Z 2 -COHOMOLOGY AND SPECTRAL PROPERTIES OF FLAT

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1 Z -COHOMOLOGY AND SPECTRAL PROPERTIES OF FLAT MANIFOLDS OF DIAGONAL TYPE S. CONSOLE, R. J. MIATELLO, AND J. P. ROSSETTI Abstract. We study the cohomology groups with Z -coefficients for compact flat Riemannian manifolds of diagonal type M Γ = Γ\R n by explicit computation of the differentials in the Lyndon-Hochschild-Serre spectral sequence. We obtain expressions for H j (M Γ, Z ), j =, and give an effective criterion for the non-vanishing of the second Stiefel-Whitney class w (M Γ ). We apply the results to exhibit isospectral pairs with special cohomological properties, for instance we give isospectral 5-manifolds with different H (M Γ, Z ), and isospectral 4-manifolds M, M having the same Z -cohomology where w (M) = 0 and w (M ) 0. We compute the Z -cohomology of all generalized Hantzsche- Wendt n-manifolds for n = 3, 4, 5 and we study H and w for a large n- dimensional family, K n, with explicit computation for a subfamily of examples due to Lee-Szczarba. Introduction Let Γ be a Bieberbach group, M Γ the associated compact flat Riemannian manifold and let F be the holonomy group of Γ. The main goal of this paper is to make a step in the computation of the cohomology groups H (M Γ, Z ) and to find connections with spectral geometry of M Γ under the assumption F = Z k. Since M Γ is an Eilenberg-Mc Lane space, its cohomology coincides with the group cohomology of Γ, that is H (M Γ, R) = H (Γ, R) for any ring of coefficients R. If m is the characteristic of R and ( F, m) =, then it is well-known that H (Γ, R) = (Z n R) F, the F -invariants in the holonomy representation (see [Hi]). However, if ( F, m) the Lyndon-Hochschild-Serre (LHS) spectral sequence does not degenerate at E and hence, in order to compute the cohomology it is necessary to investigate deeper the maps in the spectral sequence. A typical such case occurs when m = and the holonomy group F = Z k. We shall follow the approach in [ChV] by studying the differentials that occur in the LHS spectral sequence. For a large class of flat manifolds, we will obtain explicit formulas for H (Γ, Z ), which in many cases can be efficiently computed. Since the E p,q -term of the LHS spectral sequence involves Hp (Z k, Z ) (see (.4)) and since the cohomology H (Z k, Z ) is a polynomial algebra in k indeterminates, it will turn out that the first and second cohomology of M Γ can be expressed in terms of the ranks of the differentials which, in turn, can be obtained from the images of what we call Z -class polynomials, which are polynomials in the variables x,..., x k obtained by a simple rule (see Subsection.4, Remark.4). The main results in Section are stated in Theorems.5 and.7. Key words and phrases. cohomology, Bieberbach group, spectral geometry. First named author partially supported by GNSAGA of INdAM and MIUR (Italy).

2 S. CONSOLE, R. J. MIATELLO, AND J. P. ROSSETTI In Section 3, we will use again the Z -class polynomials to give a useful criterion to decide on the non-vanishing of the second Stiefel-Whitney class w of M Γ (see Thm. 3.3). In Section 4, we apply the results in the previous sections to study the Z - cohomology of families and pairs having similar spectral properties. We concentrate primarily on the generalized Hantzsche-Wendt manifolds, GHW manifolds for short (see [RS]). These are the n-dimensional flat manifolds with holonomy group Z n ; the orientable ones (a minority) are called HW manifolds and are generalizations of the classical Hantzsche-Wendt 3-manifold (or didicosm) in dimension 3 (see [MR]). For HW manifolds, the cohomology groups H j (M Γ, R) vanish for R = Q, Z p, p odd, 0 < j < n. In particular, these manifolds are rational cohomology spheres. Some related results have been obtained by A. T. Vasquez (see [Va]) showing examples of non vanishing of w, B. Putrycz ([Pu]) computing the abelianization of HW manifolds for n > 3, and K. Dekimpe and N. Petrosyan ([DP]), who give an algorithm that allows to compute integral homology of HW manifolds, and effectively compute it for all of them in dimension 7. We now summarize the new examples concerning non-homeomorphic manifolds having similar spectral properties that are given in this paper: () We determine the Z -cohomology of all GHW manifolds in dimensions 3, 4 and 5, listing all isospectral classes (see Example.8, Appendix 4.4 and Table 4.4). () For n = 4, we exhibit p-isospectral pairs for all p, M, M, that have the same cohomology but such that w (M) 0 and w (M ) = 0 (see manifolds labelled (,, 0) and (, 0, ) in Thm 4.4; see also [MP]). (3) We consider all 4-dimensional flat manifolds of diagonal type with F Z or F Z 3 and we show several isospectral or p-isospectral pairs, with p 3, having different Z -cohomology groups and where some of them have different lengths of closed geodesics (Appendix 4.4). (4) We find, for n = 5, many isospectral pairs with F Z 4 having different H (M Γ, Z ) and having the same H (M Γ, Z ) (Subsection 4.). Such examples are not possible to obtain in dimension 4. (5) We obtain expressions for H (M Γ, Z ) and H (M Γ, Z ) for an infinite family, K n, studied in [MPR], and we compute it explicitly for the subfamily introduced by Lee-Szczarba in [LS] (Theorems 4.4 and 4.5). Finally, we include several problems that are of interest to us (see Problems 3.5, 4. and 4.3).. Preliminaries.. Compact flat manifolds and Bieberbach groups. A crystallographic group is a discrete, cocompact subgroup Γ of the isometry group of R n, I(R n ) = O(n) R n. If Γ is also torsion-free, then Γ is a Bieberbach group. Such Γ acts properly discontinuously and freely on R n, thus M Γ = Γ\R n is a compact flat Riemannian manifold with fundamental group Γ. Any compact flat manifold arises in this way. The translations in Γ form a normal maximal abelian subgroup of finite index, L Λ, Λ a lattice in R n. The quotient F := Λ\Γ gives the linear holonomy group of the Riemannian manifold M Γ.

