefly recall how
it looks. There are generators a1, . . . , an in degree one, and the graded basis for K∗ can
be identified with the usual basis for the exterior algebra they generate. The differential
is given by the following formula: if ai1...ip = ai1 . . . aip is a basis element in Kp, then
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 5
d(ai1...ip) =
(−1)j−1(xij − 1)ai1...îj ...ip.
Now the cohomology of the free abelian group L is precisely an exterior algebra on n one-
dimensional generators, which in fact can be identified with the dual elements a∗1, . . . , a
In particular we see that the cochain complex HomR[L](K∗, R) has zero differentials, and
hence K∗ is a special free resolution of R over R[L] (this resolution also appears in [7]
pp. 96–97).
We now consider how to construct a compatible G–action on K∗, given a G–module
structure on L.
Theorem 3.1. If G acts on the lattice L, let K∗ = K(x1 − 1, . . . xn − 1) denote the
special free resolution of R over R[L] defined using the Koszul complex associated to the
elements x1 − 1, . . . , xn − 1, where {x1, . . . , xn} form a basis for L. Suppose that there is
a homomorphism τ : G → Aut(K1) such that for every g ∈ G and a ∈ K1 it satisfies
dτ(g)(a) = d(a)g
where d : K1 → K0 is the usual Koszul differential, and d(a)
g ∈ K0 = R[L]. Then τ
extends to K∗ using its DGA structure and so defines a compatible G–action on K∗.
Proof. First we observe that τ(g) acts on K0 = R[L] via the original G–action, i.e.
τ(g)(x) = xg for any x ∈ K0. Next we define the action on the basis of K∗ as a graded
R[L]–module, namely:
τ(g)(ai1 . . . aip) = τ(g)(ai1) . . . τ(g)(aip).
If α ∈ R[L] and u ∈ K∗, we define τ(g)(αu) = α
gτ(g)(x). By linearity and the DGA
structure of K∗, this will define τ : G → Aut(K∗), with the desired properties. �
Generally speaking it can be quite difficult to construct a compatible action; however
there is an important special case where it is quite straightforward.
Theorem 3.2. Let φ : G → Σn denote a group homomorphism, where Σn denotes the
symmetric group on n elements. Let G act on Zn via2 this homomorphism. Then the
associated Koszul complex K∗ admits a compatible G–action.
2Such a module is called a permutation module.
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 6
Proof. By Lemma 2.3 we can assume that G is a subgroup of Σn, hence it will suffice to
prove this for Σn itself. If we take generators a1, . . . , an for the Koszul complex corre-
sponding to the elements x1, . . . , xn in the underlying module L, then we can define τ as
follows:
τ(σ)(ai) = aσ(i).
This obviously defines a permutation representation onK1, and compatibility follows from
the fact that for all ai, 1 ≤ i ≤ n and σ ∈ Σ we have
dτ(σ)(ai) = d(aσ(i)) = xσ(i) − 1 = (xi − 1)
This completes the proof. �
Aside from permutation representations, it is difficult to construct general examples
of compatible actions. However if G is a cyclic group then we can handle an important
additional type of module.
Proposition 3.3. Let the cyclic group G = 〈t|tn = 1〉 act on Zn−1 by:
ξ1 : t 7→
0 1 0 . . . 0 0
0 0 1 . . . 0 0
. . .
0 0 0 . . . 1 0
−1 −1 −1 . . . −1 −1
∈ GLn−1(Z).
If x1, . . . , xn is the canonical basis under which the action is represented by the matrix
above, then the free resolution K∗ = K∗(x1−1, . . . , xn−1) admits an action of G compatible
with ξ1, which can be defined by:
τ(t)(a1) = −x
n−1an−1 τ(t)(ak) = −x
n−1(an−1 − ak−1), 1 < k ≤ n− 1.
Proof. The proof is a straightforward calculation verifying that τ defines a compatible
action. First we verify that τn = 1. For this we observe that if A is the matrix in GLn−1(Z)
representing the generator t, then expressed in terms of the basis {a1, . . . , an} we have
that τ(t) = x−1n−1A. If we iterate this action and use the fact that τ(g)(αu) = α
gτ(g)(u)
then we obtain
τ(t)n = (x−1n−1)
tn−1(x−1n−1)
tn−2 . . . (x−1n−1)
t(x−1n−1)A
n = 1.
