Spin$^c$ structures on Hantzsche-Wendt manifolds

Using a combinatorial description of Stiefel-Whitney classes of closed flat manifolds with diagonal holonomy representation, we show that no Hantzsche-Wendt manifold of dimension greater than three does not admit a spin$^c$ structure.


Introduction
Hantzsche-Wendt manifolds are examples of flat manifolds, i.e. closed Riemannian manifolds with vanishing sectional curvature. They are generalizations of the three-dimensional flat orientable manifold defined in [5] and, following [16], we say that: An orientable n-dimensional flat manifold is Hantzsche-Wendt if and only if its holonomy group is an elementary abelian 2-group of rank n − 1.
Every n-dimensional flat manifold X occurs as a quotient space of the action of Γ on the euclidean space R n , where Γ is a Bieberbach group, i.e. a torsionfree, co-compact and discrete subgroup of the group Isom(R n ) = O(n) ⋉ R n of isometries of R n . X is an Eilenberg-MacLane space of type K(Γ, 1). By Bieberbach theorems (see [18]), Γ is defined by the following short exact sequence where ι(Z n ) is the maximal abelian normal subgroup of Γ, G is finite and coincides with the holonomy group of X. Moreover, by conjugations in Γ, G acts in a natural way on Z n , giving it the structure of a G-module.
Taking into account the above definition we will say that a Bieberbach group Γ ⊂ Isom + (R n ) = SO(n) ⋉ R n , defined by (1.1), is a Hantzsche-Wendt group and X = R n /Γ is a Hantzsche-Wendt manifold (HW-group and HW-manifold for short) if G ≃ C n−1 2 . Among many properties of HW-manifolds which were objects of research one can list the following: they exist only in odd dimensions [11], they are rational homology spheres [19] and cohomologically rigid [13]. If Γ is a HW-group then it is an epimorphic image of a certain Fibonacci group [10] and if its dimension is greater than or equal to 5, then its commutator and translation subgroups coincide [14]. One of the crucial -for the purposes of this paper -properties of HW-manifolds (HW-groups) is the one described in [16]: they are diagonal, i.e. there exists a Z-basis B of the G-module Z n such that gb = ±b for every b ∈ B and g ∈ G.
Now, let n ≥ 3. The fundamental group π 1 (SO(n)) of the special orthogonal group SO(n) is of order 2. The spin group Spin(n) is its double cover -and the universal cover in fact. Let λ n : Spin(n) → SO(n) be the covering map. A spin structure on a smooth orientable manifold X is an equivariant lift of its frame bundle via λ n . Its existence is equivalent to the vanishing of the second Stiefel-Whitney class w 2 (X) of X, see [3, page 40]. In the case when X is flat, it is closely connected to the Sylow 2-subgroup of its holonomy group [2] and can be determined by an algorithm [8]. The three-dimensional HW-manifold has a spin structure (see [7,Theorem VII.1]). But this is the only case -by [12,Example 4.6] no other HW-manifold admits any spin structure.
In the case when there are no spin structures, one can consider their complex analogue. We have that Spin c (n) := Spin(n) × S 1 / (−1, −1) = Spin(n) × C2 S 1 is the double cover of SO(n) × S 1 for which the spin c structure is defined -in analogy to the spin case -with the covering mapλ n : Spin c (n) → SO(n) × S 1 given byλ The manifold X has a spin c structure if and only if w 2 (X) is the mod 2 reduction of some integral cohomology class z ∈ H 2 (X, Z), see [3, page 49]. We immediately get that existence of spin structures determines existence of spin c structures -in fact the former induces the latter, but not the other way around. For example, by an unpublished work [20] all orientable 4-manifolds have some spin c structures, but by [15], 3 of the 27 flat ones don't have any.
In this paper we prove that every HW-manifold of dimension greater than or equal to 5 does not admit any spin c structure. Note that some examples of non-spin c HW-manifolds were given in [4].
The tools that we use have been introduced in [13] and used for example in [9]. They proved their effectiveness in cohomology-related properties of diagonal manifolds.
The structure of the paper is as follows. Sections 2 and 3 give a quick glance on a way of the encoding diagonal manifolds and their Stiefel-Whitney classes by certain matrices. This has been already presented in more detail in [13] and [9]. In Section 4 we give one of two theorems on conditions equivalent to the existence of spin c structures on HW-manifolds. For our further analysis we introduce HW-matrices. This description of HW-manifolds was introduced in [13] and is in fact one-to-one with the one given in [11]. Technical Section 6 gives us some properties and formulas for matrices that we work with. The second theorem on conditions equivalent to the existence of spin c structures on HW-manifolds is given in Section 7. After that we give a very specific form to a matrix which describes a (possible) spin c HW-manifold and at last we show that this form can never occur. This proves that no HW-manifold can admit a spin c structure.