3 Z -COHOMOLOGY AND SPECTRA OF FLAT MANIFOLDS 3 Any γ I(R n ) can be written uniquely γ = BL b,where B O(n) and L b is a translation by b R n. The restriction to Γ of the canonical projection r : I(R n ) O(n), r(bl b ) = B, is a homomorphism with kernel Λ L Λ, and r(γ) = F is a finite subgroup of O(n) called the point group of Γ. In algebraic terms, Γ is an extension of F by Λ, i.e., there is an exact sequence (.) 0 Λ Γ r F. Furthermore, since L Λ is a normal subgroup of Γ, and (BL b )L λ (BL b ) = L Bλ, then Λ is B-stable for any γ = BL b Γ. This action defines an integral representation of F called the holonomy representation... Diagonal Bieberbach groups. In this article we will mainly work with Bieberbach groups such that the holonomy representation is of diagonal type, that is, there is an orthogonal basis B of the lattice Λ such that the elements in the point group F diagonalize in B with eigenvalues ±. We shall usually take Λ = Z n and B the canonical basis {e,..., e n } of R n. For such Γ we have F = Z k and we may also assume that b Zn for any element γ = BL b Γ (see [MR]). Furthermore, if Γ = B i L bi, i =... k; L Z n, with Bi = Id, then we can assume that b j e i = 0 or for every i n, j k. Although this definition clearly imposes a restriction on Γ, this is still a very rich class of Bieberbach groups (see for instance [MPR]). A standard way to represent Bieberbach groups of diagonal type is by using the column notation, i.e., writing the diagonal matrices in columns and as a subscript, the corresponding translation vector modulo the lattice. Example. (Hantzsche-Wendt 3-manifold or didicosm). Let M Γ be the flat manifold of dimension n = 3, with holonomy group Z, with generators of the corresponding group described as follows: let B = diag(,, ), B = diag(,, ), b = e + e 3, b = e + e ; i.e., Γ = B L b, B L b ; L Z n. It is clear that M Γ is orientable since det B = for every BL b Γ. In column notation B B or B B B B. It is easy to see that M Γ is a rational homology sphere, that is, the first and second Betti numbers are equal to zero. In the next section we will compute the Z - cohomology of M Γ and of other low dimensional flat manifolds (see Example.8)..3. Cohomology of (Bieberbach) groups. We recall that the cohomology of a group G with coefficients in a G-module A, denoted by H (G, A), can be calculated by means of the so called reduced bar resolution (see [Br, Ch. ]) that gives rise to a cochain complex defined as follows. Let C m (G, A) be the G-module of functions f : G G... G A }{{} m times

4 4 S. CONSOLE, R. J. MIATELLO, AND J. P. ROSSETTI with coboundary given by (df) (g,..., g m+ ) = g f(g, g 3,..., g m+ ) m + ( ) i f(g,..., g i, g i g i+, g i+,..., g m+ ) i= +( ) m+ f(g,..., g m ) The cohomology of this complex gives H (G, A), which has the structure of a graded ring, with product given by the standard cup product. One has that H 0 (G, A) = A G, the G-invariants in A, and furthermore the second cohomology group H (G, A) classifies the isomorphism classes of all extensions of G by A in which the induced action of G on A agrees with the given action. Going back to our setting, we consider a compact flat manifold M Γ with fundamental group Γ. Thus M Γ is an Eilenberg-Mac Lane space of type K(Γ, ), hence, for any coefficient ring R H (M Γ, R) = H (Γ, R), where, in the right hand side, R is regarded as a trivial Γ-module. Given Γ, we recall that a group extension of the form (.) is determined by a cohomology class β H (F, Λ), which is defined as follows. Consider the map α : F Λ\R n, such that α(b) = [Bb] for any BL b Γ, where [Bb] denotes the class of Bb in Λ\R n. Note that α is well defined since B(Λ) = Λ. In particular, given BL b, CL c Γ one has α(bc) = [(BC)(C b + c)]. We observe that in our context diagonal type Bieberbach groups the action of B does not change the class [b], i.e. α(b) = [Bb] = [b], and also α(bc) = [(BC)(C b + c)] = [b + c]. We fix α : F R n, a lift of α, so that α(b) Bb mod Λ, for any B F, and we furthermore assume that α(i) = 0. The map α represents a cohomology class in H (F, Λ\R n ) and, by the long exact sequence in cohomology, we have the isomorphism H (F, Λ\R n ) = d H (F, Λ), where d is the Bockstein homomorphism. The group Γ in the extension (.) is determined uniquely, up to equivalence, by the class β = dα H (F, Λ) represented by the map F F Λ : (.) (B, C) B α(c) α(bc) + α(b), which takes values in Λ, since B α(c) α(bc) + α(b) B Cc BC(C b + c) + Bb 0 mod Λ. We will abuse notation by writing the map in (.) by β(b, C) or dα(b, C). In the computation of group cohomology, we will make use of the LHS spectral sequence, which has terms E p,q = H p (Λ\Γ, H q (Λ, Z )), which converges to H p+q (Γ, Z ). Therefore, the following fact will be systematically used in our computations of differentials. (.3) H (Z n, A) = (Z n ) A, for any trivial Z n -module A. Furthermore, we will need a result due to Borel on the group cohomology ring of G = Z k with Z -coefficients. Theorem.. (see [Mi, Thm. 7.] or [AM, Thm. 4.4, p. 69]) As a graded algebra H (Z k, Z ) = Z [x,..., x k ], with generators x,..., x k in dimension. In particular H (Z k, Z ) = Hom (Z k, Z ),

5 Z -COHOMOLOGY AND SPECTRA OF FLAT MANIFOLDS 5 generated by the homomorphisms x i (i =,... k) mapping the i-th generator of Z k to and all other generators to 0. Thus, H (Z k, Z ) can be identified with the space of degree homogeneous polynomials in k variables, with Z -coefficients..4. Computation of the class β. Let now Γ be a Bieberbach group with holonomy group F = Z k and let β be the associated class as defined in (.). This class determines a Z -cohomology class (.4) β H (Z k, Λ Z ) = ( H (Z k, Z ) ) n. The components β i of β are homogeneous polynomials of degree two that will be very important for us. They will be called the Z -class polynomials of Γ, or -class polynomials, for short. In the next proposition we show that the β i can be computed by using the ring structure of H (Z k, Z ), given in Theorem.. Indeed, we have Proposition.3. Let Γ be a Bieberbach group of diagonal type and holonomy group Z k = B,..., B k. The l-th coordinate of β is given by (.5) βl = x i + x i x j, i : B i e l = e l b il = i : b il = where e,..., e n is the standard basis of R n. j i B j e l = e l Proof. As a first observation, we note that x i is represented by a map Zk Z k Z with value on (B i, B i ) and value zero on the other generators. Hence the coefficient of x i in β l is β l (B i, B i ). Using the above formula (.) for the differential of α, we have β(b i, B i ) = dα(b i, B i ) = B i α(b i ) + α(b i ) = l b il (B i e l + e l ). Thus the coefficient equals if and only if b il = and B ie l = e l. This verifies the coefficients of terms of the form x i in formula (.5). In order to verify the coefficient of the mixed terms x i x j (j i) in (.5), we observe that for i j β(b i, B j ) = dα(b i, B j ) = B i α(b j ) α(b i B j ) + α(b i ) = B i b j B j b i b j + b i = ( bjl (B i e l e l ) b il (B j e l e l ) ) l The coefficient of x i x j in β l is β l (B i, B j ). The above formula shows that this equals mod, for i, j such that: b il = and B je l = e l, or b jl = and B ie l = e l, but not both simultaneously. This completes the proof. Remark.4. The result in the proposition can be described in more graphic terms. In column notation, the l-component β l is a polynomial obtained from the l-th row of the diagram, by including the term x i if the entry (l, i) is, and including the term x i x j, for i j, if the entries (l, i), (l, j) in any order, are either (, u), or ( u, u), where u is either 0 or. Visually, we have

