ON CERTAIN GEOMETRIC AND HOMOTOPY
PROPERTIES OF CLOSED SYMPLECTIC MANIFOLDS
Ra'ul Ib'a~nez, Yuli Rudyak, Aleksy Tralle and Luis Ugarte
Abstract. The paper deals with relations between the Hard Lefschetz prope*
*rty,
(non)vanishing of Massey products and the evenness of odd-degree Betti nu*
*mbers
of closed symplectic manifolds. It is known that closed symplectic manif*
*olds can
violate all these properties (in contrast with the case of Kaehler manifo*
*lds). However,
the relations between such homotopy properties seem to be not analyzed. *
*This
analysis may shed a new light on topology of symplectic manifolds. In th*
*e paper,
we summarize our knowledge in tables (different in the simply-connected a*
*nd in
symplectically aspherical cases). Also, we discuss the variation of symp*
*lectically
harmonic Betti numbers on some 6-dimensional manifolds.
1. Introduction
Homotopy properties of closed symplectic manifolds attract the attention of *
*ge-
ometers since the classical papers of Sullivan [S] and Thurston [Th]. On one ha*
*nd,
ös ft" homotopy techniques help in the solution of many äh rd" problems in sym-
plectic geometry, cf. [G1, McD, RT1, TO]. On the other hand, it is still unknown
if there are specific homotopy properties of closed manifolds dependent on the *
*ex-
istence of symplectic structures on them. It turns out that symplectic manifol*
*ds
violate many specific homotopy conditions shared by the Kähler manifolds (which
form a subclass of symplectic manifolds). In particular, if M is a closed Kähl*
*er
manifold then the following holds:
(1) all the odd-degree Betti numbers b2i+1(M) are even;
(2) M has the Hard Lefschetz property;
(3) all Massey products (of all orders) in M vanish.
It is well known (and we shall see it below) that closed symplectic manifolds v*
*iolate
all the homotopy properties (1) - (3). However, it is not clear whether propert*
*ies
(1) - (3) are independent or not, in case of closed symplectic manifolds or cer*
*tain
classes of these ones. In other words, can a combination of the type
(1) - (2) - non-(3)
be realized by a closed symplectic manifold (possibly, with prescribed properti*
*es).
The knowledge of an answer to this question might shed a new light on the whole
understanding of closed symplectic manifolds.
In Theorem 3.1 we have summarized our knowledge by writing down the corre-
sponding tables. We have considered two classes of symplectic manifolds: the cl*
*ass
______________
1991 Mathematics Subject Classification. 53C15, 57R19.
Typeset by AM S-T*
*EX
1
2 RA'UL IB'A~NEZ, YULI RUDYAK, ALEKSY TRALLE AND LUIS UGARTE
of symplectically aspherical symplectic manifolds and the class of simply-conne*
*cted
symplectic manifolds. Recall that a symplectically aspherical manifold is a sy*
*m-
plectic manifold (M, !) such that !|ß2(M) = 0, i.e.
Z
f# ! = 0
S2
for every map f : S2 ! M. In view of the Hurewicz Theorem, a closed symplec-
tically aspherical manifold always has a non-trivial fundamental group. It is *
*well
known that symplectically aspherical manifolds play an important role in geomet*
*ry
and topology of symplectic manifolds, [F, G2, H, RO, RT2].
The next topic of the paper is about symplectically harmonic forms on closed
symplectic manifolds. Brylinski [B] and Libermann (Thesis, see [LM]) have intro-
duced the concept of a symplectic star operator * on a symplectic manifold. In
a sense, it is a symplectic analog of the Hodge star operator which is defined *
*in
terms of the given symplectic structure !. Using this operator, one defines a s*
*ym-
plectic codifferential ffi := (-1)k+1 (*d*), deg ffi = -1. Now we define symple*
*ctically
harmonic differential forms ff by the condition
ffiff = 0, dff = 0.
