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Revista de la Unión Matemática Argentina

Print version ISSN 0041-6932On-line version ISSN 1669-9637

Rev. Unión Mat. Argent. vol.46 no.2 Bahía Blanca July/Dec. 2005

 

On the cohomology ring of flat manifolds with a special structure

I.G. Dotti * and R.J. Miatello *

* Dedicated to the memory of our colleague and friend Angel Larotonda.
Research partially supported by grants from CONICET, ANPCyT and SecytUNC (Argentina).

1. Introduction

A Riemannian manifold is said to be Kähler if the holonomy group is contained in U (n)  . It is quaternion Kähler if the holonomy group is contained in Sp(n)Sp(1)  . It is known that quaternion Kähler manifolds of dimension ≥ 8  are Einstein, so the scalar curvature s  splits these manifolds according to whether s > 0, s = 0  or s < 0  . Ricci flat quaternion Kähler manifolds include hyperkähler manifolds, that is, those with holonomy group contained in Sp(n)  . Such a manifold can be characterized by the existence of a pair of integrable, anticommuting complex structures, compatible with the Riemannian metric, and parallel with respect to the Levi-Civita connection (see [Be], for instance).

The simplest model of hyperkähler manifolds is provided by   4n ℝ  with the standard flat metric and a pair J,K  of orthogonal anticommuting complex structures. This hyperkähler structure descends to the 4n  -torus T Λ := Λ \ℝ4n  , for any lattice Λ  in ℝ4n.  If M Γ =  Γ \ℝ4n  is a compact flat manifold such that the holonomy action of F  = Λ \Γ  centralizes (resp. normalizes) the algebra generated by J,K  , then M Γ   inherits a hyperkähler (resp. quaternion Kähler) structure.

In [DM] (see also [JR] and [BDM]) we described a doubling construction for Bieberbach groups which allows to give rather simple examples of quaternion Kähler flat manifolds which admit no Kähler structure.

The purpose of the present paper is to study the real cohomology ring of low dimensional compact flat manifolds endowed with one of these special structures. In particular, we will determine the structure of this ring in the case of all 4  -dimensional Kähler flat manifolds and all 8  -dimensional compact flat kyperkähler manifolds. We shall make use of the known classification of space groups in dimension 4  , given in [BBNWZ], and of the classification of flat hyperkähler 8-manifolds due to L. Whitt ([Wh]). It turns out that the integral holonomy groups of hyperkähler 8-manifolds are obtained by doubling the holonomy groups of the Kähler flat 4  -manifolds and as a consequence we will show that the cohomology ring is an exterior algebra in generators of degree one and two.

In [Sa], [Sa2] and [Sa3], Salamon obtains a family of linear relations among the Betti numbers of general hyperkähler manifolds (see Remark 4.2). In Section 5 we give several examples (5.1 - 5.3) showing that these relations may not hold in the quaternion Kähler case.

As a second interesting class we study the hyperkähler manifolds obtained by doubling (twice) a Hantzsche-Wendt type manifold (see [MR]). This gives, for any m  ≥ 3  , a 4n  -dimensional compact flat hyperkähler manifold with holonomy group  n- 1 ℤ2   . We will show that the cohomology ring is generated by the F  -invariant forms of degree 2  and 3  , giving a procedure to find the relations. In particular we shall see that this algebra has a complicated structure and, even in the simplest case (n = 3  ) is far from being an exterior algebra, as seen in the 8  -dimensional case.

The interest in understanding the structure of the cohomology ring of hyperkähler and quaternion Kähler flat manifolds was stimulated by the study of the Betti numbers of hyperkähler manifolds in the work of Salamon (see [Sa], [Sa2], [Sa3]) and Verbitsky ([Ve]).

2. Hyperkähler and quaternionic Kähler structures on flat manifolds

We first recall some basic notions on compact flat manifolds (see [Ch] or [Wo]). A compact connected flat Riemannian manifold has euclidean space   n ℝ  as its universal covering space and a Bieberbach group Γ  as fundamental group (i.e. a discrete cocompact subgroup Γ  of I(ℝn)  which is torsion-free). If v ∈ ℝn  , let Lv  denote translation by v.  By Bieberbach's first theorem, if Γ  is a crystallographic group then Λ  = {v  : Lv ∈ Γ } is a lattice in ℝn  . We will identify the lattice Λ  with the translation lattice {Lv : v ∈ Λ},  a normal and maximal abelian subgroup of Γ  . The quotient F  = Λ \Γ  is a finite group, the point group (or holonomy group) of Γ  . When Γ  is torsion free, the geometric interpretation of Λ \Γ  is that of the holonomy group of the flat Riemannian manifold M  .

Let Γ  be a Bieberbach group with holonomy group F  and translation lattice Λ ⊂  ℝn  . Let φ : F → ℝn  be a 1  -cocycle modulo Λ  , that is, φ  satisfies φ(B1B2)  =  B-2 1φ(B1) + φ(B2)  , modulo Λ  , for each B1, B2 ∈ F  . Then φ  defines a cohomology class in H1(F ;Λ \ℝn) ≃ H2(F  ;Λ)  and one may associate to φ  a crystallographic group with holonomy group F  and translation lattice Λ  . Furthermore, this group is torsion-free if and only if the class of φ  is a special class (see [Ch]).

Definition 2.1. Let Γ  be a Bieberbach group with holonomy group F  and translation lattice       n Λ ⊂ ℝ  . Let           n φ : F →  ℝ  be any 1  -cocycle modulo Λ  . We let dφΓ  be the subgroup of     2n I(ℝ  )  generated by elements of the form  B 0 [0 B] L(φ(B),b)   and L(λ,μ)   , for γ = BLb  ∈ Γ  and (λ, μ) ∈ Λ ⊕ Λ.

We point out that if the holonomy group F  of Γ  centralizes a complex structure on ℝn  , then M Γ   is Kähler. We now review a procedure to construct compact flat manifolds endowed with a Kähler, hyperkähler or quaternionic Kähler structure. We refer to [DM] for the details. This method will be used in later sections.

