ON COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS WITH CYCLIC HOLONOMY OF SPLIT TYPE

NANSEN PETROSYAN AND BARTOSZ PUTRYCZ

Abstract. We disprove a conjecture stating that the integral cohomology of any n-dimensional crystallographic group ℤ^{n} ⋊ ℤ_{m} admits a decomposition:

H^{∗}(ℤ^{n} ⋊ ℤ_{m}) ≅ ⨁_{i+j=∗} H^{i}(ℤ_{m}, H^{j}(ℤ^{n}))

by providing a complete list of counterexamples up to dimension 6. This finishes the computations of the cohomology of 6-dimensional crystallographic groups arising as orbifold fundamental groups of certain Calabi-Yau toroidal orbifolds. We also find a counterexample with odd order holonomy, m = 9, in dimension 8.

1. Introduction

An n-dimensional crystallographic group Γ is a discrete subgroup of isometries of ℝ^{n} acting properly discontinuously and cocompactly on ℝ^{n}. By the first Bieberbach theorem (see [4]), every such group has a normal subgroup L of translations which is a uniform lattice of ℝ^{n} and the holonomy group Γ/L is finite.

In [2], there is a complete structure theorem on the cohomology of crystallographic groups with cyclic holomony of prime order. When such a group Γ contains a torsion element, i.e. Γ = ℤ^{n} ⋊ ℤ_{p}, the theorem asserts that the integral cohomology of Γ is given by the cohomology of ℤ_{p} with coefficients in the cohomology of the lattice L. Also, it is conjectured that a similar decomposition holds for the cohomology of Γ = L ⋊ G for any finite cyclic group G.

Conjecture 1.1 ([2, 5.2]). Suppose that G is a finite cyclic group and L a finitely generated ℤG-lattice; then for any k ≥ 0 we have

H^{k}(L ⋊ G, ℤ) ≅ ⨁_{i+j=k} H^{i}(G, H^{j}(L, ℤ)).

We show that the conjecture already fails for a 4-dimensional crystallographic groups with holonomy ℤ_{4} (see Section 3.1). This is the lowest possible dimension of a crystallographic group for which the conjecture is not true. In Section 3.2, we also compute the cohomology of all crystallographic groups of dimensions up to 6 which do not satisfy the conjecture. There are 2 in dimension 4 both with holonomy ℤ_{4}. In dimension 5, there are 5 with holonomy ℤ_{4} and one with holonomy ℤ_{8}. In dimension 6, there are 13 with holonomy ℤ_{4}, 7 with holonomy ℤ_{8} and 9 with holonomy ℤ_{12}. Using this classification, in Section 4.1, we finish the computations of the cohomology of 6-dimensional crystallographic groups which arise as orbifold fundamental groups of certain toroidal orbifolds discussed in [2, Sec. 6]. Also, we give an example of an 8-dimensional crystallographic group with holonomy ℤ_{9} which is the first counterexample with odd order holonomy.

Our approach is straightforward, as we compute both sides of the conjectured equation and immediately observe that they are not isomorphic. The method of computations is based on the so-called twisted tensor product construction introduced by Wall in [13]. Roughly stated, given an arbitrary group extension 1 → L → Γ → G → 1 and free resolutions B_{∗} and C_{∗} of ℤ over ℤL and ℤG respectively, by inducting B_{∗} to a resolution Ind_{L}^{Γ}B_{∗} over ℤΓ and then tensoring with C_{∗} over ℤG, assuming trivial right action of G on Ind_{L}^{Γ}B_{∗}, one obtains an augmented chain complex of free ℤΓ-modules. Wall then proves that one can recursively construct new differentials of the complex to obtain an acyclic complex.

Key words and phrases. crystallographic group, group cohomology, spectral sequence.

Both authors were supported by the Research Fund K.U.Leuven.

The first author was also supported by the FWO-Flanders Research Fellowship.

This method of computing the cohomology of crystallographic groups has already been implemented in GAP (see [7, Package HAP]). In Section 2, we discuss how we adapt the algorithm to the case of crystallographic groups of split type. This shortens the computing times and allows us to find all counterexamples to the conjecture up to dimension 6.

Recently, using different methods, a 6-dimensional counterexample to the conjecture with holonomy ℤ_{4} has been found by Langer and Lück ([8, 0.6]). In fact, they show that there is a counterexample to the conjecture for every holonomy group whose order is divisible by 4. They also verify the conjecture with an extra assumption that the action of G on L is free away from the origin ([8, 0.5]).

In [9, 1.2], it is conjectured that the Lyndon-Hochschild-Serre spectral sequence associated to ℤ^{n} ⋊ ℤ_{m} collapses at E_{2} not only with integral coefficients, but more generally for all coefficient modules A that are ℤ-free of finite rank having trivial ℤ^{n}-action. We can easily show that several of the counterexamples to Conjecture 1.1 are also counterexamples to this conjecture. In Section 3.2 (see Theorem 3.2), we present a 3-dimensional counterexample to Conjecture [9, 1.2]. Interestingly, this is the first known example of a crystallographic group with cyclic holonomy of split type whose associated Lyndon-Hochschild-Serre spectral sequence collapses with integral coefficients but does not collapse for some other coefficients A.

2. The cruxes of the method

Before presenting the main steps used in our computations, first we discuss two notions that are essential to this method.

2.1. Twisted tensor product. Let 1 → L → Γ → G → 1 be an arbitrary extension of groups. Suppose (B_{r}, r ≥ 0) and (C_{s}, s ≥ 0) are free ℤL and ℤG-resolutions of ℤ, respectively and denote by ∂_{∗} the differential of C_{∗}.

The induced module Ind_{L}^{Γ}B_{∗} = ℤΓ⊗_{ℤL} B_{∗} is free over ℤΓ. Since induction is an exact functor, Ind_{L}^{Γ}B_{∗} with the differentials induced from those of B_{∗} becomes a free ℤΓ-resolution of ℤG.

Next, let us endow each module Ind_{L}^{Γ}B_{∗} with the trivial right G-action and define:

A_{r,s} := Ind_{L}^{Γ}B_{r} ⊗_{ℤG} C_{s}.