Let *hr(M, !) denote the space of all symplectically harmonic forms on M. Clea*
*rly,
the space Hkhr(M) := khr=( khr\ Im d) is a subspace of the de Rham cohomology
space Hk (M).
Here we also have an interesting relation between geometry and homotopy theo*
*ry.
For example, Mathieu [M] proved that Hkhr(M, !) = Hk (M) if and only if M has
the Hard Lefschetz property. We will also see that the Lefschetz map
Lk : Hm-k (M) ! Hm+k (M), dim M = 2m
(multiplication by [!]k) plays an important role in studying of Hkhr(M, !).
We set hk(M, !) = dim Hkhr(M, !). According to Yan [Y], the following questi*
*on
was posed by Boris Khesin and Dusa McDuff.
Question: Are there closed manifolds endowed with a continuous family !t of
symplectic structures such that hk(M, !t) varies with respect to t?
Yan [Y] constructed a closed 4-dimensional manifold M with varying h3(M).
So, he answered affirmatively the above question.
(Actually, Proposition 4.1 from [Y] is wrong, the Kodaira-Thurston manifold *
*is
a counterexample, but its Corollary 4.2 from [Y] is correct because it follows *
*from
our Lemma 4.4. Hence, the whole construction holds.)
However, the Yan's proof was essentially 4-dimensional. Indeed, Yan [Y] wrot*
*e:
öF r higher dimensional closed symplectic manifolds, it is not clear how to *
*answer
the question in the beginning of this section",
i.e. the above stated question.
In this note we prove the following result (Theorem 4.6): There exists at le*
*ast one
6-dimensional indecomposable closed symplectic manifold N with varying h5(N).
HOMOTOPY PROPERTIES OF SYMPLECTIC MANIFOLDS 3
Moreover, Yan remarked that there is no 4-dimensional closed symplectic nil-
manifolds M with varying dim H*hr(M). On the contrary, our example is a certain
6-dimensional nilmanifold.
Remark(November 2001). During preparartion the paper for the publication,
we made some progress in this area. Namely, now we know that, for 6-dimensional
closed manifolds, the numbers hi for i = 3, 4, 5 can vary, and there are precis*
*ely ten
6-dimensional nilmanifolds with varying hi, [IRTU1,IRTU2].
2. Preliminaries and notation
Given a topological space X, let (MX , d) be the Sullivan model of X, that i*
*s,
a certain natural commutative DGA algebra over the field of rational numbers Q
which is a homotopy invariant of X, see [DGMS, TO, S] for details. Furthermore,
if X is a nilpotent CW -space of finite type then (MX , d) completely determines
the rational homotopy type of X.
A space X is called formal if there exists a DGA-morphism
æ : (MX , d) ! (H*(X; Q), 0)
inducing isomorphism on the cohomology level. Recall that every closed Kähler
manifold in formal [DGMS].
We refer the reader to [K, Ma, RT1] for the definition of Massey products. I*
*t is
well known and easy to see that Massey products yield an obstruction to formali*
*ty
[DGMS, RT1, TO]. In other words, if the space is formal then all Massey products
must be trivial. Thus, all the Massey products in every Kähler manifold vanish.
We need also the following result of Miller [Mi]:
2.1. Theorem. Every closed simply-connected manifold M of dimension 6 is
formal. In particular, all Massey products in M vanish.
The next homotopy property related to symplectic (in particular, Kähler) str*
*uc-
tures is the Hard Lefschetz property. Given a symplectic manifold (M2m , !), we
denote by [!] 2 H2(M) the de Rham cohomology class of !. Furthermore, we de-
note by L! : k(M) ! k+2 (M) the multiplication by ! and by L[!]: Hk (M) !