Diagonal flat manifolds
In this section we give a combinatorial description of diagonal flat manifolds. This language is essential in the analysis of the Steifel-Whitney classes of such manifolds.
we shall denote the element in the i-th row and j-th column of A. By A i we shall understand the i-th row of A.
Remark 2.2. Let k ∈ N. Cyclic groups of order k with multiplicative and additive structure will be denoted by C k and Z k := Z/k, respectively. Note that in the natural way Z k is ring and possibly -a field.
Suppose Γ is a Bieberbach group defined by the short exact sequence (1.1). As mentioned in the introduction, conjugations in Γ define a G-module Z n . To be a bit more precise, corresponding representation ρ : G → GL n (Z) is called an integral holonomy representation of Γ and it is given by the formula where z ∈ Z n , g ∈ G and γ ∈ Γ is such that π(γ) = g. We will call Γ diagonal or of diagonal type if the image of ρ is a subgroup of the group of diagonal matrices of GL(n, Z). It follows that G = C k 2 for some 1 ≤ k ≤ n−1. Let S 1 = R/Z. As in [13] and [9], we consider the automorphisms g i : S 1 → S 1 , given by It is easy to see that D ∼ = C 2 × C 2 and g 3 = g 1 g 2 . We define an action of D n on T n by (t 1 , . . . , t n )(z 1 , . . . , z n ) = (t 1 z 1 , . . . , t n z n ), for (t 1 , . . . , t n ) ∈ D n and (z 1 , . . . , z n ) ∈ T n = S 1 × · · · × S 1 n .
Any minimal set of generators of a group C d 2 ⊆ D n defines a (d × n)-matrix with entries in D which in turn defines a matrix A with entries in the set Note that elements of V are written in italic. Definition 2.3. The structure of an additive group on V is given by for i, j, k ∈ V. This way V = Z 2 ⊕ Z 2 is in the natural way a Z 2 -vector space. We have the following characterization of the action of C d 2 on T n and the associated orbit space T n /C d 2 via the matrix A. When the action of C d 2 on T n defined by (2.2) is free, we will say that the associated matrix A is free and we will call it the defining matrix of T n /C d 2 . In addition, when C d 2 is the holonomy group of T n /C d 2 , we will say that A is effective.

Stiefel-Whitney classes of diagonal flat manifolds
The goal of this section is to introduce a notation and some basic results on Stiefel-Whitney classes of diagonal flat manifolds. For more precise description see [9] and [13]. Let n ∈ N and Γ be an n-dimensional diagonal Bieberbach group, given by the extension (1.1), with non-trivial holonomy group G = C d 2 (d > 0). Let A ∈ V d×n be a defining matrix of the corresponding flat manifold X = R n /Γ = T n /C d 2 . It is well-known that be the induced cohomology ring homomorphism. By [9, Proposition 3.2] the total Stiefel-Whitney class is given by (3.1) In the above formula for every 1 ≤ j ≤ n, α j , β j ∈ H 1 (C d 2 , Z 2 ) are the cocycles defined by and the linear homomorphisms α, β ∈ Hom Z2 (V, Z 2 ) are uniquely defined by the following rules be the induced group cohomology homomorphism (restriction of π * to the i-th gradation), for 0 ≤ i ≤ n. Using again [9, Proposition 3.2] and the five-term exact sequence for the extension (1.1) (see [9, Formula (7)]) we get Lemma 3.1. π * (1) is injective and the kernel of π * (2) is spanned by Remark 3.2. Note that the polynomials sw, α j , β j , θ j , where 1 ≤ j ≤ n, can be defined for any matrix A ∈ V d×n . To emphasize this connection or in the case when it won't be clear from the context, we will add the superscript A to them and write sw A for example.