Set α_{s} = rk_{ℤG}(C_{s}) and denote by B the graded complex ⨁_{r} B_{r} and let ϵ be augmentation of B. Then

D_{s} := ⨁_{r} A_{r,s} = Ind_{L}^{Γ}B ⊗_{ℤG} C_{s}

is a direct sum of α_{s} copies of Ind_{L}^{Γ}B, which together with augmentation ϵ_{s} := (Id ⊗_{ℤL}ϵ)^{α_{s}} onto C_{s} entails a free ℤΓ-resolution of C_{s}. Lastly, we denote by d_{0} the differential of each complex D_{s} and define:

A := ⨁_{s} D_{s} = ⨁_{r,s} A_{r,s}

graded by r + s.

The following crucial result was proven in [13]. In fact, its proof will comprise the main steps of the algorithm which we will discuss later.

Theorem 2.1 ([13, Lem. 2, Th. 1]). There exist ℤΓ-homomorphism d_{k}: A_{r,s} → A_{r+k−1,s−k} (k ≥ 1, s ≥ k) such that (i) ϵ_{s−1}d_{1} = ∂ϵ_{s}: A_{0,s} → C_{s−1} (ii) ∑_{i=0}^{k} d_{i}d_{k−i} = 0, for each k, (where d_{k|A_{r,s}} is interpreted as zero if r = k = 0 or if s < k.) Moreover, with the differential d = ∑_{k=0}^{∞} d_{k}, the complex (A, d) is acyclic and hence it yields a free ℤΓ-resolution of ℤ.

2.2. Contracting homotopies. Let (Q, d) be an acyclic chain complex. It will be often necessary to take preimages of d for elements which are in ker d = im d. A suitable computational method for this is by using a contracting homotopy. More about this approach can be found in [6, Section 3].

A contracting homotopy of an acyclic complex Q is a chain map h: Q_{i} → Q_{i+1} such that hd + dh = Id. Then for each y ∈ ker d, we have dh(y) = y. So h maps such an element y to its preimage under d.

Contracting homotopies are often easy to construct. To obtain a contracting homotopy for a ℤ[ℤ^{n}]-resolution B of ℤ, we will use the standard formula given in [3, p. 214], which provides a contracting homotopy for a tensor product of acyclic complexes equipped with contracting homotopies.

Let L = ℤ^{n} and Γ = L ⋊ G. We need to explain how to define a contracting homotopy on the induced complex Ind_{L}^{Γ}B from a given contracting homotopy h on B. Since every element of Ind_{L}^{Γ}B can be written as a direct sum of elements of the form (1, g) ⊗_{ℤL} y for (1, g) ∈ Γ and y ∈ B, we define the contracting homotopy by:

(2.1)  f: Ind_{L}^{Γ}B → Ind_{L}^{Γ}B, (1, g) ⊗_{ℤL} y ↦ (1, g) ⊗_{ℤL} h(y).

2.3. The steps involved. We are now ready to describe the key steps of the algorithm used to compute the cohomology of an n-dimensional crystallographic group Γ = L ⋊ G.

To obtain the free ℤL-resolution B_{∗} of ℤ, we tessellate ℝ^{n} into standard n-cubes of length 1. This defines an L-equivariant CW-structure on ℝ^{n} and the associated chain complex yields the desired resolution.

We denote by t_{i} for 1 ≤ i ≤ n the generators of L = ℤ^{n} which correspond to translations by 1 in the coordinate i. We denote by e the origin of ℝ^{n}, by e_{i} for 1 ≤ i ≤ n the 1-dimensional segment from e to t_{i}e, and by e_{i_{1}i_{2}...i_{m}} the m-dimensional cube spanned by e_{i_{1}}, e_{i_{2}} … e_{i_{m}}. Then

B_{m} = ⟨e_{i_{1}...i_{m}}, 1 ≤ i_{1} < ⋯ < i_{m} ≤ n⟩_{ℤL} for 0 ≤ m ≤ n

and the differentials of B_{∗}, denoted by d∗^{B}, are given by:

dm^{B}(e_{i_{1}...i_{m}}) = ∑_{j=1}^{m}(−1)^{j−1}(t_{i_{j}} − 1)e_{i_{1}...î_{j}...i_{m}}.

Next, we need a free ℤG-resolution C_{∗} of ℤ.

Remark 1. In our computations, the holonomy will always be a finite cyclic group, i.e. G = ⟨x | x^{q} = 1⟩. In this case, we will take for C_{∗} the standard 2-periodic resolution C_{i} = ℤG for all

i ≥ 0 and ∂_{i+1} : C_{i+1} ⟶^{x−1} C_{i} when i is even and ∂_{i+1} : C_{i+1} ⟶^{x^{q−1}+···+x+1} C_{i} when i is odd.

Now, using twisted tensor product construction, we obtain a free ℤΓ-resolution (A, d) of ℤ as follows:

(i) As discussed, we construct the resolutions (C_{∗}, ∂) and (Ind_{L}^{Γ}B_{∗}, Id ⊗_{ℤL} d∗^{B}), and free ℤΓmodules A_{r,s} for each 0 ≤ r ≤ n and 0 ≤ s and set A_{m} = ⨁_{r+s=m} A_{r,s}.

(ii) For n = 1, we define a contracting homotopy ^{1}h on the ℤ[ℤ]-free resolution (B_{∗}, d∗^{B}, n = 1) by ^{1}h(1) = e and

1h(t_{1}^{j}e) = \begin{cases}∑_{i=0}^{j−1} t_{1}^{i}e_{1} & j > 0 \\ − ∑_{i=1}^{−j} t_{1}^{−i}e_{1} & j < 0 \\ 0 & j = 0. \end{cases}

For each k ≥ 1, since (B_{∗}, d∗^{B}, n = k +1) is isomorphic to the tensor product of (B_{∗}, d∗^{B}, n = k) and the above resolution, we will define a contracting homotopy h : B → B by the recursive formula (see [3, p. 214]):

k+1h = ^{k}h ⊗ ι + (^{k}hϵ) ⊗^{1}h,

where ι is the identity map on (B_{∗}, d∗^{B}, n = 1) and ^{k}h is the contracting homotopy on

(B_{∗}, d∗^{B}, n = k).

A contracting homotopy f on Ind_{L}^{Γ}B is defined as in (2.1). (iii) Let r = 0 and β be a generator of A_{0,s}. We define d_{1}(β) = f(∂(ϵ_{s}(β))) ⊆ A_{0,s−1}.