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 7
This follows from the fact that the characteristic polynomial of A is the cyclotomic poly-
nomial p(z) = 1+ z + · · ·+ zn−1, hence we have that p(A) = 0 on the underlying module
L and so in multiplicative notation we have that ut
· · · · ·ut ·u = 1 for any u ∈ L.
Next we verify compatibility:
τ(t)d(a1) = τ(t)(x1 − 1) = x
n−1 − 1
dτ(t)(a1) = d(−x
n−1an−1) = −x
n−1(xn−1 − 1) = x
n−1 − 1.
Similarly for all 1 < k ≤ n− 1 we have that:
τ(t)d(ak) = τ(t)(xk − 1) = xk−1x
n−1 − 1
= −x−1n−1(xn−1 − 1− xk−1 − 1) = d(−x
n−1(an−1 − ak−1)) = dτ(t)(ak).
For G = Z/n, the module which gives rise to the matrix in 3.3 is the augmentation
ideal IG, which has rank equal to n − 1. The following proposition is an application of
the results in this section.
Proposition 3.4. Let G = Z/n, and assume that L is a ZG–lattice such that
L ∼= M ⊕ IGt
where M is a permutation module. Then, for any coefficient ring R, the special free
resolution K∗ over R[L] admits a compatible G–action.
Proof. This follows from applying Lemma 2.2 to Theorem 3.2 and Proposition 3.3. �
For our cohomology calculations it will be practical to use the coefficient ring R = Z(p),
where p is a prime. In this situation, for G = Z/p (see [10]) there are only three distinct
isomorphism classes of indecomposable RG–lattices, namely R (the trivial module), IG
(the augmentation ideal) and RG, the group ring. Moreover, if L is any finitely generated
ZG–lattice, we can construct a ZG-homomorphism f : L′ → L such that
• L′ ∼= Zr ⊕ ZGs ⊕ IGt
• f is an isomorphism after tensoring with R.
We shall call L′ a representation of type (r, s, t).
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 8
4. Applications to Cohomology
We are now ready to prove our main result.
Theorem 4.1. Let G = Z/p, where p is any prime. If L is any finitely generated ZG–
lattice, and Γ = L⋊G is the associated semi–direct product group, then for each k ≥ 0
Hk(Γ,Z) ∼=
i+j=k
H i(G,∧j(L∗))
where ∧j(L∗) denotes the j-th exterior power of the dual module L∗ = Hom(L,Z).
Proof. First, let us prove the analogous result for the cohomology with coefficients in
R = Z(p). We make the assumption that L is a module of type (r, s, t). We need to verify:
Hk(Γ, R) ∼=
i+j=k
H i(G,∧j(L∗R))
where L∗R = L
∗ ⊗ R. In fact, we see that in the associated Lyndon-Hochschild-Serre
spectral sequence for the extension 0 → L → Γ → G → 1 with
2 (R) = H
i(G,Hj(L,R)) ⇒ Hk(Γ, R)
there are no differentials and no extension problems. This follows from applying Theorem
2.4 and the fact that the module L gives rise to a special resolution with a compatible
action by Proposition 3.4.
Now let us consider the case when L is not of type (r, s, t). As observed previously, we
can construct a ZG–lattice L′ and a map f : L′ → L such that L′ is of type (r, s, t) and
f is an isomorphism after tensoring with R. Under these conditions f will induce a map
between the spectral sequences with R–coefficients for the extensions corresponding to L
and L′. However by our hypotheses, ∧k(L′∗R) and ∧
k(L∗R) are isomorphic as RG–modules
for all k ≥ 0, with the isomorphism induced by f . Hence the corresponding E2–terms are
isomorphic, and so the spectral sequences both collapse and the result follows.
It now remains to prove the result with coefficients in the integers Z. Note that by
the universal coefficient theorem, we have H∗(Γ,Z(p)) ∼= H
∗(Γ,Z) ⊗ Z(p) hence the only
relevant discrepancy between H∗(Γ,Z(p)) and H
∗(Γ,Z) might arise from the presence of
torsion prime to p in the integral cohomology of Γ. However, a quick inspection of the
spectral sequence of the extension 0 → L → Γ → G → 1 with Z(q) coefficients shows that
there is no torsion prime to p in the cohomology, as L is free abelian and G is a p–group.