Proposition 2.2. Let Γ ,φ  and d φΓ  be as in Definition 2.1. Then

(i) dφΓ  is a Bieberbach group with holonomy group F  , translation lattice Λ ⊕  Λ  and dφΓ \ℝ2n   is a Kähler compact flat manifold.

(ii) If Γ \ℝn   has a locally invariant Kähler structure, then dφΓ \ℝ2n   is hyperkähler. In particular, if φ′ : F → ℝ2n   is any 1  -cocycle modulo Λ ⊕ Λ  , then dφ′dφΓ \ℝ4n   is hyperkähler.

Remark 2.3. Benson-Gordon have proved ([BG]) that if N  is a simply connected nilpotent Lie group, Γ  is a discrete cocompact subgroup of N  , and M  =  Γ \N  has a Kähler structure (J,g)  (with g  positive definite) then M  is a torus. The above proposition says that there are plenty of compact flat riemannian Kähler manifolds other than tori.

Remark 2.4. In general, there are many choices of φ  as in Proposition 2.2. In this paper we shall work with φ  the 1  -cocycle associated to Γ  , as in [BDM]. We will denote d φΓ  by d Γ  in this case.

For many Bieberbach groups Γ  one can enlarge dΓ  into a Bieberbach group dqΓ  in such a way that some element in the holonomy group of dqΓ  anticommutes with the complex structure J2n  in ℝ2n  . By repeating the procedure twice, one gets a Bieberbach group such that any element in the holonomy group will either commute or anticommute with each one of a pair of anticommuting complex structures, hence the quotient manifold will be a quaternion Kähler flat manifold which in general, will not be Kähler.

Definition 2.5. Let Γ  be a Bieberbach group with holonomy group F  ≃ ℤk        2   , with translation lattice Λ  and such that b ∈ 1Λ     2  for any γ =  BL   ∈ Γ         b  . Set       [I   ]      2n En  =    -I  ∈ I(ℝ  )  . Set dq(Γ ,v) = 〈dΓ ,EnL(v,0)〉,  where      n v ∈ ℝ  .

Under rather general conditions, dq(Γ ,v)  contains dΓ  as a normal subgroup of index 2, and       n v ∈ ℝ  can be chosen so that dq(Γ ,v)  is torsion free, so Mdq(Γ ,v)   is a compact flat manifold with holonomy group F  × ℤ2   having as a double cover the Kähler manifold Md Γ   . Furthermore F  commutes with J  , but En  only anticommutes with J  . If we use this construction twice we get a Bieberbach group d2(Γ ,v,u) :=  d (d (Γ ,v),u) ⊂ I(ℝ4n)  q             q  q  such that the holonomy group normalizes two anticommuting complex structures, J1,J2   , on  4n ℝ  , hence   2          4n d q(Γ ,v,u)\ℝ  will be a quaternion Kähler manifold.

In the next results we give conditions on v ∈ ℝn  that ensure that dq(Γ ,v)  is torsion free. We also note that if n  is even, Mdq(Γ ,v)   will always be orientable. This construction will be used in Section 5.

Theorem 2.6. Let Γ  as above. Then

  1. If v ∈ ℝn   is such that 2v ∈ Λ  and satisfies

    (B  - I)v ∈ Λ for each γ = BLb  ∈ Γ ,

    then dqΓ  is a crystallographic group with translation lattice Λ ⊕ Λ  and holonomy group ℤk+1  2   . Furthermore, d Γ  q  is torsion-free if and only if v∈∕ Λ  and for each γ =  BLb ∈  Γ  we have:

    (B + I)(φ(B)  + v) ∈ Λ \ (B + I)Λ, or (B -  I)b ∕∈ (B - I)Λ.

  2. If every element in the holonomy group F  commutes or anticommutes with a translation invariant complex structure and v  satisfies the conditions in (i), then d (Γ ,v)\ℝ2n   q   is quaternion Kähler.

3. Cohomology of Kähler compact flat manifolds of dimension 4

In the computation of cohomology, in this and in later sections, we will make much use of the following result of H. Hiller ([Hi]):

Theorem 3.1. Let Γ  be a Bieberbach group and M Γ = Γ \ℝn   . If K  is a field such that the characteristic of K  does not divide ∣F ∣ , then the cohomology ring of M Γ   with coefficients in K  is given by

  *           ∧  *         F H  (M  Γ ,𝕂) ≃ (   (Λ ⊗ K))  .

Let Γ  be a 4  -dimensional Bieberbach group with holonomy group F  . It is not hard to see that in order for          4 M  = Γ \ ℝ   to be Kähler, it is necessary and sufficient that F  commutes with a complex structure J  on ℝ4.  Using the classification of compact flat manifolds of dimension 4  in [BBNWZ] we see that those groups Γ i  with non trivial holonomy group which have such property have cyclic holonomy groups F  of order 2,3,4  or 6  , and have the form Γ  = 〈γ ,Λ 〉  i     i  i , with γ =  σ L  i    i bi   , 1 ≤ i ≤ 7  as follows:

               ⌊                ⌋       ⌊                ⌋                  1                         1           1                ||    1           ||       ||     1          ||            e1 F  ≃ ℤ2 : σ1 = ⌈       - 1      ⌉, σ2 = ⌈        - 1     ⌉,  b1 = b2 = 2                             - 1                      - 1                 ⌊ 1            ⌋           ⌊ 1  0  0   1 ⌋                |    1         |           | 0  1  0   1 | F  ≃ ℤ4 : σ3 = |⌈              |⌉ ,    σ4 = |⌈             |⌉ ,  b3 = b4 = e41                        0  - 1               0  0  0  - 1                        1    0               0  0  1   0

              ⌊               ⌋        ⌊                ⌋                 1  0    0   0            1  0    0    0               || 0  1    0   0 ||        || 0  1  - 1    0 ||            e1 F  ≃ ℤ3 : σ5 = ⌈ 0 0  - 1  - 1 ⌉, σ6 = ⌈ 0  0  - 1  - 1 ⌉,  b5 = b6 = 3                 0  0    1   0            0  0    1    0                ⌊              ⌋               | 1            | F  ≃ ℤ6 : σ7 = |   1         |,   b7 = e61.               ⌈       0  - 1 ⌉                       1    1

We note that in the case of the torus T 4   , the cohomology ring is an exterior algebra generated by elements of order 1, and the Poincaré polynomial is p(t) = (t + 1)4   . For general flat Kähler 4  -manifolds we have:

Theorem 3.2. If M  Γ   is a 4  -dimensional Kähler flat manifold which is not a torus, the cohomology ring is an exterior algebra in {e1,e2,η1,η2} where the ei (i = 1,2)  have degree 1 and the ηj (j = 1,2)  have degree 2  . Furthermore, in all cases one has β =  β  = 1, β =  β  = β  = 2  0    4       1    2    3  and the Poincaré polynomial is given by          2           2 p(t) = (t + 1)(t + 1)   , 1 ≤ i ≤ 7  .