For r = 1, we have that ϵ_{s−1}d_{1}d_{0} = ∂ϵ_{s}d_{0} = 0. Hence, d_{1}d_{0} : A_{1,s} → A_{0,s−1} maps into ker ϵ_{s−1} = im d_{0}.

(iv) So, for any generator β ∈ A_{1,s}, we define d_{1}(β) = −f(d_{1}(d_{0}(β))). Similar occurs for r ≥ 2 and for any generator β ∈ A_{r,s}, we define d_{1}(β) = −f(d_{1}(d_{0}(β))).

For k ≥ 2 we need to define d_{k} which satisfy the equation ∑_{i=0}^{k} d_{i}d_{k−i} = 0. Suppose, we defined d_{i} for i < k and d_{k}|A_{r−1,s} satisfying this property. It is not difficult to check that ∑_{i=1}^{k} d_{i}d_{k−i} is in ker d_{0} = im d_{0} (see Lemma 2 of [13]).

(v) Then, for a generator β ∈ A_{r,s} we take d_{k}(β) = −f(∑_{i=1}^{k} d_{i}d_{k−i})(β).

This yields the free ℤΓ-resolution (A, d). To calculate the cohomology of Γ we:

(vi) apply the functor Hom_{ℤΓ}(−, ℤ) to (A, d) to obtain a cochain complex of finitely generated

ℤ-free modules (F, δ). (vii) For each 0 ≤ i ≤ n + 1, reduce the matrix representing the boundary map δ_{i} : F_{i} → F_{i+1}

to Smith normal form and read off the cohomology group H^{i+1}(Γ) via the isomorphism:

F_{i+1}/Imδ_{i} ≅ H^{i+1}(Γ) ⊕ Imδ_{i+1}.

Remark 2. When G is a cyclic group and C_{∗} is its standard 2-periodic resolution, since the resolution (B, d^{B}) has length n, one can easily observe that the resolution (A, d) will also be 2-periodic starting from dimension n + 1. So, in all the steps we can stop the computations once we reach dimension n + 2.

3. All counterexamples up to dimension 6

3.1. First, 4-dimensional example. Here, we provide an example of a crystallographic group of the lowest possible dimension that is a counterexample to Conjecture 1.1.

Let Γ be a 4-dimensional crystallographic group L ⋊ G where G = ⟨M | M^{4} = 1⟩ is the cyclic of order 4 acting on L = ℤ^{4} by a right multiplication given by the matrix 2 in the equation (3.1) below.

Theorem 3.1. The integral cohomology of Γ is as follows:

H^{i}(Γ) = \begin{cases}ℤ & i = 1 \\ ℤ ⊕ ℤ_{4} ⊕ ℤ_{2} & i = 2 \\ ℤ ⊕ ℤ_{4} ⊕ ℤ_{2} & i = 3 \\ ℤ_{4}^{2} & i = 2k, & k ≥ 2 \\ ℤ_{2}^{4} & i = 2k + 1, & k ≥ 2 \end{cases}

The components of the right hand side of the conjectured equation in 1.1 are

H^{i}(G, H^{j}(L, ℤ)) = \begin{cases}ℤ & 0 ≤ j ≤ 3, & i = 0 \\ ℤ_{2} & j = 1, 3, & i ≥ 1 \\ ℤ_{2} & j = 2, & i ≥ 1, 2 | i \\ ℤ_{2} & j = 4, & 2 ∤ i \\ ℤ_{4} & j = 0, & i ≥ 1, 2 | i \\ ℤ_{4} & j = 2, & 2 ∤ i, \\ 0 & otherwise \end{cases}

From above we have

H^{4}(Γ, ℤ) = ℤ_{4}^{2} ≠ ℤ_{4} ⊕ ℤ_{2}^{3} = ⨁_{i+j=4} H^{i}(G, H^{j}(L, ℤ))

Therefore, Conjecture 1.1 is false.

Above result follows from applying the computational steps of Section 2. It is implemented in [11]. This specific example computation is obtained by running the file “dim4 ord4.g”.

3.2. Up to dimension 6. The algorithm for twisted tensor product is also implemented in the HAP package in the system GAP (see [7]). Our version of the algorithm is adjusted to our case and allows for more efficient computations. This gives us the possibility to compute all examples up to dimension 6.

In this section, we list all cases of crystallographic groups of dimensions up to 6 which do not satisfy Conjecture 1.1. For the list of all crystallographic groups in these dimensions we use the classification given in CARAT (see [10]).

All groups which are not listed in this section satisfy Conjecture 1.1 and so their cohomology groups are much easier and faster to compute by using the right hand side of the conjectured equation.

Up to dimension 3, all crystallographic groups of the form L ⋊ G with G being cyclic satisfy the conjecture.

In dimension 4, there are 45 non-isomorphic crystallographic groups of this type. Among these, 2 do not satisfy Conjecture 1.1. Both of them have the holonomy group of order 4. Their holonomy representation are given by the following matrices

(3.1)  1: [\begin{matrix}−1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & −1 & 1 \end{matrix}] ,  2: [\begin{matrix}0 & 1 & 0 & 0 \\ −1 & 0 & 0 & 1 \\ 0 & 0 & −1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix}] .

Let ℤ^{4}⋊G be a crystallographic group with holonomy representation given by the first matrix.

Remark 3. The cohomology groups of ℤ^{4} ⋊ G are given in the Table 1 under the number 1. Additionally, we compute that

⨁_{i+j=4} H^{i}(G, H^{j}(ℤ^{4}, ℤ)) = ℤ_{4}^{2} ⊕ ℤ_{2}^{2}.

which has order strictly bigger than the order of H^{4}(ℤ^{4} ⋊ G, ℤ).

This implies that, in the associated Lyndon-Hochschild-Serre spectral sequence, there are nonzero differentials.

Remark 4. Let us note that the holonomy representation of G is ℤ-equivalent to a direct product of representations of dimensions 1 and 3, and the 3-dimensional representation is the example ρ_{6} from [2, Section 5]. It was not known if there was a special free ℤ[ℤ^{3}]-resolution of ℤ that admitted a compatible action of ℤ_{4} via the representation ρ_{6} (see [2, 2.4, 5.1]). The example of the group ℤ^{4} ⋊ G shows such a compatible action can never exist. Otherwise, by Lemma 2.2 and Theorem 2.4 of [2], we would arrive at a contradiction. In fact, we can say more.