Bockstein maps and spin c structures
We will keep the notation of the previous section and restrict our attention to the case of Hantzsche-Wendt manifolds of dimension greater than or equal to 5. Hence n ≥ 5 is an odd integer and d = n − 1. Let β Γ andβ Γ be the Bockstein homomorphisms of cohomology groups of Γ associated to the short exact sequences respectively. If ρ : is the homomorphism induced by the mod 2 map, then we have the following commutative diagram with the row forming an exact sequence (see [ Henceβ Γ is an isomorphism and Im β Γ = Im ρ. Let β be the Bockstein homomorphism of cohomology groups of C n−1 2 associated to the extension (4.1). The homomorphism π induces the commutative diagram is a monomorphism of the elementary abelian 2-groups of rank n − 1, hence it is an isomorphism and Let sw 2 be the sum of degree 2 terms of the polynomial sw. Then w 2 (X) = π * (sw 2 ) and by definition the manifold X = R n /Γ admits a spin c structure if and only if π * (sw 2 ) ∈ Im π * β. This condition is obviously equivalent to In addition, one can easily show that for every Using Lemma 3.1, we get the following theorem: Theorem 4.1. Assume that n ≥ 5 is an odd integer and X is an n-dimensional Hantzsche-Wendt manifold. Let A ∈ V n−1×n be a defining matrix of X. Then the following conditions are equivalent: 1. X admits a spin c structure.

HW matrices
Let n ∈ N. Every n-dimensional HW-manifold X defines some matrix A ∈ V n−1×n . For the purpose of investigating spin c properties of X it will be more convenient to work with a square matrix -a HW-matrix. HW-matrices were defined in [13]. Let Z be a finite set. By P(Z) we denote the algebra (over the field Z 2 ) of subsets of Z. Just recall that the addition and multiplication in P(Z) are defined by the symmetric difference and intersection respectively: Empty set and Z are zero and one of this algebra, respectively. Let us note without a proof: is linear.
2. Every permutation of Z is an algebra automorphism of P(Z).
Remark 5.2. We will use the notation P d := P({1, . . . , d}) for d ∈ N. In a similar way we define the column sums smc S j (A) (and smc j (A)) for S ∈ P d and 1 ≤ j ≤ n. Moreover, we define a map J A : P d → P n as follows Definition 5.4. The exists the unique Z 2 -linear involution · : V → V which maps 2 to 3 . We call this map a conjugation. To be explicit, we have Definition 5.5. Let A be a matrix with coefficients in V. We call A: where A t is the transpose of A and A is the element-wise conjugate of A; • distinguished if it has 1 on the main diagonal and 2 or 3 everywhere else.
Remark 5.6. Recall that we speak about a principal submatrix of a given matrix if the sets of row and column indices which define it are the same (see [17, Definition 6.2.5] for example). We immediately get, that principal submatrices of self-conjugate and distinguished matrices are themselves self-conjugate and distinguished, respectively.  2) smc j (A) = 0 for every 1 ≤ j ≤ n; 3) J A (U ) = 0 for every U ∈ P n \{0, 1}.
The set of HW-matrices of degree n, or n-HW-matrices for short, will be denoted by H n .
By Lemma 5.7 we immediately get: Corollary 5.9. Every HW-matrix is of odd degree.
Remark 5.10. We can think of the above definition as coming from the encoding Hantzsche-Wendt groups presented in [11]. In connection to this description we note: 1. Any row of a HW-matrix may be removed and the corresponding torus quotient will remain the same. In other words, the removal will make the matrix a defining and effective one for the same HW-manifold.
3. There is an action of the group G n := C 2 ≀ S n on the set V n×n . Namely, for every A ∈ V n×n we have that (a) c k conjugates the k-th column of A, where c k ∈ C n 2 has non-trivial element of C 2 in the k-th coordinate only; Keeping the above remark in mind, we can reformulate [11, Proposition 1.5] as follows: Proposition 5.11. The HW-manifolds X and X ′ , with corresponding HWmatrices A, A ′ ∈ V n×n , are affine equivalent if and only if A and A ′ are in the same orbit of the action of the group G n .
(P8) Let M has the following block form where on the diagonal we have matrices of degree 1, 1, a and b. There exists an element of C equal to 2 . Otherwise, for every i > a + 2, we have and since D is a principal submatrix of M of odd degree, we get a contradiction with (A4).
By (P8) there exist i and j, such that 3 ≤ i ≤ a + 2 < j ≤ n and the principal submatrix ∆ of M given by indices (2, i, j) is of the form By self-conjugacy of ∆ we immediately get but this contradicts (A4).
Remark 6.2. To a logical sentence Θ we assign (in a natural way) an element [Θ] ∈ Z 2 as follows: [Θ] = 1 ⇔ Θ is true. The following lemma, which describes map J for distinguished matrices, extends [13, Proposition 3]. Lemma 6.4. Let n ∈ N, M ∈ V n×n be distinguished and S, U ∈ P n . The following hold: (6.1) Properties 2. and 3. hold by the same rule as in the proof of Lemma 5.7. This rule will be also used in the rest of the proof.
Recall Remark 2.5, by which Z 2 is a subgroup of V.
If |U | is even on the other hand, we get that . In a similar fashion as above we have Directly from the definition of HW-matrices and the above lemma we get:

