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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.46 n.2 Bahía Blanca jul./dic. 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
![dqΓ](/img/revistas/ruma/v46n2/2a09143x.png)
![Λ ⊕ Λ](/img/revistas/ruma/v46n2/2a09144x.png)
![ℤk+1 2](/img/revistas/ruma/v46n2/2a09145x.png)
![d Γ q](/img/revistas/ruma/v46n2/2a09146x.png)
![v∈∕ Λ](/img/revistas/ruma/v46n2/2a09147x.png)
![γ = BLb ∈ Γ](/img/revistas/ruma/v46n2/2a09148x.png)
![(B + I)(φ(B) + v) ∈ Λ \ (B + I)Λ, or (B - I)b ∕∈ (B - I)Λ.](/img/revistas/ruma/v46n2/2a09149x.png)
![F](/img/revistas/ruma/v46n2/2a09150x.png)
![v](/img/revistas/ruma/v46n2/2a09151x.png)
![d (Γ ,v)\ℝ2n q](/img/revistas/ruma/v46n2/2a09152x.png)
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
![* ∧ * F H (M Γ ,𝕂) ≃ ( (Λ ⊗ K)) .](/img/revistas/ruma/v46n2/2a09161x.png)
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
![σ6η = a12e1 ∧ e2 + a13e1 ∧ (- e2 - e3 + e4) + a14e1 ∧ (- e3) + a23e2 ∧ (- e3 + e4) + a24e2 ∧ (- e3) + a34(- e2 + e4) ∧ (- e3).](/img/revistas/ruma/v46n2/2a09198x.png)
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
![η = ae1 ∧ e2 ∧ e3 + be1 ∧ e2 ∧ e4 + ce1 ∧ e3 ∧ e4 + de2 ∧ e3 ∧ e4](/img/revistas/ruma/v46n2/2a09210x.png)
![a,b,c,d ∈ ℝ](/img/revistas/ruma/v46n2/2a09211x.png)
![σ6η = (- a - b + c)e1 ∧ e2 ∧ e3 + ae1 ∧ e2 ∧ e4 + ce1 ∧ e3 ∧ e4 + de2 ∧ e3 ∧ e4.](/img/revistas/ruma/v46n2/2a09212x.png)
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.
![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------------](/img/revistas/ruma/v46n2/2a09223x.png)
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
![[ ] [ ] [ ] X = 1 0 , J = 0 - 1 , L = 0 1 , 1 0 1 0 0 1](/img/revistas/ruma/v46n2/2a09229x.png)
![[0 - 1 ] [0 - 1 ] [1 0 ] D = , N = , E = . 1 1 1 - 1 1 1](/img/revistas/ruma/v46n2/2a09230x.png)
![Eij](/img/revistas/ruma/v46n2/2a09231x.png)
![2 × 2](/img/revistas/ruma/v46n2/2a09232x.png)
![1](/img/revistas/ruma/v46n2/2a09233x.png)
![(i,j)](/img/revistas/ruma/v46n2/2a09234x.png)
![0](/img/revistas/ruma/v46n2/2a09235x.png)
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:
![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}.](/img/revistas/ruma/v46n2/2a09258x.png)
![M Γ i](/img/revistas/ruma/v46n2/2a09259x.png)
![{ (t + 1)4(t4 + 6t2 + 1), for 1 ≤ i ≤ 4, p(t) = (t + 1)4(t4 + 4t2 + 1), for 5 ≤ i ≤ 11.](/img/revistas/ruma/v46n2/2a09260x.png)
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
![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)](/img/revistas/ruma/v46n2/2a09303x.png)
![M Γ 3](/img/revistas/ruma/v46n2/2a09304x.png)
![g5 ∧ h5, g7 ∧ h7, g5 ∧ g7 + h5 ∧ h7 and h5 ∧ g7 - g5 ∧ h7.](/img/revistas/ruma/v46n2/2a09305x.png)
![F ≃ ℤ3](/img/revistas/ruma/v46n2/2a09306x.png)
![ℂ](/img/revistas/ruma/v46n2/2a09307x.png)
![fj](/img/revistas/ruma/v46n2/2a09308x.png)
![σfj = fj](/img/revistas/ruma/v46n2/2a09309x.png)
![1 ≤ j ≤ 4](/img/revistas/ruma/v46n2/2a09310x.png)
![σfj = ωfj](/img/revistas/ruma/v46n2/2a09311x.png)
![¯ωfj](/img/revistas/ruma/v46n2/2a09312x.png)
![i = 5,7](/img/revistas/ruma/v46n2/2a09313x.png)
![i = 6,8](/img/revistas/ruma/v46n2/2a09314x.png)
![ω](/img/revistas/ruma/v46n2/2a09315x.png)
![f6 = f¯5](/img/revistas/ruma/v46n2/2a09316x.png)
![f = f¯ 8 7](/img/revistas/ruma/v46n2/2a09317x.png)
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:
![( ) ( ) ( )2 β1 = 4, β2 = 4 + 4 = 12, β3 = 28, β4 = 4 + 2 = 38. 2 2 2](/img/revistas/ruma/v46n2/2a09328x.png)