6 6 S. CONSOLE, R. J. MIATELLO, AND J. P. ROSSETTI B i -class polynomial x i B i B j -class polynomial u x i + x ix j u u x i x j We will apply this rule in the case of the didicosm (see Example.). We have that α is given on generators (modulo the lattice Z n ) by α(b ) = (,, 0), α(b ) = (0,, ). Thus, we have B B -class polynomial 0 0 β = x + x x β = x + x x β 3 = x x Thus, by using formula (.5) we obtain in this case β = (x + x x, x + x x, x x ).. The Lyndon-Hochschild-Serre spectral sequence As mentioned in the Introduction, the Lyndon-Hochschild-Serre spectral sequence will be the main tool in our study of the Z -cohomology of a Bieberbach group Γ. The E -term of this sequence is E p,q = H p (F, H q (Λ, R)), which converges to H p+q (Γ, R), where the coefficient ring R is regarded as a trivial Γ-module and p, q 0. It is a standard fact that for any F -module A, the order of F annihilates H p (F, A), i.e., F η = 0 for any η H p (F, A), if p > 0. Furthermore, if multiplication by F in A is an isomorphism, then H p (F, A) = 0 for p > 0 (see for instance [AM, Cor. 5.4]). Now, by (.3), H q (Λ, R) = R q) (n for any coefficient ring R. Hence, if multiplication by F on R is an isomorphism (in particular if R is a field of characteristic zero or odd), it follows that E p,q = H p (F, H q (Λ, R)) = 0, for p > 0. Hence only the first column in E does not vanish, thus the spectral sequence already degenerates at the first column of E. Therefore (see Subsection.3) (.) H q (Γ, R) = ( q (Λ Z R) ) F. In particular, in this case, H (Γ, R) is determined by the holonomy representation only. However, when multiplication by F is not surjective, the determination of the cohomology is much harder and one needs to investigate deeper into the maps in the LHS spectral sequence. In particular, this is the case if the Bieberbach group Γ has holonomy group F with F = k and R = Z. This is the main object of study in this paper.

7 Z -COHOMOLOGY AND SPECTRA OF FLAT MANIFOLDS 7.. The Charlap-Vasquez method. As mentioned, if the spectral sequence does not degenerate in E to compute cohomology we need to look at the E 3 term, obtained by computing the cohomology of the complexes in E, as the picture in Figure. shows for low dimensions. To illustrate the main difficulties to be met in our computations, we observe that E,0 = E,0 3 = E,0 4 = = E,0 and E 0, 3 = ker d 0, = E 0, 4 = = E 0,. Thus H (Γ, R) = ker d 0, E,0 = ker d 0, H (F, Z ). The second cohomology group, H (Γ, R), comes from the sum of the terms in positions (, 0), (, ), (0, ) in E. For example, E,0 3 = E,0 /Imd0, = E,0, since the differentials d 3 in the E 3 complex, issuing from, and arriving to entry (, 0) vanish. Similarly, E, 3 = ker d, = E, 4 = = E,. However, the differential d 0, 3 in E 0, 3 = ker d 0, need not vanish in general, so in order to determine this summand of H one often needs to compute the third differential. However, as we shall see, E 0, 3 vanishes for a large class of Z k -manifolds and then one can get an explicit expression for H (Γ, R) (see Lemma.6 and Theorem.7). Remark.. Of course, the first cohomology group can also be computed by using that H (M, Z) = [Γ, Γ]\Γ, together with the universal coefficient theorem. Relative to the computation of H (M Γ, R), Charlap and Vasquez [ChV] obtained an expression for the second differential d for any group extension of a group F by a finitely generated abelian group. We now introduce and explain the terms in their formula in the case when R = Z. This is the case of interest in this paper. Take P : Λ = H (Λ, Z) Hom(H q (Λ, Z ), H q (Λ, Z )) to be the map induced by Pontrjagin multiplication on H (Λ, Z ). Given that H q (Λ, H q (Λ, Z ) = Hom(H q (Λ, Z ), H q (Λ, Z )), then P maps Λ to H q (Λ, H q (Λ, Z )) and induces (by functoriality on coefficients) a homomorphism P : H (F, Λ) H (F, H q (Λ, H q (Λ, Z ))). The composition map from Hom(H q (Λ, Z ), Z ) Hom(H q (Λ, Z ), H q (Λ, Z )) Hom(H q (Λ, Z ), Z ) induces a pairing H q (Λ, Z ) H q (Λ, H q (Λ, Z )) H q (Λ, Z ).

8 8 S. CONSOLE, R. J. MIATELLO, AND J. P. ROSSETTI We shall denote by : H p (F, H q (Λ, Z )) H (F, H q (Λ, H q (Λ, Z ))) H p+ (F, H q (Λ, Z )) the cup product induced by this pairing. Theorem. ([ChV]). Let Γ be a Bieberbach group with holonomy group F. If ξ E p,q = H p (F, H q (Λ, Z )) and β H (F, Λ) is the class corresponding to the extension (.), we have (.) d p,q (ξ) = ( )p ξ ( P (β) + v q ), where v q H (F, H q (Λ, H q (Λ, Z ))) is the q-th characteristic class of the F - module Λ. One can actually write the Charlap-Vasquez formula in terms of the Pontrjagin product on Λ = Z n. Indeed, as shown in [SemC, pp. 4-09ff], for Λ = Z n the Pontrjagin product P on H (Λ, Z) = (Zn ) agrees with the wedge product. Thus P : Λ = H (Λ, Z) Hom(H q (Λ, Z ), H q (Λ, Z )) is such that z {w w w q z w w w q }, where z H (Λ, Z ) is the Z -cohomology class of z. We have the following explicit description of P (β) H (F, H q (Λ, H q (Λ, Z ))): P (β) : F F H q (Λ, H q (Λ, Z )) = Hom(H q (Λ, Z ), H q (Λ, Z )) (B, B ) {w w q β(b, B ) w w q }, where β is the class of β in Z -cohomology (see (.4)). Thus (.) can be restated in terms of the wedge product, by making explicit also the cup product multiplication. Theorem.3. Let ε,..., ε n be a basis of Λ Z = (Z n ), and let β be as in (.4). If ξ : F p Hom(H q (Λ, Z ), Z ), then d ξ : F p+ Hom(H q (Λ, Z ), Z ) is given by (.3) d ξ(v, B, B )(u) = ξ(v)( β(b, B ) u) + v q for any v F p and u H q (Λ, Z ). We shall assume from now on that the Bieberbach group Γ is an extension (.) of F = Z k by a lattice Λ = Z n and furthermore that F acts diagonally on Λ. Here we note that the action of F = Z k on H (Λ, Z ) = (Z n ) is trivial. We have Lemma.4. If F acts diagonally on Λ then v q = 0 for any q. Proof. By [ChV, Prop. 7., p. 37], v = 0 for any F -module. For rank-one modules, v = 0 by dimension reasons and, since by [ChV, Thm. 7] v is additive on direct sums, then v = 0 for a diagonal holonomy action. Now, by [ChV, Cor. of Thm. 6], if v = 0 then v q = 0 for any q. Under the assumption that F = Z k, the terms in the spectral sequence take a special form. Indeed: (.4) E p,q = H p (Z k, q (Z n ) ) = H p (Z k, Z ) q (Z n ). Theorem.5. Let Z k act diagonally on Λ = Zn. Let ε,..., ε n be a basis of Λ Z = (Z n ), and let β i be the i-th component of β (see (.5)).