Spin c structures and HW-matrices
In this section we give a necessary and sufficient condition for existence of a spin c structure on a manifold defined by a HW-matrix. Let us note an easy lemma.
where 1 ≤ i ≤ d and U ∈ P d , is an algebra homomorphism.
We will use the following properties of the map κ A : Lemma 7.2. Let d, n ∈ N and A ∈ V d×n . Then: where α ij ∈ Z 2 . For any 1 ≤ k < l ≤ d and U = {k, l} we have and in the consequence, for any U ∈ P d , Denote by a, b, c, d the number of 0 , 1 , 2 , 3 in the rows from the set U of j-th column of A, respectively. We get κ(θ j )( Hence κ(θ j )(U ) = 1 if and only if which by definition means, that j ∈ J A (U ).
Proposition 7.3. Let n > 1 be an odd integer and let A ∈ V n−1×n be distinguished. The following conditions are equivalent: where σ 2 is the elementary symmetric polynomial of degree 2 in variables x 1 , . . . , x n .
3. There exists S ∈ P n , such that for every U ∈ P n−1 the equality (in Z 2 ) holds Proof. We will omit the super and subscript A in the proof. Denote by V the subspace of Z 2 [x 1 , . . . , x n−1 ] of polynomials of degree 2. Let V s and V f be subspaces ov V generated by monomials which are and are not squares, respectively. Let p : V → V f be the projection coming from the decomposition V = V s ⊕ V f . Note that p(θ j ) = θ j − x 2 j and p(θ n ) = θ n for 1 ≤ j < n, hence condition 1. is equivalent to but directly from the formula (3.1), since n is odd, we have that p(sw 2 ) = σ 2 . Assume 1 ≤ j ≤ n and let δ j := p(θ j ). For U ∈ P n−1 we have that Indeed, if j < n, using Lemma 7.2 we get Additionally, δ n = θ n and n ∈ U , hence Suppose that σ 2 = s j δ j ∈ V δ and let S := {j : s j = 1} ∈ P n . For every U ∈ P n−1 we have κ(σ 2 )(U ) = n j=1 s j κ(δ j )(U ). [j ∈ S · (J(U ) + U )] = |S · (J(U ) + U )| 2 , formula (7.1) follows. Now assume that (7.1) holds for some S ∈ P n and every U ∈ P n−1 . By the above calculations it may me written as Recall that U is any element of P n−1 . Using this and the linearity of κ, we get κ s j δ j = κ(σ 2 ).
Definition 7.4. Let n ∈ N, M ∈ V n×n and S ∈ P n .
1. We call S a spin c set for M if for every U ∈ P n the equation holds; 2. We call S an almost spin c set for M if for every U ∈ P n−1 equation (7.4) holds.
If S is a spin c set for M , we call (M, S) a spin c pair.
Lemma 7.5. Let n ∈ N be odd, M ∈ H n and S ∈ P n .
1. If S is an almost spin c set for M , then

If
Note again, that all equations above are in Z 2 . In particular the last one holds, because n is odd. Assume now that S is an almost spin c set for M . Equation (7.4) holds for every U ∈ P n−1 . It is enough to show that it also holds whenever n ∈ U . In that case however V = 1 + U ∈ P n−1 , so we have where in the last equality we again use the fact, that n is odd.
Theorem 7.6. Let n ∈ N, n ≥ 5, M ∈ H n and let X be the HW-manifold defined by M .The following conditions are equivalent: 1. X admits a spin c structure.
2. There exists a spin c set for M .
Proof. By Lemma 7.5 existence of a spin c and an almost spin c set are equivalent conditions. Let A be a matrix composed from the first n − 1 rows of M . Clearly it is distinguished and by Remark 5.10, A is defining and effective matrix for X. In order to get the desired equivalence, notice that for every U ∈ P n−1 the equality holds, use Theorem 4.1 and Proposition 7.3.