Proof. To determine the real cohomology rings of the Kähler flat manifolds of dimension 4  , we need to compute the F  -invariants in each degree, for each Bieberbach group Γ i  in the family considered above.

We shall carry out this computation only in the case of the group Γ 6   . The other cases are similar and their verification will be left to the reader.

It is easy to see that in degree 1, the fixed space is spanned by the elements e1,e2   .

Assume now that      ∑ η =    1≤i<j≤4aijei ∧ ej  satisfies σ6η =  η  . Now

σ6η =     a12e1 ∧ e2 + a13e1 ∧ (- e2 - e3 + e4) + a14e1 ∧ (- e3) +        a23e2 ∧ (- e3 + e4) + a24e2 ∧ (- e3) + a34(- e2 + e4) ∧ (- e3).

Now, σ6η =  η  implies a12 = a12 - a13   and a13 = - a13 - a14   , thus a13 = a14 = 0  . Also, it follows that a23 = - a23 - a24 + a34   and a24 = a23   , thus 3a23 = a34   , hence the σ6   -fixed space in degree 2 is spanned by the invariant 2  -forms e1 ∧ e2   and e2 ∧ e3 + e2 ∧ e4 + 3e3 ∧ e4 = (e2 + 3e3) ∧ (e3 + e4)  , as asserted.

We now turn into degree 3. Let

η = ae1 ∧ e2 ∧ e3 + be1 ∧ e2 ∧ e4 + ce1 ∧ e3 ∧ e4 + de2 ∧ e3 ∧ e4

with a,b,c,d ∈ ℝ  . Now

σ6η = (- a - b + c)e1 ∧ e2 ∧ e3 + ae1 ∧ e2 ∧ e4 + ce1 ∧ e3 ∧ e4 + de2 ∧ e3 ∧ e4.

Thus σ6η =  η  implies a = b  , c = 3a  . Thus we get that the space of F  -invariants in degree 3 is generated by the 3  -forms e2 ∧ e3 ∧ e4   (a = 1,d = 0)  and e1 ∧ e2 ∧ (e3 + e4) + 3e1 ∧ e3 ∧ (e3 + e4) = e1 ∧ η6   . This completes the verification for Γ 6   .

In the remaining cases the invariants are computed similarly. We now give a table that lists the F  -invariants in each degree, for each group.

 Degree  1                  2                         3           4  Γ       e               e ∧ e                   e  ∧e  ∧ e        e   1,3,5,7  e1              e1∧ e2                   1e ∧ 3e ∧ e4 ----------2---------------3---4--------------------2---3---4-----------  Γ 2     e1              e1 ∧ e2                  e1 ∧ e3 ∧ e4     e ---------e2----------e3 ∧-(- e1-+-2e4)-------e2-∧e3-∧-(--e1 +-2e4)------  Γ 4     e1              e1 ∧ e2                    e1 ∧ η4       e  ---------e2--(e1 +-e2 --2e3)-∧(--e3 +-e4)-:=-η4------e2 ∧-η4------------  Γ 6     e1              e1 ∧ e2                  e2 ∧ e3 ∧ e4    e  ---------e2-----(e2 +-3e3)∧-(e3 +-e4)-:=-η6----------e1 ∧-η6------------

Here e = e1 ∧ e2 ∧ e3 ∧ e4   .

The assertions on the Betti numbers and on the structure of the ring follow immediately from the information in the table, thus the theorem follows. □

4. The cohomology ring of hyperkähler flat 8-manifolds

By doubling the 4-dimensional Bieberbach groups listed in the previous section we obtain a family of 8-dimensional hyperkähler flat manifolds. In [Wh] L. Whitt gives a full classification of such manifolds, showing there are 12 diffeomorphism classes. This classificaton shows in particular, that the holonomy representations of all such manifolds are obtained by doubling the holonomies of Kähler 4-manifolds. The goal of this section will be to determine the cohomology ring of this family. We first need to recall Whitt's classification. For simplicity of notation we shall set

     [      ]        [        ]       [      ] X  =   1  0   , J =    0  - 1  ,  L =   0  1   ,        1  0            1    0           0  1

     [0   - 1 ]        [0   - 1 ]        [1   0 ] D  =            , N  =            , E =           .        1    1            1  - 1            1  1

Furthermore, let Eij  be the 2 × 2  matrix with 1  in the (i,j)  place and 0  otherwise.