9 Z -COHOMOLOGY AND SPECTRA OF FLAT MANIFOLDS 9 (i) If ξ : (Z k )p Hom(H q (Λ, Z ), Z ), then d ξ : (Z k ) p+ Hom(H q (Λ, Z ), Z ) is given by (.5) d ξ(v, B, B )(u) = ξ(v)( β(b, B ) u) for any v (Z k ) p and u H q (Λ, Z ). (ii) The differentials d p,q are given as follows: For d 0, : E 0, = H (Z n, Z ) = (Z n ) E,0 = H (Z k, Z ) (.6) we have d 0, εi = β i i =,..., n. For d, : E, = H (Z k, Z ) Z n E3,0 = H 3 (Z k, Z ) (.7) we have d, (x i ε j ) = x i β j i =,..., k, j =,..., n, where x,..., x k is a basis of H (Z k, Z ). For d 0, : E 0, = (Z n ) E, = H (Z k, (Zn ) ) (.8) we have d 0, (εi ε j )(ε k ) = δ ik βj + δ jk βi = where {ε i ε j } (i < j) is a basis of Z n. β j if k = i, β i if k = j, 0 otherwise, Proof. Part (i) is a direct consequence of the formula for the Pontrjagin product. Relative to (ii), we will check (.7) and (.8), leaving the verification of (.6) to the reader. To check (.7), we note that x i ε j H (Z k, Z ) Z n is identified with ξ H (Z k, H (Λ, Z )) = H (Z k, Zn ), where ξ : Z k Z n f (0,..., x i (f),......, 0). }{{} at place j By expression (.3) for d, we have d ξ(v, B, B ) = ξ(v)( β(b, B )), hence d (x i ε j )(v, B, B ) = x i (v)( β j (B, B )) = x i β j (v, B, B ). Thus (.7) follows. We now verify (.8). A basis of (Z n ) is {ε i ε j : i < j}, where {ε i } is a basis of Z n. Hence, by (.3) again d (ε i ε j )(B, B )(ε k ) = ε i ε j ( β(b, B ) ε k ) This completes the proof. = ε i ε j ( l β l (B, B )ε l ε k ) = (δ ik βj + δ jk βi )(B, B ) For the computation of E 0, it is important to study the differential d 0,. To this end, the following lemma despite its somewhat restrictive assumptions is not hard to prove and it will be sufficient for the applications to be given in later sections.

10 0 S. CONSOLE, R. J. MIATELLO, AND J. P. ROSSETTI Lemma.6. Let n k(k + ). If there are n t linearly independent Z -class polynomials, then (.9) dim ker d 0, = ( ) t. In particular, if there is only one linearly dependent Z -class polynomial, then ker d 0, = {0}, thus E 0, 3 = E 0, = {0}. Proof. Step. We first show that if there are m (out of m) linearly independent Z -class polynomials, then rank d 0, ( ) m. Indeed, look at the (transpose) matrix of d 0, in block form ε ε ε ε ε ε m ε ε 3... ε ε m... ε m ε m ε β β βm ε β β3... βm 0 0 ε 3 β β ε ε m β β... βm where β j stands for the column matrix of the coefficients of the Z -class polynomial β j. By a standard elimination process it is not difficult to see that the rank of this matrix is maximal, namely ( m ). Indeed, since β,..., β m are linearly independent, we see that the first m columns have maximal rank. Continuing with the elimination process, one sees the full rank is maximal. Step. Suppose now that the first t Z -class polynomials β,..., β t vanish and the remaining n t are linearly independent. Then all columns corresponding to ε i ε j, i < j t vanish, so that the rank reduces by ( t ). Now, by applying a similar argument as before, one obtains that the matrix has maximal rank. Step 3. If instead of having t vanishing Z -class polynomials, we have that exactly n t of them are linearly independent, then one can carry out the proof in a similar way, though it is more involved... Use of the LHS spectral sequence. Let Γ be a Bieberbach group with diagonal holonomy F = Z k and lattice Λ = Zn. Our next goal is to obtain explicit formulas for the dimensions of H (M, Z ) and H (M, Z ). To this end, we will compute E p,q = H p (Z k, Z ) q (Z n ) together with the differentials. Recall that H p (Z k, Z ) can be identified with the space of homogeneous polynomials of degree p in k variables, which has dimension ( ) k+p q. Moreover, (Z n ) has dimension ( ) n q. Thus, the dimension of E p,0 dim E,0 = ( ) k+ ) and the dimension of E 0,q is ( n q). p equals ( k+p p ) (in particular

11 Z -COHOMOLOGY AND SPECTRA OF FLAT MANIFOLDS Observe also that the q-th line in the spectral sequence is obtained from the first one by multiplying by ( ) n q. The figure shows the dimensions of E p,q for the terms that are relevant for computing the first and second cohomology groups. Theorem.7. Let M be an n-dimensional compact flat manifold with diagonal holonomy Z k. Then (.0) dim H (M, Z ) = n rank d 0, + k. (.) ( ) k + dim H (M, Z ) = rank d 0, + kn rank d, + dim E 0, Furthermore, if there are n linearly independent Z -class polynomials, then ( ) k + (.) dim H (M, Z ) = rank d 0, + kn rank d,. Note that rank d 0, coincies with the number of linearly independent Z -class polynomials β l, l =,..., n (see (.6)). Proof. Computation of H (M, Z ). The relevant terms in the spectral sequence are E,0 = E,0, of dimension k, and E 0, = E 0, 3 = ker d 0,, of dimension n rank d0,. Hence, since H (M, Z ) = E 0, E,0, our assertion follows. Computation of H (M, Z ).