Theorem 3.2. Consider L = ℤ^{3} and Γ = L ⋊_{ρ_{6}} ℤ_{4}. Let A = L as a ℤΓ-module via the representation ρ_{6}. Then, in the Lyndon-Hochschild-Serre spectral sequence associated to Γ, the differential d_{2}^{0,2}(A) : E_{2}^{0,2}(A) → E_{2}^{2,1}(A) is nonzero. In particular,

H^{2}(Γ, A) ≅ ⨁̸_{i+j=2} H^{i}(ℤ_{4}, H^{j}(L, A)).

The group Γ gives a counterexample of the lowest possible dimension to a more general form of Conjecture 1.1 stated in [9, 1.2], where one allows nontrivial coefficients. This is because any group ℤ^{n} ⋊ ℤ_{m} for n ≤ 2 admits a local compatible action (see [1, 3.1]) and therefore, by a slight generalization of a theorem of Adem and Pan (see the proof of [1, 2.3]), satisfies this more general form of the conjecture.

Proof of 3.2. We apply the theory of characteristic classes introduced by Sah in [12] and further studied in [9] and [5].

Suppose, by a way of contradiction that d_{2}^{0,2}(A) = 0. One can easily check that H_{2}(A, ℤ) and H_{3}(A, ℤ), as ℤΓ-modules, are isomorphic to A and the trivial module ℤ, respectively.

Now, the characteristic class v_{2}^{2}, being in the image of the differential d_{2}^{0,2}(H_{2}(A, ℤ)), vanishes. The only other possible nonzero characteristic class that can occur on the second page of the spectral sequence is v_{2}^{3}. But, by Theorem 7.11 of [5], it follows that the order of v_{2}^{3} is a divisor of one. Hence, it also vanishes. Since, we already know that the Lyndon-Hochschild-Serre spectral sequence associated to Γ collapses with ℤ-coefficients, we can conclude that the differential d_{3}^{0,3}(H_{3}(A, ℤ)) = 0 implying that v_{3}^{3} = 0. Thus, we have shown that all characteristic classes vanish. Therefore, the Lyndon-Hochschild-Serre spectral sequence collapses at E_{2} for all coefficient modules that have a trivial L-action (see [5, 7.13]).

Since the holonomy representation of the group ℤ^{4} ⋊ G decomposes into a direct sum of ρ_{6} and the nontrivial one-dimensional representation, by Corollary 4.2 of [9] (see also [5, 7.3-5]), it follows that the Lyndon-Hochschild-Serre spectral sequence associated to the group ℤ^{4} ⋊ G collapses at E_{2} for all coefficient modules that have a trivial L-action. But this is clearly a contradiction to our computations of the 4-dimensional integral cohomology of the group ℤ^{4} ⋊ G (see Remark 3). □

In dimension 5, there are 96 non-isomorphic crystallographic groups with cyclic holonomy of split type. Out of these, 6 do not satisfy Conjecture 1.1, 5 of them with holonomy ℤ_{4} and 1 with holonomy ℤ_{8}. The list of matrices corresponding to their holonomy generators are contained in the file “allGroups.g” in [11]. Their cohomology groups are given in Table 1 with numbers from 3 to 8.

In dimension 6, there are 240 non-isomorphic crystallographic groups with cyclic holonomy of split type. Out of these, 29 do not satisfy Conjecture 1.1, 13 of them with holonomy ℤ_{4}, 7 with holonomy ℤ_{8} and 9 with holonomy ℤ_{12}. The list of matrices corresponding to their holonomy generators are contained in the file “allGroups.g” in [11]. Their cohomology groups are given in Table 1 with numbers from 9 to 37.

Remark 5. Let Γ = L ⋊ G be the group number 8 from the Table 1. We provide here the terms comprising the right hands side of the conjectured isomorphism:

H^{i}(G, H^{j}(L, ℤ)) = \begin{cases}ℤ & j = 0, 4, & i = 0 \\ ℤ^{2} & j = 2, 3, & i = 0 \\ ℤ_{8} & j = 0, & i ≥ 1, 2 | i \\ ℤ_{4} & j = 1, & 2 ∤ i \\ ℤ_{2}^{2} & j = 2, 3, & i ≥ 1, 2 | i \\ ℤ_{4} & j = 4, & i ≥ 1, 2 | i \\ ℤ_{2} & j = 5, & i ≥ 1, 2 | i \\ 0 & otherwise \end{cases}

to observe that the free ranks and the orders of the maximal finite subgroups of the groups H^{k}(L⋊G, ℤ) and ⨁_{i+j=k} H^{i}(G, H^{j}(L, ℤ)) are the same for every k. This means that the Lyndon-Hochschild-Serre spectral sequence collapses at E_{2} but there are extension problems.

Table 1: Cohomology groups of counterexamples up to dimension 6.