According to [Wh], Theorem 4.3, the holonomy group of Γ  is cyclic with generator given by σ0 = I  , or by one of the following:

                                  ⌊     |    ⌋        ⌊    |    ⌋               [         ]            I4 |0 L            I4 |0 E F ≃  ℤ2 : σ1 =  I4     0  ,    σ2 = ⌈----|0-L-⌉,  σ3 = ⌈----|0 X-⌉ ,                 0  - I4              0  |- I4           0  |- I4              ⌊  I4 L E21  ⌋          σ = ⌈     L E    ⌉,           4    ----|---12-                 0  | - I4

              ⌊    |     ⌋        ⌊ I4 |0  L  ⌋        ⌊ I4 |L  E12 ⌋                -I4-|------        |    |0  L  |        |    |L   0  | F ≃  ℤ4 : σ5 = ⌈   |J    ⌉,  σ6 = |⌈----|J-----|⌉,  σ7 = |⌈----|J------|⌉ ,                    |   J               |                    |                                        |   J                |    J

              ⌊    |       ⌋        ⌊    |        ⌋         ⌊    |       ⌋               | I4 | 0  L  |        | I4 |L   E22 |          -I4-|------- F ≃  ℤ3 : σ8 = |---|-0--L--| , σ9 = |----|L----0--| , σ10 = ⌈    |N      ⌉,               ⌈    |N      ⌉        ⌈    |N       ⌉              |    N                    |    N                |    N                  |

              ⌊    |       ⌋                 -I4|-------- F ≃  ℤ6 : σ11 = ⌈  |D      ⌉ .                    |    D

We take b1 = b2 = b3 = b4 = e21   , b5 = b6 = b7 = e14-   , b8 = b9 = b10 = e31   and b11 = e16   .

The next theorem gives the cohomology rings over ℝ  of the hyperkähler manifolds Γ \ℝ8   i   , where Γ  = 〈γ ,Λ 〉   i     i  i with γ = σ L  i    i bi   ,1 ≤  i ≤ 11  , and Γ  i  is one of the 8  -dimensional Bieberbach groups listed above.

Theorem 4.1. Let M Γ   be an 8  -dimensional hyperkähler manifold that is not a torus, where Γ = Γ i   , 1 ≤ i ≤ 11  , is one of the Bieberbach groups given above. Then the cohomology ring is an exterior algebra with generators e ,η   i j   given as follows:

If  1 ≤ i ≤ 4 : {ei : 1 ≤ i ≤ 4, degei = 1, ηj : 1 ≤ j ≤ 6; deg ηj = 2}.  If 5 ≤  i ≤ 11 : {ei : 1 ≤ i ≤ 4, deg ei = 1, ηj : 1 ≤ j ≤ 4;deg ηj = 2}.

The Poincaré polynomials of M  Γ     i   are respectively given by

      {          (t + 1)4(t4 + 6t2 + 1), for 1 ≤ i ≤ 4, p(t) =    (t + 1)4(t4 + 4t2 + 1), for 5 ≤ i ≤ 11.

Proof. In this case we will not proceed as in Theorem 3.1, but, instead, we will diagonalize the induced holonomy action of F  on  8 ℂ   .

In the case when Γ = Γ 1   , clearly the F  -invariants are an exterior algebra generated by the elements of the form ei,1 ≤ i ≤ 4  and ei ∧ ej  with 5 ≤ i < j ≤ 8  . If Γ = Γ 2   , the answer is the same, with the same generators of degree 1; as elements of degree two we have to take f ∧ f  i   j  , 5 ≤  i < j ≤ 8  , where f  = e  j    j  for j = 5,7  and           1 f6 = e6 + 2e1   ,           1 f8 = e8 + 2e3   . The cases of Γ 3   and Γ 4   are identical to that of Γ 2   .

In the next case, when F ≃ ℤ4   , the holonomy action can be diagonalized over ℂ  in a suitable basis so that σfj = fj  for 1 ≤ j ≤  4  , and σfj = ifj  (resp. -  ifj  ), for i = 5,7  (resp. for i = 6,8  ). Thus the algebra of complex F  -invariants is an exterior algebra with generators fj, 1 ≤ j ≤ 4,  and exterior products of the form fi ∧ fj  where i = 5,7  , j = 6,8  . Furthermore we have that f =  ¯f  6    5   and f  = f¯  8    7   .

We now determine the real F  -invariants in degree two. We have, over ℂ  , that the generators are      ¯ f5 ∧ f5   ,      ¯ f7 ∧ f7   ,      ¯ f5 ∧ f7   and      ¯ f7 ∧ f5   .

If we set f5 = g5 + ih5   , f7 = g7 + ih7   , with g5,g7   , h5,h7   real forms, we see that

f5 ∧ ¯f5 =                 - 2ig5 ∧ h5, f7 ∧ ¯f7 =                 - 2ig7 ∧ h7,      ¯ f5 ∧ f7 =  (g5 ∧ g7 + h5 ∧ h7) + i(h5 ∧ g7 - g5 ∧ h7), f7 ∧ ¯f5 =  - (g5 ∧ g7 + h5 ∧ h7) + i(h5 ∧ g7 - g5 ∧ h7)

Thus, we get that the real cohomology of M  Γ 3   in degree two is spanned by

g5 ∧ h5, g7 ∧ h7, g5 ∧ g7 + h5 ∧ h7 and h5 ∧ g7 - g5 ∧ h7.

Similarly, if F  ≃ ℤ3   , the holonomy action can be diagonalized over ℂ  in a basis fj  , such that σfj = fj  for 1 ≤ j ≤ 4  , and σfj = ωfj  (resp. ¯ωfj  ), for i = 5,7  (resp. for i = 6,8  ). Here ω  is a primitive root of 1 of order 3. Furthermore, again f6 = f¯5   and f  = f¯  8    7   .

Thus, in this case, the algebra of F  -invariants is an exterior algebra with generators fj, 1 ≤ j ≤ 4,  and exterior products of degree two fi ∧ fj  where i = 5,7  , j = 6,8  . The real invariants are obtained in the same way as in the case F  ≃ ℤ4.

The situation when F  ≃ ℤ6   is entirely similar except that we must take ω  to be a primitive root of 1 of order 6.

Using the above information we see that the Betti numbers of the manifolds M   Γ i   , 1 ≤ i ≤ 4  , are as follows:

             (  )    ( )                        ( )2 β1 = 4, β2 =   4  +   4   = 12,  β3 = 28, β4 =   4   +  2 = 38.                2      2                          2

whereas, for 5 ≤ i ≤ 11  :

              (  )   (  )                                    (  ) β1 = 4,  β2 =   4  +   4  =  10, β3 = 4 + 42 = 20,  β4 = 2 +  4  4 = 26.                 2      1                                      2

As a verification, we note that in both cases we have that ∑8        i    i=0(- 1) βi = 0  (as it should).