12 S. CONSOLE, R. J. MIATELLO, AND J. P. ROSSETTI The relevant terms in E are shown in black boxes in the picture. Observe that (, 0) and (, ) already stabilize at level 3, but this is not in general the case for the (0, ) term. So we have H (M, Z ) = E,0 /Im d0, ker d, E 0,. As remarked at the beginning of this section, dim E,0 = ( ) k+, thus assertion (.) follows. Finally the last assertion follows from Lemma.6 and (.). Example.8. As a warm up, we begin by computing the Z -cohomology of all 3-dimensional flat manifolds with diagonal holonomy Z k, k =,. k = : there are two groups of this type: Γ c and Γ +a, corresponding to manifolds named dicosm and first amphicosm. In column notation: Γ c : B, Γ +a : B. The Z -class polynomials are β Γc = (0, 0, x ), βγ+a = (0, x, 0). So rank d 0, = in both cases, and applying formula (.0) in Theorem.7 we have hence, by Poincaré duality also dim H (c, Z ) = dim H (+a, Z ) = 3 + = 3 dim H (c, Z ) = dim H (+a, Z ) = 3. As a way of illustration, we will also carry out the computation of H (c, Z ) and H (+a, Z ) by the method in the previous section, i.e. by using (.). Since the set of Z -class polynomials are the same for both manifolds, then H (c, Z ) = H (+a, Z ). We have already seen that rank d 0, = and, on the other hand, (.7) implies that rank d, =. It remains to determine dim E 0,. Since rank d, =, it follows that the spectral sequence stabilizes at level 3, that is E 0, 3 = E 0,, hence dim E 0, = dim ker d 0, = dim E 0, rank d 0,. Using (.8) we compute that rank d0, = (the computations are similar to those in Example.9, supra). Thus, dim E 0, = 3 =. Hence, dim H (M Γc, Z ) = = 3.

13 Z -COHOMOLOGY AND SPECTRA OF FLAT MANIFOLDS 3 k =. There are three groups of this type, Γ c, Γ +a, Γ a, corresponding, respectively, to the manifolds called didicosm, and first and second amphidicosms. Γ c : B B, Γ +a : B B, Γ a : B B. We will carry out the computations only for the didicosm or the 3-dimensional HW manifold, since the other cases are similar. The Z -class polynomials are β Γc = (x x + x, x x + x, x x ), So rank d 0, = 3 and applying formula (.0) in Theorem.7 we get that dim H (c, Z ) = and by Poincaré duality also dim H (c, Z ) =. Similarly, for the first and second amphidicosms one computes dim H (+a, Z ) = dim H (+a, Z ) = 3, dim H ( a, Z ) = dim H ( a, Z ) =. Example.9. We now give an example where we apply Theorem.7 to compute the full Z -cohomology ring of a 4-dimensional compact flat manifold with F = Z. Later (in Appendix 4.4), we give a complete table of the Z -cohomology of all 4- dimensional flat manifolds with diagonal holonomy Z k, k =, 3, that is obtained by the same method. We remark that in this example, stabilization of the LHS spectral sequence for the relevant terms to compute the first and second cohomology groups, only occurs at level 3. Consider the 4-dimensional manifold with Bieberbach group Γ 8 given in column notation by Γ 8 : B B Then, using formula (.5) we have β Γ8 = (0, 0, x x, x + x ). Since two Z -class polynomials are linearly independent, one sees immediately that rank (d 0, : Z 4 Z3 ) is. Therefore by (.0) in Theorem.7 dim H (M Γ8, Z ) = 4 + = 4. We now compute the differential d, : Z 8 Z4, which sends x i ε j x i β j = x i βj (under the standard identification) for every i, j:

14 4 S. CONSOLE, R. J. MIATELLO, AND J. P. ROSSETTI d, x ε 0 x ε 0 x ε 3 x x x ε 4 x 3 + x x d, x ε 0 x ε 0 x ε 3 x x x ε 4 x x + x 3 We see that the image of d, is generated by all possible products of Z -class polynomials by the monomials x j. Thus, in this case rank d, = 4. Note that we cannot apply (.), since two Z -class polynomials vanish. However, since E 3,0 3 = Z 4 /Im d, = {0}, we have d 0, 3 = 0, so that E 0, 3 = E 0, = ker d 0,. Let us compute this map. Although this may follow from Lemma.6, we will carry out the details to illustrate the use of the method. We have (see (.8)) For Γ 8 we have d 0, : (Z 4 ) = Z 6 ε i ε j Z ε k ( β j if k = i, β i if k = j, 0 otherwise. ) ε ε ε ε 3 ε ε 4 ε ε 3 ε ε 4 ε 3 ε 4 x ε x x x x ε x x x x ε 3 x x x x ε 4 x x x Its rank is 5, so dim ker d 0, = 6 5 =. Thus, by (.) in Theorem.7 (see also the figure) we get dim H (M Γ8, Z ) = = 6.

15 Z -COHOMOLOGY AND SPECTRA OF FLAT MANIFOLDS 5 Note that in this example the first two rows of the Bieberbach group (written in column notation) do not contain any. In this situation the flat manifold M Γ is a flat toral extension of a flat manifold M of smaller dimension (see [Ya]). In the present case M is a Klein bottle and there is a fibration T M Γ M, where the fiber is a torus. Thus the cohomology of M Γ can be computed using the Leray-Serre spectral sequence of the above fibration that converges to H (M Γ ) and it has E = H ( M, H (T )). This procedure yields the same result obtained above for the Z -cohomology of M Γ8. 3. Second Stiefel-Whitney class The characteristic classes of a compact Riemannian manifold are closely related to its geometry. We note that Pontrjagin classes for any compact flat manifold vanish, since they can be expressed in terms of the curvature tensor (Chern-Weyl theorem), however the Stiefel-Whitney classes need not vanish, a somewhat unexpected fact first noticed by Auslander and Szczarba [AS]. We recall furthermore that an oriented Riemannian manifold M admits a spin structure if and only if w (M) = 0 ([Mi]). The goal of this section will be to describe an explicit method to decide whether w (M) = 0 for a compact flat manifold M with diagonal holonomy Z k, generalizing results in [LS] and [Va] for some families of Z k -manifolds. Indeed, in Theorem 3.3 we give a simple criterion for w (M) to vanish or not, in terms of the Z -class polynomials β l. We first recall that there is a one to one correspondence between real vector bundles ξ over a paracompact n-manifold M and homotopy classes of maps M BO(n), where BO(n) is the classifying space of O(n). The inclusion of the subgroup D(n) = Z n of diagonal matrices in O(n) induces a monomorphism φ : H (BO(n), Z ) H (Z n, Z ) = Z [x,..., x n ], with image the subring Z [σ,..., σ n ], generated by the elementary symmetric functions σ j of degree j in x,..., x n. The preimage w j = (φ ) σ j generates H j (BO(n), Z ) and is called the universal j-th Stiefel-Whitney class. If ξ is a real vector bundle over M with classifying map f ξ : M BO(n), the Stiefel-Whitney classes of ξ are defined by w j (ξ) = f ξ (w j ) H j (M, Z ), for j n. When ξ is the tangent bundle of M, w j (T M) =: w j (M) is called the j-th Stiefel- Whitney class of M. Let now Γ be the Bieberbach group of a Z k -manifold of diagonal type. We observe that in this setting, we have a trivial action of F = Z k on H (Λ, Z ) = (Z n ). Our next goal is to show how to compute the second Stiefel-Whitney class w of a Z k -manifold with diagonal holonomy by using the LHS spectral sequence. Observe first that the map Γ r F induces a map µ(r) : M Γ B(F ), where B(F ) is the classifying space for F and M Γ identifies with the classifying space for Γ. Remark 3.. Since B(F ) (resp. M Γ ) is the Eilenberg-Mac Lane space of F (resp. Γ), the induced homomorphism in singular cohomology µ(r) : H (B(F ), Z ) H (M Γ, Z ) can be identified with the homomorphism r : H (F, Z ) H (Γ, Z ),