d ⋊ n  H^{1}  H^{2}  H^{3}  H^{4}
CARAT  H^{5}  H^{6}  H^{2k−1}, k ≥ 4  H^{2k}, k ≥ 4
1 4 ⋊ 4  ℤ  ℤ ⊕ ℤ_{4} ⊕ ℤ_{2}  ℤ ⊕ ℤ_{4} ⊕ ℤ_{2}  ℤ_{4} ⊕ ℤ_{2}^{3}
27-1.2  ℤ_{4} ⊕ ℤ_{2}^{3}  ℤ_{4} ⊕ ℤ_{2}^{3}  ℤ_{4} ⊕ ℤ_{2}^{3}  ℤ_{4} ⊕ ℤ_{2}^{3}
2 4 ⋊ 4  ℤ  ℤ ⊕ ℤ_{4} ⊕ ℤ_{2}  ℤ ⊕ ℤ_{4} ⊕ ℤ_{2}  ℤ_{4}^{2}
27-1.5  ℤ_{2}^{4}  ℤ_{4}^{2}  ℤ_{2}^{4}  ℤ_{4}^{2}
3 5 ⋊ 4  ℤ  ℤ^{2} ⊕ ℤ_{4} ⊕ ℤ_{2}^{2}  ℤ^{2} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}  ℤ ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{5}
81-1.2  ℤ ⊕ ℤ_{4} ⊕ ℤ_{2}^{6}  ℤ_{4}^{2} ⊕ ℤ_{2}^{6}  ℤ_{4}^{2} ⊕ ℤ_{2}^{6}  ℤ_{4}^{2} ⊕ ℤ_{2}^{6}
4 5 ⋊ 4  ℤ  ℤ^{2} ⊕ ℤ_{4} ⊕ ℤ_{2}^{2}  ℤ^{2} ⊕ ℤ_{4} ⊕ ℤ_{2}^{2}  ℤ ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{3}
81-1.5  ℤ ⊕ ℤ_{4} ⊕ ℤ_{2}^{4}  ℤ_{4}^{2} ⊕ ℤ_{2}^{4}  ℤ_{4}^{2} ⊕ ℤ_{2}^{4}  ℤ_{4}^{2} ⊕ ℤ_{2}^{4}
5 5 ⋊ 4  ℤ^{2}  ℤ^{2} ⊕ ℤ_{4} ⊕ ℤ_{2}  ℤ^{2} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{2}  ℤ ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{4}
82-1.3  ℤ_{4}^{2} ⊕ ℤ_{2}^{6}  ℤ_{4}^{2} ⊕ ℤ_{2}^{6}  ℤ_{4}^{2} ⊕ ℤ_{2}^{6}  ℤ_{4}^{2} ⊕ ℤ_{2}^{6}
6 5 ⋊ 4  ℤ^{2}  ℤ^{2} ⊕ ℤ_{4} ⊕ ℤ_{2}  ℤ^{2} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{2}  ℤ ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}
82-1.5  ℤ_{4}^{2} ⊕ ℤ_{2}^{4}  ℤ_{4}^{2} ⊕ ℤ_{2}^{4}  ℤ_{4}^{2} ⊕ ℤ_{2}^{4}  ℤ_{4}^{2} ⊕ ℤ_{2}^{4}
7 5 ⋊ 4  ℤ^{2}  ℤ^{2} ⊕ ℤ_{4}  ℤ^{2} ⊕ ℤ_{2}^{4}  ℤ ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{2}
82-1.7  ℤ_{4} ⊕ ℤ_{2}^{5}  ℤ_{4} ⊕ ℤ_{2}^{5}  ℤ_{4} ⊕ ℤ_{2}^{5}  ℤ_{4} ⊕ ℤ_{2}^{5}
8 5 ⋊ 8  0  ℤ^{2} ⊕ ℤ_{8} ⊕ ℤ_{4}  ℤ^{2}  ℤ ⊕ ℤ_{8} ⊕ ℤ_{4} ⊕ ℤ_{2}^{2}
142-1.2  ℤ_{2}^{2}  ℤ_{8}^{2} ⊕ ℤ_{2}^{4}  ℤ_{2}^{2}  ℤ_{8}^{2} ⊕ ℤ_{2}^{4}
9 6 ⋊ 4  ℤ^{2}  ℤ^{5} ⊕ ℤ_{4}  ℤ^{8} ⊕ ℤ_{2}^{2}  ℤ^{5} ⊕ ℤ_{4} ⊕ ℤ_{2}^{3}
202-1.3  ℤ^{2} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{4}  ℤ ⊕ ℤ_{4} ⊕ ℤ_{2}^{6}  ℤ_{4}^{2} ⊕ ℤ_{2}^{6}  ℤ_{4}^{2} ⊕ ℤ_{2}^{6}
10 6 ⋊ 4  ℤ  ℤ^{4} ⊕ ℤ_{4} ⊕ ℤ_{2}^{2}  ℤ^{6} ⊕ ℤ_{4} ⊕ ℤ_{2}^{2}  ℤ^{3} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{6}
203-1.2  ℤ ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{8}  ℤ_{4}^{2} ⊕ ℤ_{2}^{10}  ℤ_{4}^{2} ⊕ ℤ_{2}^{10}  ℤ_{4}^{2} ⊕ ℤ_{2}^{10}
11 6 ⋊ 4  ℤ  ℤ^{4} ⊕ ℤ_{4} ⊕ ℤ_{2}^{2}  ℤ^{6} ⊕ ℤ_{4} ⊕ ℤ_{2}^{2}  ℤ^{3} ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}^{3}
203-1.5  ℤ ⊕ ℤ_{4} ⊕ ℤ_{2}^{9}  ℤ_{4}^{4} ⊕ ℤ_{2}^{4}  ℤ_{2}^{12}  ℤ_{4}^{4} ⊕ ℤ_{2}^{4}
12 6 ⋊ 4  ℤ  ℤ^{4} ⊕ ℤ_{4}^{2}  ℤ^{6} ⊕ ℤ_{2}^{2}  ℤ^{3} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{4}