One finally easily checks that the corresponding Poincaré polynomials are respectively given by (t + 1)4(t4 + 6t2 + 1)  and (t + 1)4(t4 + 6t2 + 1)  , as asserted. □

Remark 4.2. (i) In the case when M Γ   is the torus T 8   , we have Γ =  ℤ8   , and the cohomology ring is just the exterior algebra ∧ *ℝn  ; we have      ( ) βi =  n       i for 0 ≤ i ≤ 8  .

(ii) We note that all Poincaré polynomials are divisible by        4 (t + 1)   , a fact valid for all hyperkähler manifolds ([Sa]). This fails to be true in the quaternionic Kähler case (see Example 5.2).

In [Sa] Salamon obtains a general identity for the Betti numbers of a 4m  -dimensional hyperkähler manifold. This identity reads, for n = 4,8,12  :

4β1 + β2 = 22  (n =  4)

(1)

25β1 + β3 + β4 = 46 + 10 β2  (n = 8)

(2)

48 β1 + 16β3 + β6 = 70 + 30β2 + 6β4   (n =  12)

(3)

In the next section, we shall give examples showing these identities need not hold in the quaternionic Kähler case.

5. Quaternionic Kähler manifolds

The purpose of this section is to compute the cohomology ring of some quaternionic Kähler flat manifolds which are not hyperkähler. These examples will reveal several new features.

Example 5.1. We first look at a simple 4  -dimensional manifold with holonomy group ℤ2.

Let Γ = 〈EL  e12 ,Λ 〉 where      [        ]       I2 E =       - I2 and Λ  is the canonical lattice. Note that Γ  is essentially the double of the Klein bottle group.

Consider the two anticommuting complex structures in   4 ℝ   given by

     [     ]         [        ] J1 =  J      ,  J2 =    0   I2  .           J           - I2  0

(4)

It is easy to verify that EJ1  = J1E  , EJ2 =  - J2E  , thus it follows that M Γ   is quaternionic Kähler.

Relative to the Betti numbers we have β0 = β4 = 1  , β1 = β3 =  2  , β2 = 2  , since the F  -fixed vectors in degree 2 are e1 ∧ e2,e3 ∧ e4   . Again, the algebra of invariants is an exterior algebra with generators of degree one and two: e1,e2,e3 ∧ e4   .

Thus we see that 4β1 + β2 = 10 ⁄= 22  , so Salamon's identity (1) does not always hold in the 4  -dimensional quaternionic Kähler case.

Furthermore the Poincaré polynomial is

p(t) = 1 + 2t + 2t2 + 2t3 + t4 = (t + 1)2(t2 + 1).

Thus we see that p(t)  is not divisible by       4 (t + 1)   , but only by       2 (t + 1)   . Note that this should be the case, since M Γ   is Kähler (the complex structure J1   descends). Note also that the torus  4 T   is a hyperkähler covering of M Γ   . It has Poincaré polynomial (t + 1)4   .

Example 5.2. We now look at the cohomology ring for a quaternionic Kähler 8  -manifold with holonomy group ℤ4 × ℤ2   . Let Γ = 〈AL  e2,CL  e1-,Λ〉          2     2 where Λ  is the canonical lattice and

    ⌊     |     ⌋       -I4-|------ C = ⌈     |J    ⌉           |   J ,     ⌊ I2               ⌋     |                  | A = |     - I2          |     ⌈          - I2    ⌉                      I2 .

Consider the two anticommuting complex structures on ℝ8   given by

     ⌊               ⌋          ⌊                 ⌋       J                            0   I2  0    0      |    - J        |          |- I2  0   0    0 | J1 = |⌈         J     |⌉ ,   J2 = |⌈  0   0   0  - I |⌉ .                                                  2                   - J              0   0   I2   0

(5)

Recall that as usual,      [      ]       0  - 1 J =   1   0 .

Here, note that C  commutes with both J  1   , J  2   and A  commutes with J   1   and anticommutes with J2   . Thus M Γ   is a quaternion Kähler manifold. It is also Kähler since J1   descends.

It is easy to see that, in degree 1, the F  -invariants are generated by e1,e2   and in degree 2, by e1 ∧ e2, e3 ∧ e4, e5 ∧ e6, e7 ∧ e8   . Thus β1 = 2, β2 = 4  .

In degrees 3 and 4 we find that

∧3    (ℝ8)F  = span{e1 ∧ e3 ∧ e4, e2 ∧ e3 ∧ e4, e1 ∧ e5 ∧ e6, e2 ∧ e5 ∧ e6,                   e1 ∧ e7 ∧ e8, e2 ∧ e7 ∧ e8, e3 ∧ e5 ∧ e7, e3 ∧ e5 ∧ e8,                                           e4 ∧ e5 ∧ e7, e4 ∧ e6 ∧ e8}.

∧4    8 F     (ℝ )  = span{e1  ∧ e2 ∧ e3 ∧ e4, e1 ∧ e2 ∧ e5 ∧ e6, e1 ∧ e2 ∧ e7 ∧ e8,                    e1 ∧ e3 ∧ e5 ∧ e7,e1 ∧ e3 ∧ e6 ∧ e8, e1 ∧ e4 ∧ e6 ∧ e8,                    e1 ∧ e4 ∧ e5 ∧ e7, e5 ∧ e6 ∧ e7 ∧ e8, e3 ∧ e4 ∧ e7 ∧ e8,                   e3 ∧ e4 ∧ e5 ∧ e6, e2 ∧ e3 ∧ e6 ∧ e8, e2 ∧ e4 ∧ e5 ∧ e7,                                    e2 ∧ e3 ∧ e5 ∧ e7, e2 ∧ e4 ∧ e6 ∧ e8}.

The Poincaré polynomial is given by:

p(t) = 1+ 2t+ 4t2 + 10t3+ 14t4+ 10t5 + 4t6+ 2t7+ t8 = (t+ 1)2(t6+ 3t4+ 4t3+  3t2 + 1).

We see that this cannot be the polynomial of a hyperkähler manifold, since the odd Betti numbers are not a multiple of 4  and p(t)  is not divisible by (t + 1)4   .