This completes our proof. �
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 9
In [2], Corollary 3.3, it was observed that the spectral sequence for the extension L⋊G
satisfies a collapse at E2 wthout extension problems if the same is true for all the restricted
extensions L ⋊ Gp, where the Gp ⊂ G are the p–Sylow subgroups of G. We obtain the
following
Corollary 4.2. Let G denote a finite group of square–free order, and L any finitely
generated ZG–lattice. Then for all k ≥ 0 we have
Hk(L⋊G,Z) ∼=
i+j=k
H i(G,∧j(L∗)).
We now consider a more geometric situation. Suppose that the group G = Z/p acts on
a space X which has the homotopy type of a product of circles.
Theorem 4.3. Let G = Z/p, where p is any prime. Suppose G acts on a space X
homotopy equivalent to (S1)n with XG 6= ∅, then for each k ≥ 0
Hk(EG×G X,Z) ∼=
i+j=k
H i(G,Hj(X,Z)) ∼= Hk(Γ,Z)
where Γ = π1(X)⋊G.
Proof. The space EG×GX fits into a fibration X →֒ EG×GX → BG which has a section
due to the fact that XG 6= ∅. Let Γ denote the fundamental group of EG×GX . The long
exact sequence for the homotopy groups of the fibration gives rise to a split extension
1 → π1(X) → Γ → G → 1. Since π1(X) ∼= L, a ZG–lattice, this shows that Γ ∼= L ⋊G,
where the G action is induced on L via the action on the fiber. Note that EG ×G X is
an Eilenberg-MacLane space of type K(Γ, 1). Hence, H∗(EG ×G X,Z) ∼= H
∗(Γ,Z) and
the result follows from Theorem 4.1.
Note that a special case of this result was proved in [1], namely for actions where
π1(X)⊗ Z(p) is isomorphic to a direct sum of indecomposables of rank p− 1. The terms
H i(Z/p,∧j(L∗)) can be computed if L is known up to isomorphism. In fact, all we need
is to know L up to Z/p cohomology, as this will determine its indecomposable factors (at
least up to Z(p)–equivalence).
As we mentioned in the introduction, our results complete the calculation for the co-
homology of crystallographic groups with prime order holonomy when combined with
previous work on the torsion–free case (Bieberbach groups). The terms appearing in the
formulas in Theorem 1.3 can be explicitly computed, as was observed in [8].
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 10
5. Extensions to Other Groups
In this section we explore to what degree our results can be extended to other groups.
In the case of the cyclic group of order four, the indecomposable integral representations
are easy to describe, so it is a useful test case.
Let G = Z/4, from [3] we can give a complete list of all (nine) indecomposable pair-
wise nonequivalent integral representations by the following adopted table, where a is a
generator for Z/4:
ρ1 : a → 1; ρ2 : a → −1; ρ3 : a →
; ρ4 : a →
ρ5 : a →
0 0 −1
1 0 −1
0 1 −1
; ρ6 : a →
0 1 0
−1 0 1
0 0 1
ρ7 : a →
0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0
; ρ8 : a →
0 0 −1 1
1 0 −1 1
0 1 −1 0
0 0 0 1
ρ9 : a →
0 1 0 0
−1 0 0 1
0 0 −1 1
0 0 0 1
Theorem 5.1. Let G = Z/4 and L a finitely generated ZG–lattice. If L is a direct sum
of indecomposables of type ρi for i ≤ 7, and i 6= 6, then there is a compatible action and
Hk(L⋊G,Z) =
i+j=k
H i(G,Hj(L,Z)).
Proof. For the indecomposables ρ1, ρ2, ρ3, ρ4, compatible actions are known to exist on
the associated resolutions by the results in [2]. The same is true3 for ρ5 and ρ7 by 3.3 and
3.2. Hence if L is any integral representation expressed as a direct sum of ρi, i ≤ 7 and
i 6= 6, the result follows from 2.2 and 2.4. �
3In fact ρ5 corresponds to the dual module IG
∗, but for cyclic groups there is an isomorphism IG ∼=
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 11
In the case of ρ6, ρ8 and ρ9, a compatible action is not known to exist. However, via an
explicit computation done in [14], we can establish the collapse of the spectral sequence
for the extension Γ6 associated to ρ6, yielding
H i(Γ6,Z) =
Z if i = 0, 1
Z/4⊕ Z if i = 2
Z/2⊕ Z if i = 3
Z/4⊕ Z/2 if i ≥ 4
which verifies the statement analogous to 5.1 for the cohomology of Γ6.