203-1.7  ℤ ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}^{3}  ℤ_{4}^{2} ⊕ ℤ_{2}^{6}  ℤ_{4}^{2} ⊕ ℤ_{2}^{6}  ℤ_{4}^{2} ⊕ ℤ_{2}^{6}
13 6 ⋊ 4  ℤ^{3}  ℤ^{4} ⊕ ℤ_{4} ⊕ ℤ_{2}  ℤ^{4} ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}^{3}  ℤ^{3} ⊕ ℤ_{4}^{4} ⊕ ℤ_{2}^{6}
220-1.3  ℤ ⊕ ℤ_{4}^{4} ⊕ ℤ_{2}^{10}  ℤ_{4}^{4} ⊕ ℤ_{2}^{12}  ℤ_{4}^{4} ⊕ ℤ_{2}^{12}  ℤ_{4}^{4} ⊕ ℤ_{2}^{12}
14 6 ⋊ 4  ℤ^{3}  ℤ^{4} ⊕ ℤ_{4} ⊕ ℤ_{2}  ℤ^{4} ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}^{3}  ℤ^{3} ⊕ ℤ_{4}^{5} ⊕ ℤ_{2}^{3}
220-1.5  ℤ ⊕ ℤ_{4}^{5} ⊕ ℤ_{2}^{5}  ℤ_{4}^{4} ⊕ ℤ_{2}^{8}  ℤ_{4}^{4} ⊕ ℤ_{2}^{8}  ℤ_{4}^{4} ⊕ ℤ_{2}^{8}
15 6 ⋊ 4  ℤ^{3}  ℤ^{4} ⊕ ℤ_{4}  ℤ^{4} ⊕ ℤ_{4} ⊕ ℤ_{2}^{4}  ℤ^{3} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{6}
220-1.7  ℤ ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}^{7}  ℤ_{4}^{2} ⊕ ℤ_{2}^{10}  ℤ_{4}^{2} ⊕ ℤ_{2}^{10}  ℤ_{4}^{2} ⊕ ℤ_{2}^{10}
16 6 ⋊ 4  ℤ  ℤ^{4} ⊕ ℤ_{4} ⊕ ℤ_{2}^{3}  ℤ^{4} ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}  ℤ^{3} ⊕ ℤ_{4}^{4} ⊕ ℤ_{2}^{8}
221-1.2  ℤ^{3} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{10}  ℤ_{4}^{4} ⊕ ℤ_{2}^{12}  ℤ_{4}^{4} ⊕ ℤ_{2}^{12}  ℤ_{4}^{4} ⊕ ℤ_{2}^{12}
17 6 ⋊ 4  ℤ  ℤ^{4} ⊕ ℤ_{4} ⊕ ℤ_{2}^{3}  ℤ^{4} ⊕ ℤ_{4} ⊕ ℤ_{2}^{3}  ℤ^{3} ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}^{7}
221-1.5  ℤ^{3} ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}^{5}  ℤ_{4}^{4} ⊕ ℤ_{2}^{8}  ℤ_{4}^{4} ⊕ ℤ_{2}^{8}  ℤ_{4}^{4} ⊕ ℤ_{2}^{8}
18 6 ⋊ 4  ℤ^{2}  ℤ^{3} ⊕ ℤ_{4} ⊕ ℤ_{2}^{2}  ℤ^{4} ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}^{3}  ℤ^{3} ⊕ ℤ_{4}^{4} ⊕ ℤ_{2}^{6}
225-1.2  ℤ^{2} ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}^{11}  ℤ ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}^{12}  ℤ_{4}^{4} ⊕ ℤ_{2}^{12}  ℤ_{4}^{4} ⊕ ℤ_{2}^{12}
19 6 ⋊ 4  ℤ^{2}  ℤ^{3} ⊕ ℤ_{4} ⊕ ℤ_{2}^{2}  ℤ^{4} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{4}  ℤ^{3} ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}^{5}
225-1.5  ℤ^{2} ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}^{7}  ℤ ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}^{8}  ℤ_{4}^{4} ⊕ ℤ_{2}^{8}  ℤ_{4}^{4} ⊕ ℤ_{2}^{8}
20 6 ⋊ 4  ℤ^{2}  ℤ^{3} ⊕ ℤ_{4} ⊕ ℤ_{2}  ℤ^{4} ⊕ ℤ_{4} ⊕ ℤ_{2}^{4}  ℤ^{3} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{6}
225-1.7  ℤ^{2} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{8}  ℤ ⊕ ℤ_{4} ⊕ ℤ_{2}^{10}  ℤ_{4}^{2} ⊕ ℤ_{2}^{10}  ℤ_{4}^{2} ⊕ ℤ_{2}^{10}
21 6 ⋊ 4  ℤ^{2}  ℤ^{3} ⊕ ℤ_{4} ⊕ ℤ_{2}  ℤ^{4} ⊕ ℤ_{4} ⊕ ℤ_{2}^{4}  ℤ^{3} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{4}
225-1.10  ℤ^{2} ⊕ ℤ_{2}^{10}  ℤ ⊕ ℤ_{4}^{3} ⊕ ℤ_{2}^{4}  ℤ_{4}^{2} ⊕ ℤ_{2}^{12}  ℤ_{4}^{4} ⊕ ℤ_{2}^{4}