Note also that Salamon's identity (2) is not satisfied. Indeed

25β1 + β3 + β4 = 25.2 + 10 + 14 = 74 ⁄= 46 + 10β2 =  46 + 10.4 = 86.

(6)

Example 5.3. We now look at a quaternionic square double of the Klein bottle. As shown in [DM], there are several such manifolds non diffeomorphic to each other. Since they all have the same holonomy representation it will suffice to consider only one example of this type. Let Γ = 〈EL  e , A ′′Le , B′′L e , Λ 〉          12-     -22      23 where Λ  is the canonical lattice and

    ⌊                 ⌋       E1     |     E1          | E = |⌈          E      |⌉                 1                    E1 ,       ⌊                 ⌋        I2   ′′   |    - I2         | A   = |⌈          - I    |⌉                     2                       I2 ,       [       ]   ′′    I4 B  =       - I              4 . Let J1   , J2   be the following anticommuting complex structures on ℝ8   :

     ⌊               ⌋       J                         [       ] J =  ||    J          || ,   J =    0    I4  .  1   ⌈       - J     ⌉      2    - I4  0                   - J

(7)

Here, as usual,      [      ] J =   0  - 1       1   0 and      [      ] E1 =   1  0        0  - 1 .

In this case, each of the elements in the holonomy group, either commutes or anticommutes with J1,J2   but neither of these complex structures descends; M Γ   is a quaternion Kähler manifold.

It is easy to see that, in degree 1, the F  -invariants are generated by e1   and in degree 2, this space is zero.

In degrees 3 and 4 we find that

∧3    n F    (ℝ  )  = span{e2 ∧ e3 ∧ e4, e2 ∧ e5 ∧ e6, e2 ∧ e7 ∧ e8, e3 ∧ e5 ∧ e7,                              e  ∧ e ∧ e , e ∧ e  ∧ e , e ∧ e ∧ e }.                               3    6   8   4   5    8  4    6   7

∧4     (ℝn)F = span{e1  ∧ e2 ∧ e3 ∧ e4, e1 ∧ e2 ∧ e5 ∧ e6, e1 ∧ e2 ∧ e7 ∧ e8,                     e1 ∧ e3 ∧ e5 ∧ e7,e1 ∧ e3 ∧ e6 ∧ e8, e1 ∧ e4 ∧ e5 ∧ e8,                   e1 ∧ e4 ∧ e6 ∧ e7, e5 ∧ e6 ∧ e7 ∧ e8, e3 ∧ e4 ∧ e7 ∧ e8,                    e3 ∧ e4 ∧ e5 ∧ e6, e2 ∧ e4 ∧ e6 ∧ e8, e2 ∧ e4 ∧ e5 ∧ e7,                                    e2 ∧ e3 ∧ e6 ∧ e7, e2 ∧ e3 ∧ e5 ∧ e8}

The Poincaré polynomial is given by:

                3      4    5    7   8          4 4     3    2 p(t) = 1 + t + 7t + 14t + 7t  + t + t  = (t + 1)(t  - 4t + 6t  - 3t + 1).

We see that this cannot be the polynomial of a hyperkähler manifold, since the odd Betti numbers are odd and they should be a multiple of 4  . Actually M Γ   cannot even have a Kähler structure since odd Betti numbers are odd and β  =  0   2  .

On the other hand, we observe that the Poincaré polynomial is divisible by       4 (t + 1)   and Salamon's identity (2) is satisfied. Indeed

25β1 + β3 + β4 = 25 + 7 + 14 = 46 = 46 + 10 β2.

(8)

By inspection, we see that in this (non hyperkähler) case, the algebra of invariants is not an exterior algebra. The invariants in degree 1 and 3 do not suffice to generate the invariants in degree 4 unless we include the action of the star operator.

6. Doubling Hantzsche-Wendt groups

In this section we shall compute the cohomology ring of M  6 = dΓ \ℝ6,  and M 12 = d2Γ \ℝ12   where Γ  is the classical Hantzsche-Wendt Bieberbach group in dimension 3 (see [Wo]). We shall see that the cohomology ring of M 12   is far from having the structure of an exterior algebra in this case.

Let Γ =  {ALa, BLb, Le  : 1 ≤ i ≤ 3}                    i , where      ⌊             ⌋        ⌊             ⌋        - 1                    1 A =  ⌈      - 1    ⌉   B =  ⌈    - 1      ⌉   a =  e3,b = e1+e2                  1                    - 1          2       2   .

It turns out that      3 Γ \ℝ   is the only 3  -dimensional compact flat manifold with β1 =  0.  It is called the Hantzsche-Wendt manifold ([Wo]). The Poincaré polynomial is given by p(t) = 1 + t3   and the holonomy group is ℤ22   . We shall next study the cohomology ring of Md Γ   and Md2 Γ   .

Theorem 6.1. The cohomology ring ∧ *   6 F     (ℝ  )   of Md Γ   is a graded algebra of dimension 16 generated by the elements of degree 2 : {ηi : 1 ≤ i ≤ 3} and of degree 3: {δj : 1 ≤ j ≤ 8} subject to the relations η2i = ηiδj = δi.δj = 0.

The Poincaré polynomial of Md Γ   is given by

p(t) = 1 + 3t2 + 8t3 + 3t4 + t6 = (1 + t)2(t4 - 2t3 + 6t2 - 2t + 1).

Proof. The generators of the holonomy group F  of dΓ  are

     ⌊ -1           ⌋         ⌊ 1            ⌋           -1                     -1 A′ = ⌈      1 -1    ⌉   B ′ = ⌈     -1 1     ⌉                 - 1 1                   -1 -1

The holonomy action on the exterior algebra diagonalizes in the canonical basis with eigenvalues ±  1  . Clearly there are no vectors fixed by both A ′,B ′ . Thus β1(Md Γ ) = 0.  On the other hand, the fixed vectors in degree d >  1  are:

∧2   6 F    (ℝ )  = span{e2  ∧ e5,e3 ∧ e6,e1 ∧ e4},  thus β2 =  3  .