16 6 S. CONSOLE, R. J. MIATELLO, AND J. P. ROSSETTI where Z is regarded as a trivial module in both cases. Remark also that r agrees with the inflation map, since the Z -action is trivial (see e.g. [We, Section 6.7] for the definition of inflation). The composition of morphisms Γ r F = Z k i D(n) = Z n O(n) induces a map of M into BO(n) which is the classifying map for the tangent bundle of M. Let x,..., x k be a basis of H (Z k, Z ) and let x,..., x n be the standard basis of H (D(n), Z ). The classes ω l = i (x l ) = m a ml x m, are called the -weights of the map i. Let σ j (ω,..., ω n ) be the j-th elementary symmetric function in ω,..., ω n. The following proposition is a direct consequence of the results in [BH, Section ] (cf. also [LS, (.)]). Proposition 3.. Let Γ r F = Z k be the map in (.). Then (3.) w j (M) = r σ j (ω,..., ω n ). We now focus on the case of w (M). For this purpose, we consider the exact sequence associated to the LHS spectral sequence (cf. [Ev, Cor. 7..3]), H (Λ, Z ) d0, H (F, Z ) r H (Γ, Z ) and recall that inflation agrees with r in this case. We thus have (3.) Im d 0, = ker r. Hence, from (3.) and (.6), we obtain: Theorem 3.3. Let M Γ be a n-dimensional compact flat manifold with diagonal holonomy Z k. Then w 0 if and only if the symmetric function σ (ω,..., ω n ) is not a sum of Z -class polynomials. Example 3.4. We apply the criterion in the theorem to determine w in the case of the 3-dimensional manifolds with holonomy group Z, in the notation of Example.8. We first compute the -weights in the case of Γ c. We have ω = x, ω = x, ω 3 = x + x. Thus σ (ω, ω, ω 3 ) = x + x + x x. Recall that β = (x x + x, x x + x, x x ), so that σ is the sum of the three Z -class polynomials. Hence w = 0. For Γ +a the -weights are: ω = 0, ω = x, ω 3 = x + x. Thus σ (ω, ω, ω 3 ) = x x + x. Furthermore β = (x, x + x x, 0), so that σ is not the sum of Z -class polynomials. Hence w (M Γ+a ) 0.

17 Z -COHOMOLOGY AND SPECTRA OF FLAT MANIFOLDS 7 In the case of Γ a the -weights are the same as for Γ +a. Recall that β = (x, x + x x, x x ), Thus, σ (ω, ω, ω 3 ) = x x + x in this case is the sum of the two Z -class polynomials β and β 3, hence w (M Γ a ) = 0. Problem 3.5. Find a method (or extend the method here) to compute the second Stiefel-Whitney class for flat manifolds with holonomy group Z k that are not of diagonal type. 4. Cohomology and spectral properties In this section we will exhibit several families of flat manifolds of diagonal type with a variety of spectral and cohomological properties. Two manifolds are said to be p-isospectral if they have the same spectrum with respect to on the Hodge Laplacian p acting on p-forms. In this case both manifolds must have the same Betti numbers, since betti p (M) equals the multiplicity of the zero eigenvalue of p. It is not very easy to exhibit p-isospectral manifolds for all 0 p n having different cohomological properties. In this article we will exhibit several simple examples of Sunada-isospectral manifolds in dimension n = 4, F Z, having different Z -cohomology, actually all H i s are different for i 3 (in Appendix see for instance, manifolds M 7 and M 0, or M 8 and M 9 ). Much more difficult to construct is a pair, M, M, p-isospectral for all p, where M and M have the same H and different H with Z -coefficients. In the following subsection, we will use the formulas in Theorem.7 to produce such an example. 4.. An illustrative example. We will first construct a pair of 5-manifolds with holonomy group Z 4 having the same first Z -cohomology, but different second Z -cohomology, that are p-isospectral for all p. In both cases the second Stiefel- Whitney class does not vanish. The groups are taken from the CARAT (Aachen) list (see [CS]). Namely, they are the first and fourth in this list, there denoted by #g, the -th torsion free group in Z-class group 8.., and #g4, 4-th torsion free group in Z-class group 8... Their generators are described in column notation in the following tables: #g : B B B 3 B 4 #g4 : B B B 3 B 4 The computations were first carried out with the aid of a computer program and then also checked by hand. We have: (a) the differential d 0, : Z 5 Z0 has rank 5 for both groups, (b) the differential d, : Z 0 Z 0 has rank 8 for #g and 9 for #g4, (c) the differential d 0, :: Z 0 Z 50 has (maximal) rank 0 for both groups.

18 8 S. CONSOLE, R. J. MIATELLO, AND J. P. ROSSETTI The Z -class polynomials β i, i 5 are given by β = β = x 3x 4 + x 4 β = x x 4 + x β = x x 4 + x β 3 = β 3 = x x 3 + x x 4 + x β 4 = β 4 = x x 3 + x 3 β 5 = β 5 = x x + x Since they are linearly independent for both groups, then rank d 0, = 5, which shows (a). Moreover Lemma.6 applies, so that d 0, has maximal rank, which implies (c). To verify (b) is more complicated since one has to compute the rank of a 0 0 matrix with Z -entries. Actually the elements in the image of d, are the third degree polynomials in four variables given by the products of the Z -class polynomials by the monomials x l, namely x l βi, x l β i for l =,..., 4, i =,..., 5. Now, for #g we have the relations { x β4 + x β + x β3 + x 3 β + x 3 β3 = 0, x β + x β3 + x 3 β + x 4 β + x 4 β3 = 0, A computation shows that 8 out of the 0 polynomials x l βi are linearly independent. In the case of #g4 we have only one relation x β + x β + x β 5 + x 4 β 5 = 0, and another long computation shows that 9 polynomials out of the 0 polynomials x l β i are linearly independent. Thus (b) follows. Now, applying formulas (.0) and (.) in Theorem.7, we obtain that the first Z -cohomology group has dimension 4 for both groups, while the second has dimension 7 for the first group, and 6 for the second. We now show that the second Stiefel-Whitney class w H (M, Z ) does not vanish. To see that this is not zero, by Theorem 3.3, we need to verify that w is not a sum of Z -class polynomials. The two weights are, for both groups, ω = x 3, ω = x 4, ω 3 = x 3 + x 4, ω 4 = x, ω 5 = x so the second symmetric polynomial in the ω j s is given in both cases by the homogeneous polynomial σ = x x + x 3 x 4 + x 3 + x 4. Suppose σ is a sum S of Z -class polynomials. Then S must contain the Z - class polynomials β = β and β 4 = β 4, otherwise the monomials x 3 and x 4 would not appear. But then S must contain the monomial x x 3 which does not appear neither in σ nor in any other Z -class polynomial, so that it cannot be cancelled out by adding Z -class polynomials. Concerning the spectral properties of the corresponding manifolds, they are both very similar. Indeed, a computation using the definition in (5.) shows that the Sunada numbers of the two groups coincide, therefore both manifolds are p- isospectral for 0 p n. In fact, one shows that the only non-vanishing Sunada numbers in this case are c, = c, = c, = c 3, = 3 and c 3, = c 4, = c 4, =