22 6 ⋊ 8  0  ℤ^{3} ⊕ ℤ_{8} ⊕ ℤ_{4} ⊕ ℤ_{2}  ℤ^{4}  ℤ^{3} ⊕ ℤ_{8} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{3}
467-1.2  ℤ_{2}^{4}  ℤ ⊕ ℤ_{8}^{3} ⊕ ℤ_{2}^{8}  ℤ_{2}^{4}  ℤ_{8}^{4} ⊕ ℤ_{2}^{8}
23 6 ⋊ 8  ℤ  ℤ^{2} ⊕ ℤ_{8} ⊕ ℤ_{4}  ℤ^{4} ⊕ ℤ_{8} ⊕ ℤ_{4} ℤ^{3} ⊕ ℤ_{8} ⊕ ℤ_{4} ⊕ ℤ_{2}^{2}
471-1.2 ℤ ⊕ ℤ_{8} ⊕ ℤ_{4} ⊕ ℤ_{2}^{4}  ℤ_{8}^{2} ⊕ ℤ_{2}^{6}  ℤ_{8}^{2} ⊕ ℤ_{2}^{6}  ℤ_{8}^{2} ⊕ ℤ_{2}^{6}
24 6 ⋊ 8  ℤ  ℤ^{2} ⊕ ℤ_{8} ⊕ ℤ_{2}  ℤ^{4} ⊕ ℤ_{4}^{2}  ℤ^{3} ⊕ ℤ_{8} ⊕ ℤ_{2}^{3}
471-1.3  ℤ ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{4}  ℤ_{8} ⊕ ℤ_{4} ⊕ ℤ_{2}^{6}  ℤ_{8} ⊕ ℤ_{4} ⊕ ℤ_{2}^{6}  ℤ_{8} ⊕ ℤ_{4} ⊕ ℤ_{2}^{6}
25 6 ⋊ 8  ℤ  ℤ^{2} ⊕ ℤ_{8} ⊕ ℤ_{2}  ℤ^{4} ⊕ ℤ_{4}^{2}  ℤ^{3} ⊕ ℤ_{8} ⊕ ℤ_{2}^{3}
471-1.6  ℤ ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{4}  ℤ_{8} ⊕ ℤ_{4} ⊕ ℤ_{2}^{6}  ℤ_{8} ⊕ ℤ_{4} ⊕ ℤ_{2}^{6}  ℤ_{8} ⊕ ℤ_{4} ⊕ ℤ_{2}^{6}
26 6 ⋊ 8  ℤ  ℤ^{2} ⊕ ℤ_{8} ⊕ ℤ_{2}  ℤ^{4} ⊕ ℤ_{4}^{2}  ℤ^{3} ⊕ ℤ_{8} ⊕ ℤ_{4} ⊕ ℤ_{2}
471-1.5  ℤ ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{4}  ℤ_{8}^{2} ⊕ ℤ_{2}^{4}  ℤ_{4}^{2} ⊕ ℤ_{2}^{6}  ℤ_{8}^{2} ⊕ ℤ_{2}^{4}
27 6 ⋊ 8  ℤ  ℤ^{2} ⊕ ℤ_{8}  ℤ^{4} ⊕ ℤ_{2}^{2}  ℤ^{3} ⊕ ℤ_{8} ⊕ ℤ_{2}
471-1.7  ℤ ⊕ ℤ_{4} ⊕ ℤ_{2}^{3}  ℤ_{8} ⊕ ℤ_{4} ⊕ ℤ_{2}^{2}  ℤ_{4} ⊕ ℤ_{2}^{5}  ℤ_{8} ⊕ ℤ_{4} ⊕ ℤ_{2}^{2}
28 6 ⋊ 8  0  ℤ^{3} ⊕ ℤ_{8} ⊕ ℤ_{2}^{2}  ℤ^{2}  ℤ^{3} ⊕ ℤ_{8}^{2} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{3}
475-1.2  ℤ_{2}^{2}  ℤ ⊕ ℤ_{8}^{3} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{6}  ℤ_{2}^{2}  ℤ_{8}^{4} ⊕ ℤ_{4}^{2} ⊕ ℤ_{2}^{6}
29 6 ⋊ 12  ℤ  ℤ^{2} ⊕ ℤ_{12} ⊕ ℤ_{6}  ℤ^{2} ⊕ ℤ_{12} ⊕ ℤ_{6} ℤ ⊕ ℤ_{12}^{2} ⊕ ℤ_{6}^{3} ⊕ ℤ_{2}
365-1.3 ℤ ⊕ ℤ_{12}^{2} ⊕ ℤ_{6}^{3} ⊕ ℤ_{2}  ℤ_{12}^{2} ⊕ ℤ_{6}^{4} ⊕ ℤ_{2}^{2}  ℤ_{12}^{2} ⊕ ℤ_{6}^{4} ⊕ ℤ_{2}^{2}  ℤ_{12}^{2} ⊕ ℤ_{6}^{4} ⊕ ℤ_{2}^{2}
30 6 ⋊ 12  ℤ  ℤ^{2} ⊕ ℤ_{12} ⊕ ℤ_{6}  ℤ^{2} ⊕ ℤ_{12} ⊕ ℤ_{6} ℤ ⊕ ℤ_{12}^{3} ⊕ ℤ_{6} ⊕ ℤ_{3}
365-1.5 ℤ ⊕ ℤ_{12} ⊕ ℤ_{6}^{4} ⊕ ℤ_{2}  ℤ_{12}^{4} ⊕ ℤ_{3}^{2}  ℤ_{6}^{6} ⊕ ℤ_{2}^{2}  ℤ_{12}^{4} ⊕ ℤ_{3}^{2}
31 6 ⋊ 12  ℤ  ℤ^{2} ⊕ ℤ_{12} ⊕ ℤ_{2}  ℤ^{2} ⊕ ℤ_{4} ⊕ ℤ_{2}  ℤ ⊕ ℤ_{12}^{2} ⊕ ℤ_{2}^{4}
365-1.9 ℤ ⊕ ℤ_{12} ⊕ ℤ_{4} ⊕ ℤ_{2}^{4}  ℤ_{12}^{2} ⊕ ℤ_{2}^{6}  ℤ_{12}^{2} ⊕ ℤ_{2}^{6}  ℤ_{12}^{2} ⊕ ℤ_{2}^{6}
32 6 ⋊ 12  ℤ  ℤ^{2} ⊕ ℤ_{12} ⊕ ℤ_{2}  ℤ^{2} ⊕ ℤ_{4} ⊕ ℤ_{2} ℤ ⊕ ℤ_{12}^{2} ⊕ ℤ_{4} ⊕ ℤ_{2}
365-1.11  ℤ ⊕ ℤ_{12} ⊕ ℤ_{2}^{5}  ℤ_{12}^{2} ⊕ ℤ_{4}^{2}  ℤ_{6}^{2} ⊕ ℤ_{2}^{6}  ℤ_{12}^{2} ⊕ ℤ_{4}^{2}
33 6 ⋊ 12  ℤ  ℤ^{2} ⊕ ℤ_{12} ⊕ ℤ_{2}  ℤ^{2} ⊕ ℤ_{12} ⊕ ℤ_{6} ℤ ⊕ ℤ_{12}^{2} ⊕ ℤ_{6}^{2} ⊕ ℤ_{2}^{2}
366-1.4 ℤ ⊕ ℤ_{12}^{2} ⊕ ℤ_{6}^{3} ⊕ ℤ_{2}  ℤ_{12}^{2} ⊕ ℤ_{6}^{4} ⊕ ℤ_{2}^{2}  ℤ_{12}^{2} ⊕ ℤ_{6}^{4} ⊕ ℤ_{2}^{2}  ℤ_{12}^{2} ⊕ ℤ_{6}^{4} ⊕ ℤ_{2}^{2}
34 6 ⋊ 12  ℤ  ℤ^{2} ⊕ ℤ_{12} ⊕ ℤ_{2}  ℤ^{2} ⊕ ℤ_{12} ⊕ ℤ_{6}  ℤ ⊕ ℤ_{12}^{3} ⊕ ℤ_{6}
366-1.5 ℤ ⊕ ℤ_{12} ⊕ ℤ_{6}^{4} ⊕ ℤ_{2}  ℤ_{12}^{4} ⊕ ℤ_{3}^{2}  ℤ_{6}^{6} ⊕ ℤ_{2}^{2}  ℤ_{12}^{4} ⊕ ℤ_{3}^{2}
35 6 ⋊ 12  ℤ  ℤ^{2} ⊕ ℤ_{12} ⊕ ℤ_{2}  ℤ^{2} ⊕ ℤ_{12} ⊕ ℤ_{2}  ℤ ⊕ ℤ_{12}^{2} ⊕ ℤ_{2}^{4}
366-1.10  ℤ ⊕ ℤ_{12}^{2} ⊕ ℤ_{2}^{4}  ℤ_{12}^{2} ⊕ ℤ_{2}^{6}  ℤ_{12}^{2} ⊕ ℤ_{2}^{6}  ℤ_{12}^{2} ⊕ ℤ_{2}^{6}
36 6 ⋊ 12  ℤ  ℤ^{2} ⊕ ℤ_{12} ⊕ ℤ_{2}  ℤ^{2} ⊕ ℤ_{12} ⊕ ℤ_{2} ℤ ⊕ ℤ_{12}^{2} ⊕ ℤ_{4} ⊕ ℤ_{2}
366-1.11 ℤ ⊕ ℤ_{12} ⊕ ℤ_{6} ⊕ ℤ_{2}^{4}  ℤ_{12}^{2} ⊕ ℤ_{4}^{2}  ℤ_{6}^{2} ⊕ ℤ_{2}^{6}  ℤ_{12}^{2} ⊕ ℤ_{4}^{2}
37 6 ⋊ 12  0  ℤ^{3} ⊕ ℤ_{12} ⊕ ℤ_{3}  ℤ^{2}  ℤ^{3} ⊕ ℤ_{12} ⊕ ℤ_{6} ⊕ ℤ_{3}^{3}
562-1.2  ℤ_{2}^{2}  ℤ ⊕ ℤ_{12} ⊕ ℤ_{6}^{2} ⊕ ℤ_{3}^{5}  ℤ_{2}^{2}  ℤ_{12}^{2} ⊕ ℤ_{6}^{2} ⊕ ℤ_{3}^{5}