![5 ≤ i ≤ 11](/img/revistas/ruma/v46n2/2a09329x.png)
![( ) ( ) ( ) β1 = 4, β2 = 4 + 4 = 10, β3 = 4 + 42 = 20, β4 = 2 + 4 4 = 26. 2 1 2](/img/revistas/ruma/v46n2/2a09330x.png)
![∑8 i i=0(- 1) βi = 0](/img/revistas/ruma/v46n2/2a09331x.png)
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
![p(t) = 1 + 2t + 2t2 + 2t3 + t4 = (t + 1)2(t2 + 1).](/img/revistas/ruma/v46n2/2a09365x.png)
![p(t)](/img/revistas/ruma/v46n2/2a09366x.png)
![4 (t + 1)](/img/revistas/ruma/v46n2/2a09367x.png)
![2 (t + 1)](/img/revistas/ruma/v46n2/2a09368x.png)
![M Γ](/img/revistas/ruma/v46n2/2a09369x.png)
![J1](/img/revistas/ruma/v46n2/2a09370x.png)
![4 T](/img/revistas/ruma/v46n2/2a09371x.png)
![M Γ](/img/revistas/ruma/v46n2/2a09372x.png)
![(t + 1)4](/img/revistas/ruma/v46n2/2a09373x.png)
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
![∧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}.](/img/revistas/ruma/v46n2/2a09395x.png)
![∧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}.](/img/revistas/ruma/v46n2/2a09396x.png)
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).](/img/revistas/ruma/v46n2/2a09397x.png)
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
![∧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](/img/revistas/ruma/v46n2/2a09417x.png)
![∧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}](/img/revistas/ruma/v46n2/2a09418x.png)
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).](/img/revistas/ruma/v46n2/2a09419x.png)
![4](/img/revistas/ruma/v46n2/2a09420x.png)
![M Γ](/img/revistas/ruma/v46n2/2a09421x.png)
![β = 0 2](/img/revistas/ruma/v46n2/2a09422x.png)
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
![p(t) = 1 + 3t2 + 8t3 + 3t4 + t6 = (1 + t)2(t4 - 2t3 + 6t2 - 2t + 1).](/img/revistas/ruma/v46n2/2a09444x.png)
Proof. The generators of the holonomy group of
are
![⌊ -1 ⌋ ⌊ 1 ⌋ -1 -1 A′ = ⌈ 1 -1 ⌉ B ′ = ⌈ -1 1 ⌉ - 1 1 -1 -1](/img/revistas/ruma/v46n2/2a09447x.png)
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:
![2 2 ηi = 0, ηiηj = ηjηi, δi = 0, δiδj = - δjδi, ηkδm = δmηk](/img/revistas/ruma/v46n2/2a09480x.png)
![ηkδm](/img/revistas/ruma/v46n2/2a09481x.png)
![k,m](/img/revistas/ruma/v46n2/2a09482x.png)
Proof. The holonomy group is generated by:
![[A ′ ] [B ′ ] A ′′ = ′ B ′′ = ′ A B](/img/revistas/ruma/v46n2/2a09483x.png)
![F](/img/revistas/ruma/v46n2/2a09484x.png)
![∧ *(ℝ12).](/img/revistas/ruma/v46n2/2a09485x.png)
![β1 = 0.](/img/revistas/ruma/v46n2/2a09486x.png)
![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 ′′.](/img/revistas/ruma/v46n2/2a09487x.png)
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:
![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).](/img/revistas/ruma/v46n2/2a09518x.png)
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.](/img/revistas/ruma/v46n2/2a09519x.png)
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
![2 2 ηi = 0, ηiηj = ηjηi, δj = 0, δjδi = - δiδj, ηiδi = δjηi.](/img/revistas/ruma/v46n2/2a09550x.png)
![ηiδj](/img/revistas/ruma/v46n2/2a09551x.png)
![ηkδm](/img/revistas/ruma/v46n2/2a09552x.png)
![i,j,k,m](/img/revistas/ruma/v46n2/2a09553x.png)
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:
![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 .](/img/revistas/ruma/v46n2/2a09593x.png)
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[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.
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[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