19 Z -COHOMOLOGY AND SPECTRA OF FLAT MANIFOLDS 9 c 5,0 =. Furthermore it is not difficult to check by the methods in [MR3] that both manifolds are length isospectral, that is, they have the same lengths of closed geodesics counted with multiplicities. Problem 4.. In [MR3, Ex. 3.4], a pair of Sunada isospectral manifolds with the same lengths of closed geodesics but different multiplicities is given. A straightforward computation shows that these manifolds have different cohomology over Z. It would be of interest to find a pair of manifolds with the same spectral properties, but where they both have the same Z -cohomology. 4.. Hantzsche-Wendt manifolds. A Hantzsche-Wendt group [MR] is an n- dimensional diagonal orientable Bieberbach group Γ with F = Z n. The point (or holonomy) group is {B : Be i = ±e i, i n, det B = } and necessarily n must be odd. The corresponding manifold M Γ = Γ\R n is called a Hantzsche- Wendt (or HW) manifold. Although the holonomy action is fixed, by varying the translation vectors and one gets a large class of non-homeomorphic flat manifolds (see [MR]). They all are rational homology spheres, but their Z -cohomology is far from being zero and can be different for different manifolds. It is convenient to choose generators of F, B i, with B i (e i ) = e i and B i e j = e j, if j i, for i n. In column notation an HW group has the following form. B B B 3... B n Here, each can be either 0 or. In dimension n = 3 there is just one HW manifold, namely the didicosm (see Examples.,.8 and 3.4. In dimension n = 5 there are two isomorphism classes. The corresponding Bieberbach groups can be described as follows: Γ 5 : B B B 3 B 4 Γ 5 : B B B 3 B 4 Using Theorem.7 one can verify that in both cases dim H (M, Z ) = 4 and dim H (M, Z ) = 6. We now look at the second Stiefel-Whitney class. The -weights are ω = x + x 3 + x 4, ω = x + x 3 + x 4, ω 3 = x + x + x 4, ω 4 = x + x + x 3, ω 5 = x + x + x 3 + x 4

20 0 S. CONSOLE, R. J. MIATELLO, AND J. P. ROSSETTI Thus σ (ω,..., ω 5 ) = x x + x x 3 + x x 4 + x x 3 + x x 4 + x 3 x 4. The Z -class polynomials are given by: x + x x + x x 3 + x x 4 + x x 3 + x x 4, x x + x x 3 + x + x x 3 + x x 4 + x 3 x 4, x x 3 + x x 4 + x x 3 + x x 4 + x 3 + x 3x 4, x x 4 + x x 4 + x{ 3 x 4 + x 4, x x together with + x x 3 + x x 4 for Γ 5, x x + x x 3 + x x 4 for Γ 5. Now, assume we have a linear system that one gets by imposing the condition that σ = l<m x lx m is equal to a linear combination (with coefficients 0 or ) of Z - class polynomials. Since the first polynomial is the only one containing x, the first coefficient must be zero, and so on, up to the fourth coefficient. So, only the fifth coefficient is left, but clearly neither of x x + x x 3 + x x 4 nor x x + x x 3 + x x 4 can be equal to l<m x lx m. Therefore, it follows that the second Stiefel-Whitney class does not vanish in either case. By using the methods above, we give a very simple proof of the following consequence of a result in [Pu], where the first homology with integer coefficients is computed. Proposition 4.. For any n-dimensional HW manifold, dim H (M, Z ) = n. Proof. Let M n be an HW manifold of dimension n. The result follows from (.0) and the fact that the differential d 0, has maximal rank n, since the Z -class polynomials are linearly independent. To see this, it is enough to observe that the first Z -class polynomial is the only one containing the monomial x, the second is the only one containing x, and so on up to x n. The last Z -class polynomial is nonzero and does not contain any monomial of the form x j so that it is necessarily independent from the others. Problem 4.3. () Determine the Z -cohomology of an arbitrary HW manifold. Determine the structure of the cohomology ring. () Identify the HW manifolds having most regular Z -cohomology rings, that is, having smallest possible dimensions. (3) Connect properties of the cohomology with properties of the associated directed graphs as in [MR]. (4) Interpret isospectrality of HW manifolds in terms of properties of the graphs The family K n. We now consider a family K n (see [MPR] and also [RS]) which is a subfamily of the generalized Hantzsche-Wendt manifolds. The GHW manifolds consist of those Bieberbach groups in dimension n having holonomy group Z n. These groups form a very large class that includes the examples constructed in [LS] and also the HW groups discussed in the previous subsection as a minority. For instance, in dimension n = 4 there are (resp. 0) GHW (resp. HW) groups, while if n = 5 there are 3 (resp. ) GHW (resp. HW) groups. Note that a GHW group is of type HW if and only if it is orientable. We refer to [RS], [MPR] and [DDM] for more facts on these groups. For instance, in [MPR] it is shown that all GHW manifolds are isospectral on forms, that is, for the full Hodge-Laplacian acting on p-forms for all 0 p n. This also holds for the family K n.

21 Z -COHOMOLOGY AND SPECTRA OF FLAT MANIFOLDS In this subsection we will obtain cohomological results on the family K n and as a byproduct, we will construct isospectral 4-manifolds M, M, having the same Z -cohomology but where w (M) = 0 and w (M ) 0. We first define the family K n. Let Γ n be the Bieberbach group having holonomy group Z n and lattice Z n, with generators given as follows: B B B B n a a3 a n a3 a n a3 n a4 n where B,... B n are generators of Z n and the coefficients a ij are either or 0. We note that if the coefficients are all equal to zero, we get the examples given by Lee-Szczarba in [LS]. Denote by M n the compact flat manifold with Bieberbach group Γ n. Our first goal in this subsection is to prove the following theorem. Theorem 4.4. If M n is a GHW manifold in the family K n, then For n = 4 (a) if a = a 3 =, then w (M 3 ) = 0 (for any value of a 3 ); (b) if either a or a 3 is zero, then w (M 3 ) 0 (for any value of a 3 ). (c) there exist isospectral pairs of manifolds in K 4 such that one has vanishing w and the other not. For n 5: w (M n ) 0 for any choice of the coefficients a ij. Proof. For n = 4 we write x := a, y := a 3, z := a 3, so that we denote the groups by (x, y, z): The Z -class polynomials are B B B 3 x y z x(x x + x ) + y(x 3 + x x 3 ), x + x x + z(x x 3 + x 3 ), x + x x 3, x 3. We now look at the second Stiefel-Whitney class. Let x i : Z n Z be the homomorphism sending B i to and all B j with j i to 0. Then i x = x,..., i x n = x n, i x n = 0. Thus the -weights are ω = x,..., ω n = x n, ω n = 0 and σ (ω,..., ω n ) = σ (x,..., x n ).

