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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).
A Riemannian manifold is said to be Kähler if the holonomy group is contained in . It is quaternion Kähler if the holonomy group is contained in . It is known that quaternion Kähler manifolds of dimension are Einstein, so the scalar curvature splits these manifolds according to whether or . Ricci flat quaternion Kähler manifolds include hyperkähler manifolds, that is, those with holonomy group contained in . 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 with the standard flat metric and a pair of orthogonal anticommuting complex structures. This hyperkähler structure descends to the -torus , for any lattice in If is a compact flat manifold such that the holonomy action of centralizes (resp. normalizes) the algebra generated by , then 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 -dimensional Kähler flat manifolds and all -dimensional compact flat kyperkähler manifolds. We shall make use of the known classification of space groups in dimension , 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 -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 , a -dimensional compact flat hyperkähler manifold with holonomy group . We will show that the cohomology ring is generated by the -invariant forms of degree and , 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 () is far from being an exterior algebra, as seen in the -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 as its universal covering space and a Bieberbach group as fundamental group (i.e. a discrete cocompact subgroup of which is torsion-free). If , let denote translation by By Bieberbach's first theorem, if is a crystallographic group then is a lattice in . We will identify the lattice with the translation lattice a normal and maximal abelian subgroup of . The quotient is a finite group, the point group (or holonomy group) of . When is torsion free, the geometric interpretation of is that of the holonomy group of the flat Riemannian manifold .
Let be a Bieberbach group with holonomy group and translation lattice . Let be a -cocycle modulo , that is, satisfies , modulo , for each . Then defines a cohomology class in and one may associate to a crystallographic group with holonomy group and translation lattice . Furthermore, this group is torsion-free if and only if the class of is a special class (see [Ch]).
Definition 2.1. Let be a Bieberbach group with holonomy group and translation lattice . Let be any -cocycle modulo . We let be the subgroup of generated by elements of the form and , for and
We point out that if the holonomy group of centralizes a complex structure on , then 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 and be as in Definition 2.1. Then
(i) is a Bieberbach group with holonomy group , translation lattice and is a Kähler compact flat manifold.
(ii) If has a locally invariant Kähler structure, then is hyperkähler. In particular, if is any -cocycle modulo , then is hyperkähler.
Remark 2.3. Benson-Gordon have proved ([BG]) that if is a simply connected nilpotent Lie group, is a discrete cocompact subgroup of , and has a Kähler structure (with positive definite) then 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 as in Proposition 2.2. In this paper we shall work with the -cocycle associated to , as in [BDM]. We will denote by in this case.
For many Bieberbach groups one can enlarge into a Bieberbach group in such a way that some element in the holonomy group of anticommutes with the complex structure in . 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 be a Bieberbach group with holonomy group , with translation lattice and such that for any . Set . Set where .
Under rather general conditions, contains as a normal subgroup of index 2, and can be chosen so that is torsion free, so is a compact flat manifold with holonomy group having as a double cover the Kähler manifold . Furthermore commutes with , but only anticommutes with . If we use this construction twice we get a Bieberbach group such that the holonomy group normalizes two anticommuting complex structures, , on , hence will be a quaternion Kähler manifold.
In the next results we give conditions on that ensure that is torsion free. We also note that if is even, will always be orientable. This construction will be used in Section 5.
Theorem 2.6. Let as above. Then
then is a crystallographic group with translation lattice and holonomy group . Furthermore, is torsion-free if and only if and for each we have:
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 be a Bieberbach group and . If is a field such that the characteristic of does not divide , then the cohomology ring of with coefficients in is given by
Let be a -dimensional Bieberbach group with holonomy group . It is not hard to see that in order for to be Kähler, it is necessary and sufficient that commutes with a complex structure on Using the classification of compact flat manifolds of dimension in [BBNWZ] we see that those groups with non trivial holonomy group which have such property have cyclic holonomy groups of order or , and have the form , with , as follows:
We note that in the case of the torus , the cohomology ring is an exterior algebra generated by elements of order 1, and the Poincaré polynomial is . For general flat Kähler -manifolds we have:
Theorem 3.2. If is a -dimensional Kähler flat manifold which is not a torus, the cohomology ring is an exterior algebra in where the have degree 1 and the have degree . Furthermore, in all cases one has and the Poincaré polynomial is given by , .