The column d ⋊ n states that the group is of dimension d and has the holonomy group of order n. The column CARAT contains the names of the integral representations of the holonomy groups (ℤ-classes) in the classification given in the system CARAT [10]. Each name needs to be preceded by “min.”. Note that CARAT uses left action for the holonomy group, thus holonomy representation has to be transposed before identification in CARAT.

4. Other consequences

4.1. Unresolved cases from [2]. Several crystallographic groups that were considered in [2] were not known to satisfy Conjecture 1.1. In section 5 of the same paper, the authors studied all crystallographic groups with holonomy ℤ_{4} of split type whose holonomy representations are indecomposable. Out of total 9 such groups, there were two 4 dimensional examples, encoded ρ_{8} and ρ_{9}, which were not known to satisfy Conjecture 1.1. We verify that the example of ρ_{8} satisfies the conjecture. The example of ρ_{9} is the same as the one considered in Section 3.1, so also number 2 from Table 1. Hence, it does not satisfy the conjecture.

In section 6 of [2], in relation to certain 6-dimensional Calabi-Yau toroidal orbifolds arising in string theory, some crystallographic groups were considered. It was shown, that out of possible 18 such groups only two, denoted ℤ_{8}^{(5)} and ℤ_{12}^{(6)} were not known to admit local compatible actions.

So, their cohomology was not computed. The 5-dimensional group ℤ_{8}^{(5)} is the same as the group number 8 from Table 1. The 6-dimensional group ℤ_{12}^{(6)} is isomorphic to the group number 37. Thus, both groups do not satisfy Conjecture 1.1.

4.2. Cyclic holonomy group of odd non-prime order. All previous counterexamples to Conjecture 1.1 have holonomy of order divisible by 4. We provide counterexample with odd order holonomy ℤ_{9}.

The first occurrence of a crystallographic group with holonomy ℤ_{9} is in dimension 6. From Section 3.2 we see that it satisfies the conjecture.

We find a counterexample in dimension 8 where the holonomy is generated by the matrix:

[\begin{matrix}−1 & 0 & −1 & −1 & −1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & −1 & −1 \\ −1 & −1 & 0 & 0 & −1 & −1 & 0 & 0 \\ −1 & −1 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \end{matrix}] .

by calculating that

H^{4}(ℤ^{8} ⋊ G, ℤ) = ℤ^{8} ⊕ ℤ_{9}^{2} ⊕ ℤ_{3}^{4} ≠ ℤ^{8} ⊕ ℤ_{9} ⊕ ℤ_{3}^{6} = ⨁_{i+j=4} H^{i}(G, H^{j}(ℤ^{8}, ℤ)).

The computation of this specific example is obtained by running the file “dim8 ord9.g” from [11].

References

[1] A. Adem, and J. Pan. Toroidal orbifolds, Gerbes and group cohomology. Trans. Amer. Math. Soc., 358(9):3969–3983, 2006.

[2] A. Adem, J. Ge, J. Pan, and N. Petrosyan. Compatible actions and cohomology of crystallographic groups. J. Algebra, 320(1):341–353, 2008.

[3] H. Cartan, and S. Eilenberg. Homological algebra. Princeton University Press, Princeton, N. J., 1956.

[4] L. S. Charlap. Bieberbach groups and flat manifolds. Universitext. Springer-Verlag, New York, 1986.

[5] D. Degrijse, and N. Petrosyan. Characteristic Classes for Cohomology of Split Hopf Algebra Extensions. J. Algebra, 322:366–385, 2011.

[6] G. Ellis. Homological algebra programming. In Computational group theory and the theory of groups, volume 470 of Contemp. Math., pages 63–74. Amer. Math. Soc., Providence, RI, 2008.

[7] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.4.12, 2008.

[8] M. Langer, and W. Lück. On the group cohomology of the semi-direct product ℤ^{n} ⋊_{ρ} ℤ/m and a conjecture of Adem–Ge–Pan–Petrosyan. J. Pure Appl. Algebra, 216(6):1318 – 1339, 2012.

[9] N. Petrosyan. Cohomology of split group extensions and characteristic classes. J. Algebra, 321:2916–2925, 2009.

[10] W. Plesken, and T. Schulz. Counting crystallographic groups in low dimensions. Experiment. Math., 9(3):407– 411, 2000.

[11] B. Putrycz. Package ccgch – Implementation of twisted tensor product for crystallographic groups of split type and cyclic holonomy and implementation of checks of Conjecture 1.1 up to dimension 6.

   http://bart.mat.ug.edu.pl/ccgch. 2012.

[12] C.-H.. Sah. Cohomology of split group extensions. J. Algebra, 255–302, 1974.

[13] C. T. C. Wall. Resolutions for extensions of groups. Proc. Cambridge Philos. Soc., 57:251–255, 1961.

Department of Mathematics, Katholieke Universiteit Leuven, Kortrijk, Belgium

  E-mail address: Nansen.Petrosyan@kuleuven-kortrijk.be

Department of Mathematics, Katholieke Universiteit Leuven, Kortrijk, Belgium, Institute of Mathematics, University of Gdańsk, Gdańsk, Poland

  E-mail address: Bartosz.Putrycz@mat.ug.edu.pl