22 S. CONSOLE, R. J. MIATELLO, AND J. P. ROSSETTI We consider the linear system obtained by letting l<m x lx m be equal to a Z -linear combination of Z -class polynomials: x x + x x 3 + x x 3 = λ x 3 + λ (x + x x 3 ) + λ 3 [x + x x + z(x x 3 + x 3 )] +λ 4 [x(x x + x ) + y(x 3 + x x 3 )]. By comparing the x components we get that λ 3 = 0. Taking this into account we have λ + xλ 4 = 0 (from the x -component) λ + yλ 4 = 0 (from the x 3 -component) xλ 4 = (from the x x -component). yλ 4 = (from the x x -component) λ = (from the x x 3 -component) Hence we get Thus, there are two cases to consider: λ = λ =, λ 3 = 0, xλ 4 =, yλ 4 =. (): if x = y =, then the system has the solution λ = λ = λ 4 =, λ 3 = 0. So, x x + x x 3 + x x 3 is a sum of Z -class polynomials and w = 0 (for any choice of a 3 ); (): if x or y is zero, then the system has no solution. Therefore w 0 (for any z). The computation of the Z -cohomology can be done as usual. We shall list the results in the following table. Table. Family K 4 (x,y,z) betti Z = bettiz 3 betti Z w Sunada n. isospectral pairs ( 0 0 ) (0, 0, 0) ( ) (, 0, 0) ( ) (,, 0) ( ) (0,, 0) ( ) (0, 0, ) ( ) (, 0, ) ( ) (0,, ) ) (,, ) ( Note that the manifolds (,, 0) and (, 0, ), marked with, have the same Z - cohomology, but, as we have just shown, they have different second Stiefel-Whitney classes. Furthermore we claim that they are isospectral, as we shall see, by computing the Sunada numbers c s,t. The groups have the form

23 Z -COHOMOLOGY AND SPECTRA OF FLAT MANIFOLDS 3 B B B 3 B B B 3 We note that B has exactly one in its fixed space, spanned by e, e 3, e 4, hence B contributes to c 3,. Now, B has exactly two s in its fixed space, spanned by e, e 3, e 4, hence B contributes to c 3,. In the same way, B 3 contributes to c 3,. If we look at B B, the fixed space is spanned by e 3, e 4, of dimension, and it has in its third coordinate, hence it contributes to c,. Taking into account the contributions of B B 3 to c,, of B B 3 to c,, of B B B 3 to c, and of Id, we can see that c 3, = c, =, c 3, = c, = c, = c 4,0 =, and the remaining numbers c s,t are all equal to zero. The analogous calculation for the second group yields the same values for the c s,t, hence the corresponding manifolds are isospectral. We leave this verification to the reader. We look now at the second Stiefel-Whitney class w of K n for n = 5. B B B 3 B 4 a a3 a4 a3 a4 a34 where the coefficients a ij are either or 0. The Z -class polynomials are P (x,..., x 4 ) = a (x x + x ) + a 3(x x 3 + x 3 ) + a 4(x x 4 + x 4 ), P (x,..., x 4 ) = x + x x + a 3 (x x 3 + x 3) + a 4 (x x 4 + x 4), P 3 (x,..., x 4 ) = x + x x 3 + a 34 (x 3 x 4 + x 4 ), P 4 (x,..., x 4 ) = x 3 + x 3 x 4, P 5 (x,..., x 4 ) = x 4. Let us solve the linear system obtained by assuming that l<m x lx m is in the kernel of r, i.e., x x + x x 3 + x x 4 + x x 3 + x x 4 + x 3 x 4 = λ P (x,..., x 4 ) + λ P (x,..., x 4 ) + λ 3 P 3 (x,..., x 4 )+ +λ 4 P 4 (x,..., x 4 ) + λ 5 P 5 (x,..., x 4 ). Since P (x,..., x 4 ) is the only polynomial containing x, λ = 0. Taking this into account, we see that there is no monomial x x 4 on the right hand side. Thus, there are no solutions, for any values of the coefficients. So the second Stiefel-Whitney class (given by r ( l<m x lx m )) does not vanish. With entirely similar arguments, one shows that in the general case of dimension n, w 0, for any value of the coefficients. To conclude the subsection we consider the case of the manifold M n = LS, i.e. the case when all a ij s equal 0, as in the examples given by Lee-Szczarba. We will compute explicitly the second cohomology group.

24 4 S. CONSOLE, R. J. MIATELLO, AND J. P. ROSSETTI Theorem 4.5. (4.) dim H (LS, Z ) = n, dim H (LS, Z ) = Proof. In this case the Z -class polynomials are ( ) n. β = 0,..., β i = x i + x i x i+,..., β n = x n. Thus rank d 0, = n and applying formula (.0) in Theorem.7 we get dim H (LS, Z ) = n (n ) + n = n. Relative to the second assertion, we note that we are in the hypotheses of (.), so we just need to compute rank d,. It is not difficult to see that all polynomials x j βi are linearly independent. Thus rank d, = (n ). Hence ( ) ( ) n n dim H (M, Z ) = (n ) + (n )n (n ) = GHW manifolds in dimension 5. In this subsection we give the cohomology with Z -coefficients of all GHW 5-manifolds obtained by using Theorem.7 and with the help of a computer program, We will list also all the isospectral pairs. By definition, these manifolds have holonomy group Z 4, they are all of diagonal type and correspond to three different holonomy representations that will be described below. They are given in Table 4.4 following the enumeration in [CS] and listed according to their first and second Z -cohomology groups. By Poincaré duality, this yields all the cohomological information, since for Z -coefficients all manifolds behave as orientable ones. Table. Cohomology classes of GHW manifolds in dimension 5. betti Z betti Z List of manifolds 4 5 7, 4, 6,, 58, 6, 67, 69, 74, 77, 84, 85, 04, 05, 06, 07,, 5, 7,8,, 4 6, 3, 4, 6, 8, 9, 0,, 3, 7, 8, 0,, 3, 36, 37, 38, 39, 40, 4, 44, 45, 50, 5, 5, 53, 57, 59, 60, 6, 65, 66, 68, 7, 73, 75, 76, 78, 80, 8, 8, 83, 90, 9, 9, 93, 96, 97, 98, 00, 0, 0, 09,, 3, 4, 6, 9 4 7, 5,, 5, 9, 8, 9, 30, 3, 4, 43, 46, 47, 48, 49, 54, 55, 56, 63, 64, 70, 7, 79, 86, 87, 88, 89, 94, 95, 99, 03, 08, 0, 0, , 5, 6, 7, 3, 33, 34, 35 We conclude the subsection by listing all the isospectral pairs among these 3 manifolds, obtained by comparison of the Sunada numbers, calculated with the help of a computer program. We will write =4=6 to mean that the manifolds numbered, 4 and 6 are isospectral, while, if a number is missing, it means that the manifold is not isospectral to any other manifold in the list.