Proof. To determine the real cohomology rings of the Kähler flat manifolds of dimension , we need to compute the -invariants in each degree, for each Bieberbach group in the family considered above.
We shall carry out this computation only in the case of the group . 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 .
Assume now that satisfies . Now
Now, implies and , thus . Also, it follows that and , thus , hence the -fixed space in degree 2 is spanned by the invariant -forms and , as asserted.
We now turn into degree 3. Let
Thus implies , . Thus we get that the space of -invariants in degree 3 is generated by the -forms and . This completes the verification for .
In the remaining cases the invariants are computed similarly. We now give a table that lists the -invariants in each degree, for each group.
Here .
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
According to [Wh], Theorem 4.3, the holonomy group of is cyclic with generator given by , or by one of the following:
We take , , and .
The next theorem gives the cohomology rings over of the hyperkähler manifolds , where with ,, and is one of the -dimensional Bieberbach groups listed above.
Theorem 4.1. Let be an -dimensional hyperkähler manifold that is not a torus, where , , is one of the Bieberbach groups given above. Then the cohomology ring is an exterior algebra with generators given as follows:
Proof. In this case we will not proceed as in Theorem 3.1, but, instead, we will diagonalize the induced holonomy action of on .
In the case when , clearly the -invariants are an exterior algebra generated by the elements of the form and with . If , the answer is the same, with the same generators of degree 1; as elements of degree two we have to take , , where for and , . The cases of and are identical to that of .
In the next case, when , the holonomy action can be diagonalized over in a suitable basis so that for , and (resp. ), for (resp. for ). Thus the algebra of complex -invariants is an exterior algebra with generators and exterior products of the form where , . Furthermore we have that and .
We now determine the real -invariants in degree two. We have, over , that the generators are , , and .
If we set , , with , real forms, we see that
Thus, in this case, the algebra of -invariants is an exterior algebra with generators and exterior products of degree two where , . The real invariants are obtained in the same way as in the case
The situation when is entirely similar except that we must take to be a primitive root of 1 of order 6.
Using the above information we see that the Betti numbers of the manifolds , , are as follows:
One finally easily checks that the corresponding Poincaré polynomials are respectively given by and , as asserted. □
Remark 4.2. (i) In the case when is the torus , we have , and the cohomology ring is just the exterior algebra ; we have for .
(ii) We note that all Poincaré polynomials are divisible by , 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 -dimensional hyperkähler manifold. This identity reads, for : (1)
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 -dimensional manifold with holonomy group
Let where and is the canonical lattice. Note that is essentially the double of the Klein bottle group.
Consider the two anticommuting complex structures in given by (4)
It is easy to verify that , , thus it follows that is quaternionic Kähler.
Relative to the Betti numbers we have , , , since the -fixed vectors in degree 2 are . Again, the algebra of invariants is an exterior algebra with generators of degree one and two: .
Thus we see that , so Salamon's identity (1) does not always hold in the -dimensional quaternionic Kähler case.
Furthermore the Poincaré polynomial is
Example 5.2. We now look at the cohomology ring for a quaternionic Kähler -manifold with holonomy group . Let where is the canonical lattice and
, .
Consider the two anticommuting complex structures on given by (5)
Recall that as usual, .
Here, note that commutes with both , and commutes with and anticommutes with . Thus is a quaternion Kähler manifold. It is also Kähler since descends.
It is easy to see that, in degree 1, the -invariants are generated by and in degree 2, by . Thus .
In degrees 3 and 4 we find that
The Poincaré polynomial is given by:
We see that this cannot be the polynomial of a hyperkähler manifold, since the odd Betti numbers are not a multiple of and is not divisible by .