∧3   6 F    (ℝ )  = span{ei ∧ ej ∧ ek :  exactly one of i,j,k  is 1  or 4  and one is 3  or 6}.

That is:

∧   3(ℝ6)F = span{e1  ∧ e3 ∧ e2, e1 ∧ e3 ∧ e5, e1 ∧ e6 ∧ e2, e1 ∧ e6 ∧ e5, e4 ∧ e3 ∧ e2, e4 ∧ e3 ∧ e5,  e4 ∧ e6 ∧ e2,  e4 ∧ e6 ∧ e5} . Hence β3 = 8  .

∧4( ℝ6)F = ∧2( ℝ6)F ∧ ∧2( ℝ6)F =  span{e  ∧ e  ∧ e ∧ e ,e  ∧ e ∧ e ∧ e ,e  ∧ e ∧ e  ∧ e }                                          2   5    3   6  2    5   1    4 3    6   1    4 . Thus β4 = 3  .

∧5   6 F      ∧6    6F    (ℝ )  = 0,    (ℝ )  =  ℝ e1 ∧ e2 ∧ e3 ∧ e4 ∧ e5 ∧ e6.

The remaining assertions in the theorem follow immediately from the above description of the F  -invariants. □

We now look at the case of d2Γ  . By Proposition 2.2, M 122 = d2 Γ \ℝ12   dΓ   is a 12  -dimensional hyperkähler manifold with holonomy group F  = ℤ2 .        2

Theorem 6.2. The even cohomology ring ∧   ev(M  12)F   is an exterior algebra generated by η ,...,η  ∈  ∧2 (ℝ12)F  1      18   i.e., the ℝ  -algebra generated by {η : 1 ≤ i ≤ 18}   i subject to the relations  2 ηi = 0,ηiηj = ηjηi.  The full cohomology ring ∧ *  12 F   (ℝ   )   is generated by elements of degree two { ηi : 1 ≤ i ≤ 18} and degree 3 { δj : 1 ≤ i ≤ 64} , subject to the relations:

 2                   2 ηi = 0, ηiηj = ηjηi, δi = 0, δiδj = - δjδi, ηkδm =  δmηk

and furthermore ηkδm   may (or may not) be equal to zero. This vanishing can be explicitly given in terms of k,m  (but is complicated).

Proof. The holonomy group is generated by:

      [A  ′    ]           [B  ′     ] A ′′ =         ′      B ′′ =        ′             A                     B

Again, we need to compute the F  -invariants in ∧   *(ℝ12).  Clearly, we have no fixed vectors of degree one, hence β1 =  0.  Now consider the three complementary sets:
 S1 = {e3,e6,e9,e12}=  set of basis vectors fixed by A ′′,                                                      ′′ S2 =  {e1,e4,e7,e10}=   set of basis vectors fixed by B , S3 =  {e2,e5,e8,e11}=   basis vectors fixed by neither of A ′′,B ′′.

In degree 2, we note that ei ∧ ej  is F  -invariant if and only if ei,ej  lie both in one of S   1   , S   2   or S  3   . Thus ∧2(ℝ12)F  = span{e  ∧ e  : {e ,e} ⊂ S , k = 1,2,3}                    i   j    i  j     k has dimension  (4) 3 2  = 18.

Similarly, we see that the fixed vectors in higher degrees can be expressed in terms of the sets Si  :

∧3    (ℝ12)F = span{ei ∧ ej ∧ ek :  exactly one of ei,ej,ek  lies in S1   , one in S2   and one in S3   }. Thus      ( ) β3 =  4 3 = 64.       1

∧4( ℝ12)F = span{e  ∧ e  ∧ e ∧ e  :                    i   j    k   m  an even number of the e ,e,e ,e  i  j k  m  lie in each one of S1,S2,S3}.  Thus,        (4)    (4)2 β4 = 3  4 +  3 2  = 3 + 108 =  111.

∧   5(ℝ12)F = span{ei ∧ ej ∧ ek ∧ em ∧ eh :  an odd number of the e′i  s lie in each one of S   1   , S   2   , S }.  3  Hence β  =  3(4)(4)2 = 3 × 64 = 192   5     3  1  .

∧6   12 F    (ℝ  )  = span{ei ∧ ej ∧ ek ∧ em ∧ eh ∧ em :  an even number of the  ′ ei  s lie in each Si, i = 1,2,3} . Hence       ( )   ( )3 β6 = 6 42 +   42  = 36 + 216 =  252  .

Thus, the Poincaré polynomial p(t)  is given by:

       2     3       4       5       6       7      8      9      10   12 1 + 18t + 64t  + 111t  + 192t +  252t + 192t  + 111t  + 64t  + 18t  + t            =  (t + 1)4(t8 - 4t7 + 28t6 - 28t5 + 10t4 - 28t3 + 28t2 - 4t + 1).

As a verification, note that Salamon's identity (3) holds:

48.0 + 16.64 + 252 =   1024 + 252 =  1276  70 + 30.18 + 6.111 =   70 + 540 + 666 = 1276.

We now look at the cohomology ring. By the description of the invariants it is clear that ∧2(ℝ12)F  generates ∧2k( ℝ12)F  for any 0 < k ≤  6  , while it is not hard to check that ∧2k+1   12F      (ℝ   )  can be generated by ∧2   12 F    (ℝ  )  and ∧3    12 F    (ℝ  )  . The relations in the statement can also be easily verified.

Thus, the cohomology ring is generated as an algebra by ∧2    (ℝ12)F  and ∧3    (ℝ12)F  , as claimed. □

Remark 6.3. There is a natural generalization of the previous example. It is known that for any n  odd, there exists a large family of n  -dimensional Bieberbach groups with holonomy group  n-1 ℤ2   , and such that the corresponding flat manifold M Γ   is a rational homology sphere, i.e. all Betti numbers except β0, βn  are equal to zero (see [MR]). These manifolds generalize the classical 3  -dimensional Hantzsche-Wendt manifold ([Wo]) and are called HW  -manifolds, for short. The argument in the proof of Theorem 6.1 can be adapted to any odd dimension n  and gives a similar result on the cohomology ring of   2 Md Γ   , for any HW  -group Γ  .