Standard forms of spin c pairs
Recall that in Remark 5.10 we have defined the action of the group G n = C 2 ≀S n on the space V n×n , for every n ∈ N. We will show that in fact it can act on spin c pairs.
Lemma 8.1. Let n ∈ N, M ∈ V n×n , S ∈ P n be such that (M, S) is a spin c pair. Then for every σ ∈ S n , (σM, σS) is also a spin c pair.
Proof. Let U ∈ P n and σ ∈ S n . Using an easy observation that J σM (U ) = σJ M (σ −1 U ) and Lemma 5.1, we get Note, with the assumptions of the above lemma, that G n acts on P n by permutations, using the canonical epimorphism G n → S n . Moreover, if g ∈ G n is an element which acts by conjugations of columns only, then J gM = J M , since 1 = 1 . We immediately get Corollary 8.2. Let n ∈ N, M ∈ V n×n , S ∈ P n be such that (M, S) is a spin c pair. Then for every g ∈ G n , (gM, gS) is also a spin c pair. for every two-element set U ∈ P n . Then there exists an integer k, such that 2k ≥ n and in the orbit G n M there exists a matrix M ′ in the following block form where A and B are self-conjugate of degree k and n − k, respectively. Moreover There exists a permutation σ ∈ S n , which fixes 1 and such that M ′ = σM ′′ is of the block form where A, B are of degrees k, n − k respectively and the equation (8.1) holds. Let U = {i, j} for 1 ≤ i < j ≤ n. By our assumptions and Lemma 6.4 we have Consider two cases: Lemma 8.4. Let n ∈ N and M ∈ V n×n be distinguished in the following block form where A, B are of degrees k, l, respectively. Assume that k > 0 and: Then C consists only of elements equal to 2 .
Proof. If l = 0, there is nothing to prove. Assume that l > 0, take i ≤ k and j > k. The principal submatrix of M defined by indices 1, i, j is of the form: We can deduce some further restrictions on a standard form of a matrix. Lemma 8.6. Keeping the notation from the above definition, let (M, S) be a spin c pair in the standard form and k + l < m ≤ n. Then, in the block form where a ∈ {2 , 3 } is such that the equation holds.
Proof. Let i ≤ k + l. 3) follows from the fact that smr S 1 = 1 + (k + l − 1)2 and smr S m = ka + la. By the following lemma, certain spin c pairs can be transformed to standard forms.
Lemma 8.7. Let n ≥ 3 be an odd integer and M ∈ V n×n be distinguished. Let S be a spin c set for M . If J M (U ) = 0 for U ⊂ S and |U | = 3, then there exists g ∈ G n such that (gM, gS) is a spin c pair in a standard form.
Proof. By Corollary 8.2 (gM, gS) is a spin c pair for any g ∈ G n . Our goal is to show that (M, S) can be transformed to a pair in the standard form.
By 9 Spin c structures on HW-manifolds By the results of previous sections we know that the existence of a spin c structure on a HW-manifold is equivalent to the existence of a spin c set for its HW-matrix. We will show that this never happens in dimensions greater than 3.
Lemma 9.1. Let n ≥ 5 be an odd integer and M ∈ H n . There does not exist a spin c set S for M such that |S| = n.
Proof. If such a set S exists, then by our assumptions J M (U ) = 0 for |U | = 3 and by Lemma 8.7 we can assume that (M, S) is in a standard form: where the degrees of A, B equal k, l respectively, k ≥ l and: where the blocks on the diagonal are of degrees k, l, 1, 1, respectively and k ≥ l.
Since k + l = n − 2 is odd, k = l + 1 mod 2. By Lemma 8.6 we have Finally we are ready to state the main result of the paper: Theorem 9.5. Let X be a Hantzsche-Wendt manifold of dimension n ≥ 5. Then X does not admit a spin c -structure.
Proof. This follows directly from Theorem 7.6 and Proposition 9.4.