Note also that Salamon's identity (2) is not satisfied. Indeed (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 where is the canonical lattice and
, , . Let , be the following anticommuting complex structures on : (7)
Here, as usual, and .
In this case, each of the elements in the holonomy group, either commutes or anticommutes with but neither of these complex structures descends; is a quaternion Kähler manifold.
It is easy to see that, in degree 1, the -invariants are generated by and in degree 2, this space is zero.
In degrees 3 and 4 we find that
The Poincaré polynomial is given by:
On the other hand, we observe that the Poincaré polynomial is divisible by and Salamon's identity (2) is satisfied. Indeed (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 and where is the classical Hantzsche-Wendt Bieberbach group in dimension 3 (see [Wo]). We shall see that the cohomology ring of is far from having the structure of an exterior algebra in this case.
Let , where .
It turns out that is the only -dimensional compact flat manifold with It is called the Hantzsche-Wendt manifold ([Wo]). The Poincaré polynomial is given by and the holonomy group is . We shall next study the cohomology ring of and .
Theorem 6.1. The cohomology ring of is a graded algebra of dimension 16 generated by the elements of degree 2 : and of degree 3: subject to the relations
The Poincaré polynomial of is given by
Proof. The generators of the holonomy group of are
The holonomy action on the exterior algebra diagonalizes in the canonical basis with eigenvalues . Clearly there are no vectors fixed by both . Thus On the other hand, the fixed vectors in degree are:
thus .
exactly one of is or and one is or
That is:
. Hence .
. Thus .
The remaining assertions in the theorem follow immediately from the above description of the -invariants. □
We now look at the case of . By Proposition 2.2, is a -dimensional hyperkähler manifold with holonomy group
Theorem 6.2. The even cohomology ring is an exterior algebra generated by i.e., the -algebra generated by subject to the relations The full cohomology ring is generated by elements of degree two and degree 3 , subject to the relations:
Proof. The holonomy group is generated by:
In degree 2, we note that is -invariant if and only if lie both in one of , or . Thus has dimension
Similarly, we see that the fixed vectors in higher degrees can be expressed in terms of the sets :
exactly one of lies in , one in and one in }. Thus
an even number of the lie in each one of Thus,
an odd number of the s lie in each one of , , Hence .
an even number of the s lie in each . Hence .
Thus, the Poincaré polynomial is given by:
As a verification, note that Salamon's identity (3) holds:
We now look at the cohomology ring. By the description of the invariants it is clear that generates for any , while it is not hard to check that can be generated by and . The relations in the statement can also be easily verified.
Thus, the cohomology ring is generated as an algebra by and , as claimed. □
Remark 6.3. There is a natural generalization of the previous example. It is known that for any odd, there exists a large family of -dimensional Bieberbach groups with holonomy group , and such that the corresponding flat manifold is a rational homology sphere, i.e. all Betti numbers except are equal to zero (see [MR]). These manifolds generalize the classical -dimensional Hantzsche-Wendt manifold ([Wo]) and are called -manifolds, for short. The argument in the proof of Theorem 6.1 can be adapted to any odd dimension and gives a similar result on the cohomology ring of , for any -group .
Indeed, for any odd, one shows that the cohomology ring of any -manifold is generated by . Actually, it is an exterior algebra generated by , subject to the relations .
The full cohomology ring is generated by and . It has generators of degree 2, and of degree 3. They satisfy the following relations
The basis is split into complementary sets , with , and where is the fixed set of , for .
By arguing as in the case we see that
Hence we obtain
Similarly
: an even number of 's lie in each for each .
: an odd number of 's lie in each for each .
We note that this implies in particular that for odd,
Let us illustrate the previous discussion by computing the cohomology for
Clearly we have . Furthermore
both lie in one of the Thus
such that each contains one of the That is, .
Similarly:
,
,
,
,
= ,
,
.
Therefore, we finally obtain:
[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