Indeed, for any n  odd, one shows that the cohomology ring of any HW  -manifold ∧ev   4n F    (ℝ   )  is generated by ∧2    4n F    (ℝ   )  . Actually, it is an exterior algebra generated by { ηi : 1 ≤ i ≤ 6n} , subject to the relations η2i = 0, ηiηj = ηjηi, i ⁄= j  .

The full cohomology ring ∧( ℝ4n)F  is generated by ∧2(ℝ4n)F  and ∧3( ℝ4n)F  . It has generators {η : 1 ≤ i ≤ 6n}   i of degree 2, and { δ : 1 ≤ j ≤ 4n}    j of degree 3. They satisfy the following relations

  2                  2 ηi = 0, ηiηj = ηjηi, δj = 0, δjδi = - δiδj, ηiδi = δjηi.

Furthermore, we note that there are more relations, linking ηiδj  with ηkδm  , for different i,j,k,m  .

The basis     4n {ei}i=1   is split into n  complementary sets S1,S2,...,Sn  , with Si = {ei,ei+3,ei+6, ei+9} , and where Si  is the fixed set of   ′′ B i  , for i = 1,...n  .

By arguing as in the case n = 3  we see that

∧   2(ℝ4n)F = span{er  ∧ es : both er,es lie in the same set Si}.  Hence we obtain        ( ) β2 = n  4 =  6n.         2

Similarly

∧2k   4n F     (ℝ   )  = span{er1 ∧ ...∧ er2k   : an even number of erj   's lie in each Si} for each i  .

∧2k+1   4n F       (ℝ  )  = span{er1 ∧ ...∧ er2k+1   : an odd number of eij   's lie in each Si} for each i  .

We note that this implies in particular that β =  0,  i  for i  odd, i < n.

Let us illustrate the previous discussion by computing the cohomology for n = 5.

Clearly we have β0 = 1, β1 = 0  . Furthermore

∧2   20 F    (ℝ  )  ≃ span{er ∧  es :  both er,es  lie in one of the Si 1 ≤ i ≤ 5}.  Thus        (4) β2 = 5  2 = 30.

∧3   20 F    (ℝ  )  ≃ span{er1 ∧ er2 ∧ er3   such that each Si (1 ≤  i ≤ 5)  contains one of the erj}.  That is, β3 =  0  .

Similarly:

      (4)  (5)(4)2 β4 = 5 4 +   2 2   = 365  ,

     ( ) β5 =  4 5 = 1024       1  ,

       (4)(4)   (5)(4)3 β6 = 20 4  2 +   3 2   = 2280  ,

     ( )()( ) β7 =  5  4  44 = 20 × 256 =  5120       1  3  1  ,

     (5)(4)(4)    (4)3(4)   (5)(4)4 β8 =  2  4  4 + 5 2   4  +  4  2   = = 10 + 5 × 216 +  30 × 216 = 7570  ,

     ( )() ( ) β9 =  52  43 241 3 = 10240  ,

      (5)    (4)   (5)(4)(4)(4)3  (4)5 β10 =  23 ×  2  +  1  4  3  2  +  2   = 10 × 18 + 20 × 216 + 36 × 216 =  122276  .

Therefore, we finally obtain:

p(t) =  1 + t20 + 30(t2 + t18) + 365(t4 + t16) + 1024(t5 + t15) + 2280(t6 + t14) +                            7   13          8   12           9   11          10                     5120(t  + t  ) + 7570(t + t  ) + 10240(t + t  ) + 12276t  .

References

[Be]    Besse A.,L., Einstein Manifolds, Ergebnisse der Math. 10, Springer Verlag, 1987.

[BBNWZ]    Brown,H., Bülow, R., Neubüser, J. Wondratschok, H., Zassenhaus H., Crystallographic Groups of Dimension Four, John Wiley, New York, 1978.

[BDM]    Barberis, L., Dotti, I., Miatello, R., Clifford structures on certain locally homogeneous manifolds, Annals Global Analysis and Geometry 13 (1995) 289-301.

[BG]    Benson, Ch., Gordon, C., Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), 513-518.

[Ch]    Charlap, L., Bieberbach groups and flat manifolds, Springer Verlag 1986.

[DM]    Dotti, I.G., Miatello, R.,J., Quaternionic Kähler flat manifolds, Differential Geometry and Applications 15 (2001) 59-77.

[Hi]    Hiller, H., Cohomology of Bieberbach groups, Mathematika 32 (1985) 55-59.

[JR]    Johnson, F.E., Rees, E., Kähler groups and rigidity phenomena, Proc. Camb. Phil. Soc 109 (1991) 31-44.

[MR]    Miatello, R.J., Rossetti J.P., Isospectral Hantzsche-Wendt manifolds, Jour. für die reine angewandte Mathematik 515 (1999) 1-23.

[Sa]    Salamon, S., On the cohomology of Kähler and hyperkähler manifolds, Topology 35 (1996) 137-155.

[Sa2]    Salamon, S., Riemannian manifolds and holonomy groups, Pitman Res. Notes in Math. 201 (1989).

[Sa3]    Salamon, S., Spinors and cohomology, Rend. Sem. Univ. Pol. Torino 50 (1992) 393-410.

[Ve]    Verbitskii, M., Action of the Lie algebra of SO(5) on the cohomology of hyperkähler manifolds, Functional Anal. Appl. 24 (1991) 229-230.

[Wh]    Whitt, L., Quaternionic Kähler manifolds, Transactions of the Amer. Math. Soc. 272 (1982) 677-692.

[Wo]    Wolf, J.A., Spaces of constant curvature, New York, Mc Graw Hill 1967.

 

I.G. Dotti
FaMAF-CIEM, Universidad Nacional de Córdoba,
Córdoba 5000, Argentina
idotti@mate.uncor.edu

R.J. Miatello
FaMAF-CIEM, Universidad Nacional de Córdoba,
Córdoba 5000, Argentina
miatello@mate.uncor.edu

Recibido: 6 de abril de 2006
Aceptado: 7 de agosto de 2006

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