Indeed, for all the examples of semidirect products we have considered so far, there is
a collapse at E2 in the Lyndon-Hochschild-Serre spectral sequence of the group extension
0 → L → G⋊ L → G → 1 and therefore we can make the following:
Conjecture 5.2. Suppose that G is a finite cyclic group and L a finitely generated ZG–
lattice; then for any k ≥ 0 we have
Hk(G⋊ L,Z) =
i+j=k
H i(G,Hj(L,Z)).
In [15] examples were given of semi-direct products of the form L ⋊ (Z/p)2 where the
associated mod p Lyndon-Hochschild-Serre spectral sequence has non–zero differentials.
This relies on the fact that for G = Z/p × Z/p, there exist ZG–modules M which are
not realizable as the cohomology of a G–space. These are the counterexamples to the
Steenrod Problem given by G.Carlsson (see [5]), where M can be identified with L∗. No
such counterexamples exist for finite cyclic groups Z/N , which means that disproving our
conjecture will require a different approach.
We should also mention that by using results due to Nakaoka (see [12], pages 19 and
50) we know that the spectral sequence for a wreath product Zn ⋊ G, where G ⊂ Σn
(the symmetric group) acting on Zn via permutations will always collapse at E2, without
extension problems. This can be interpreted as the fact that a strong collapse theorem
holds for all permutation modules and all finite groups G. A simple proof of this result
can be obtained by applying Theorem 2.4 to Proposition 3.2.
As suggested by [15], the results here can be considered part of a very general problem,
which is both interesting and quite challenging:
Problem: Given a finite group G, find suitable conditions on a ZG–lattice L so that the
spectral sequence for L⋊G collapses at E2.
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 12
6. Application to Computations for Toroidal Orbifolds
Interesting examples arise from calculations for six–dimensional orbifolds, where the
usual spectral sequence techniques become rather complicated. Here our methods provide
an important new ingredient that allows us to compute rigorously beyond the known
range. An important class of examples in physics arises from actions of a cyclic group G =
Z/N on T6. In our scheme, these come from six-dimensional integral representations of
Z/N . However, the constraints from physics impose certain restrictions on them (see [11],
[16]). If θ ∈ GL6(Z) is an element of order N , then it can be diagonalized over the complex
numbers. The associated eigenvalues, denoted α1, α2, α3, should satisfy α1α2α3 = 1, and
in addition all of the αi 6= 1. The first condition implies that the orbifold T
6 → T6/G
is a Calabi–Yau orbifold, and so admits a crepant resolution. These more restricted
representations have been classified4 in [11], where it is shown that there are precisely 18
inequivalent lattices of this type.
It turns out that calculations are focused on computing the equivariant cohomology
H∗(EG ×G T
6,Z) (see [2] and [4] for more details). As was observed in [2], we can
compute the group of gerbes associated to the orbifold X = [T6/G] via the isomorphism
Gb(X ) ∼= H3(EG×G T
6,Z) ∼= H3(Z6 ⋊G,Z),
whence our methods can be used to obtain some fairly complete results in this setting.
Before proceeding we recall that as in Corollary 4.2 the collapse of the spectral sequence
for an extension L ⋊ G will follow from the existence of compatible Sylp(G) actions on
the Koszul complex K∗ for every prime p dividing |G|. If these exist we shall say that K∗
admits a local compatible action.
Theorem 6.1. Among the 18 inequivalent integral representations associated to the six–
dimensional orbifolds T6/Z/N described above, only two of them are not known to admit
(local) compatible actions. Hence for those 16 examples there is an isomorphism
Hk(EZ/N ×Z/N T
6,Z) ∼=
i+j=k
H i(Z/N,Hj(T6,Z))
Proof. Consider the defining matrix of an indecomposable action of Z/N on Zn with
determinant one, expressed in canonical form as
4In the language of physics, they show that there exist 18 inequivalent N = 1 supersymmetric string
theories on symmetric orbifolds of (2, 2)–type without discrete background.