Hk+2 (M) the induced homomorphism in the de Rham cohomology H*(M). As
usual we write L instead of L! or L[!]if there is no danger of confusion. We say
that a symplectic manifold (M2m , !) has the Hard Lefschetz property if, for ev*
*ery
k, the homomorphism
Lk : Hm-k (M) ! Hm+k (M)
is surjective. In view of the Poincar'e duality, for closed manifolds M it me*
*ans
that every Lk is an isomorphism. We need also the following result of Gompf [G1,
Theorem 7.1].
2.2. Theorem. For any even dimension n 6, finitely presented group G and
integer b there is a closed symplectic n-manifold M with ß1(M) ~=G and bi(M) b
for 2 i n - 2, such that M does not satisfy the Hard Lefschetz condition.
Furthermore, if b1(G) is even then all degree-odd Betti numbers of M are even.
We denote such manifold M by M(n, G, b).
4 RA'UL IB'A~NEZ, YULI RUDYAK, ALEKSY TRALLE AND LUIS UGARTE
2.3. Remark. Theorem 7.1 in [G1] is formulated in a slightly different way, but
the proof is based on constructing of M by some "symplectic summation" in a way
to violate the Hard Lefschetz property.
In our explicit constructions we will need some particular classes of manifo*
*lds,
namely, nilmanifolds, resp. solvmanifolds. These are homogeneous spaces of the
form G= , were G is a simply connected nilpotent, resp. solvable Lie group and
is a co-compact discrete subgroup (i.e. a lattice). The most important informat*
*ion
for us is the following (see e.g. [TO] for the proofs):
2.4. Recollection. (i) Let g be a nilpotent Lie algebra with structural constan*
*ts
cijkwith respect to some basis, and let {ff1, ..., ffn} be the dual basis of g**
*. Then the
differential in the Chevalley-Eilenberg complex ( *g*, d) is given by the formu*
*la
X ij
dffk = - ck ffi^ ffj.
1 i 2 H*(M), ff, fi 2 H*(M), then ]CP m posses*
*ses a
non-trivial Massey triple product even for m - n = 3.
Proof. (i) and (ii) are proved in [McD], (i) and (iii) are proved in [RT1].
HOMOTOPY PROPERTIES OF SYMPLECTIC MANIFOLDS 5
2.6. Theorem. There exists a closed simply-connected symplectic manifold V ,
dim V = 8 such that b3(V ) = 1 and all the Massey products in V are trivial.
Proof. Let (M, !) be a 4-dimensional symplectic manifold with b1(M) = 1. The
existence of such manifolds follows from results of Gompf [G1]. Without loss of
generality we can assume that the symplectic form on M is integral. We embed
M in CP5 symplectically and denote by X the result of the blow up along this
embedding. So, we have a commutative diagram
fM ----! X
? ?
q?y ?yp
M ----! CP5
where q : fM ! M is a locally trivial bundle with the fiber CP2 and p : X \ fM !
CP5 \ M is a diffeomorphism.
Let CP11be a projective line in CP5 which does not meet M. Then the subman-
ifold p-1 (CP11) gives us a class a 2 H2(X). Similarly, the inclusion CP12 CP2
Mf X gives us a class b 2 H2(X). It is well known that {a, b} is a basis of H*
*2(X),
[MS, TO].
Let æ 2 H2(X) be the cohomology class of p*!0. It is clear that æ(a) 6= 0 and
æ(b) = 0. It is well known that X possesses a symplectic form whose cohomology
class [ ] is æ + varepsilonoe for varepsilon small enough. It is also easy to s*
*ee that
b3(X) = b1(M) = 1, see loc. cit.
Let be the normal bundle of the inclusion i : fM X, and let U 2 H*(T ) *
*be
the Thom class of . Consider the Browder-Novikov collapsing map c : X ! T
and set oe = c*U. It is easy to see that oe(a) = 0 and oe(b) 6= 0. Furthermor*
*e, for
every x 2 H*(X) we have
(*) oex = c*(U(i*x)).