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 13
0 . . . 0 v1
1 0 . . . 0 v2
0 1 0 . . . 0 v3
. . .
0 . . . 1 vn
where v1 = ±1.
In [11] it was determined that the matrices that specify the indecomposable modules
appearing as summands for the N = 1 supersymmetric Z/N -orbifolds can be given as
follows, where the vectors represent the values (v1, v2, . . . , vn):
Indecomposable matrices relevant for N = 1 supersymmetry
n = 1 n = 2 n = 3 n = 4
Z/2(1) : (−1) Z/3(2) : (−1,−1) Z/4(3) : (−1,−1,−1) Z/6(4) : (−1, 0,−1, 0)
Z/4(2) : (−1, 0) Z/6(3) : (−1, 0, 0) Z/8(4) : (−1, 0, 0, 0)
Z/6(2) : (−1, 1) Z/12(4) : (−1, 0, 1, 0)
n = 5 n = 6
Z/6(5) : (−1,−1,−1,−1,−1) Z/7(6) : (−1,−1,−1,−1,−1,−1)
Z/8(5) : (−1,−1, 0, 0,−1) Z/8(6) : (−1, 0,−1, 0,−1, 0)
Z/12(6) : (−1,−1, 0, 1, 0,−1)
We will show that all of these, except possibly Z/8(5) and Z/12(6), admit local compat-
ible actions. The examples of rank two or less were dealt with in [2]; for N = 2, 3, 6, 7 the
result follows directly from 4.1 and 4.2. The case Z/4(3) was covered in 5.1. We will deal
explicitly with the cases Z/8(4), Z/12(4) and Z/8(6).
(1) The group Z/8 acts on Z4 with generator represented by the matrix:
0 0 0 −1
1 0 0 0
0 1 0 0
0 0 1 0
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 14
We define a compatible action by the following formulas:
τ(t)(a1) = −x
4 a4, τ(t)(a2) = a1, τ(t)(a3) = a2, τ(t)(a4) = a3.
(2) The group Z/12 acts on Z4 with generator represented by the matrix:
0 0 0 −1
1 0 0 0
0 1 0 1
0 0 1 0
For this example it suffices to construct a compatible action for Z/4 (with generator rep-
resented by the matrix T 3) as we already know that a compatible action exists restricted
to Z/3. Now we have that Z/4 acts on Z4 with
T 3 =
0 −1 0 −1
0 0 −1 0
0 1 0 0
1 0 1 0
which is a matrix whose square is −I. This implies that the module is a sum of two copies
of the faithful rank two indecomposable (see §5), for which a compatible action is known
to exist (as explained in Theorem 5.1), and so this case is taken care of.
(3) The group Z/8 acts on Z6 with generator represented by the matrix:
0 0 0 0 0 −1
1 0 0 0 0 0
0 1 0 0 0 −1
0 0 1 0 0 0
0 0 0 1 0 −1
0 0 0 0 1 0
The formulas for a compatible action are given by
τ(t)(a1) = −x
6 a6, τ(t)(a2) = a1, τ(t)(a3) = x
6 (a2 − a6)
τ(t)(a4) = a3, τ(t)(a5) = x
6 (a4 − a6), τ(t)(a6) = a5.
We have shown that (local) compatible actions exist for all representations constructed
using indecomposables other than Z/8(5) and Z/12(6). However, these indecomposables
can only appear once in the list due to dimensional constraints, namely in the form
Z/8(5) ⊕ Z/2(1) and Z/12(6) itself. Thus our proof is complete. �
COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 15
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[14] Petrosyan, N. Jumps in cohomology of groups, periodicity, and semi–direct products, Ph.D. Dis-
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189–214.
Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2,
Canada
E-mail address : adem@math.ubc.ca
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
E-mail address : gejq04@mails.tsinghua.edu.cn
Institute of Mathematics, Academia Sinica, Beijing 100080, China
E-mail address : pjz@amss.ac.cn
Department of Mathematics, Indiana University, Bloomington IN 47405, USA
E-mail address : nanpetro@indiana.edu
	1. Introduction
	2. Preliminary Results
	3. Construction of Compatible Actions
	4. Applications to Cohomology
	5. Extensions to Other Groups
	6. Application to Computations for Toroidal Orbifolds
	References