Notice that in H*(X) we have æ5 6= 0 6= æ2oe2. Indeed, æ5 = p*([!0])5, and t*
*he
restriction of æ2oe2 on fM is (up to a non-zero multiplicative constant) the to*
*p class
,q*! 2 H8(Mf ), where , restricts to a non-zero element of the fiber CP2.
2.7. Lemma. In X we have
æ3oe = 0, æ4 \ [X] = k1a, æ2oe2 \ X = k2b, k1, k2 6= 0.
Proof. The equality æ3oe = 0 follows from the equalities (*) and !3 = 0.
Furthermore,
æ \ (æ4 \ [X]) = æ5 \ [X] 6= 0, oe \ (æ4 \ [X] = (æ4oe) \ [X] = 0
and hence æ4 \ [X] = k1a, k1 6= 0. Finally,
æ \ (æ2oe2 \ [X]) = (æ3oe2) \ [X] = 0
6 RA'UL IB'A~NEZ, YULI RUDYAK, ALEKSY TRALLE AND LUIS UGARTE
and hence æ2oe2 \ [X] = k2b, k2 2 R. Thus, k2b 6= 0 since æ2oe2 6= 0.
By the routine arguments, we can assume that is an integral form (by choos*
*ing
a suitable varepsilon). Because of the Donaldson Theorem [[]D, Theorem 1 and
Proposition 39, there is a closed symplectic submanifold V of X of codimension*
* 2
(i.e. dim V = 8) such that the homology class j*[V ] is dual to N[ ] for N la*
*rge
enough; here j : V ! X denotes the inclusion. In other words, j*[V ] = (~æ + ~o*
*e) \
[X] for some ~, ~ 2 R. Furthermore, j* : ßi(V ) ! ßi(X) is an isomorphism for
i 3 and an epimorphism for i = 4. So, according to the Hurewicz-Whitehead
theorem, the homomorphism j* : Hi(V ) ! Hi(X) is an isomorphism for i 3 and
an epimorphism for i = 4. In particular, b3(V ) = 1.
We set u = j*æ and v = j*oe.
2.8. Lemma. The R-vector space H6(V ) has dimension 2 and is generated by u3
and u2v.
Proof. We have H6(V ) = H2(V ) = H2(X) = R2. Furthermore, by Lemma 1,
j*(u3 \ [V ])= æ3 \ (j*[V ]) = æ3 \ (~æ + ~oe) \ [X] = (~æ4 + ~æ3oe) \ [X]
= ~æ4 \ [X] = ~0a, ~06= 0.
Similarly,
j*(u2v \ [V ])= æ2oe \ (j*[V ]) = æ2oe \ (~æ + ~oe) \ [X] = (~æ3oe + ~æ2oe2) *
*\ [X]
= ~æ2oe2 \ [X] = ~0b, ~06= 0.
Thus, u3 and u2v are linearly independent over R.
Now we prove that every Massey product in H*(V ) is trivial, i.*
*e., it
contains zero provided that it is defined. Notice that all the products u2, uv*
* and
v2 are non-zero because j* : H4(X) ! H4(V ) is a monomorphism.
Case 1. H5(V ), i.e., ff| = |fi| = |fl| = 2. Then the Masse*
*y product
is not defined, since fffi 6= 0.
Case 2. H6(V ). Clearly, if ff 6= 0 6= fl and *
*is defined then
|fi| = 3. Furthermore, if ff = u or fl = u then H6(V ) = (ff, fl), and so 0 2 <*
*ff, fi, fl>.
So, it remains to consider the case . But every Massey triple produc*
*t of
the form , |x| = 2, |y| = 3 contains zero. You can see it directly or *
*use the
equlity
= -
from [K, Theorem 8].
Case 3. H7(V ) = 0. Trivial.
Case 4. H8(V ) = R. Then there exist z 2 H*(X) with ffz 6= 0,*
* and
so (ff, fl) = H8(V ). Thus, is trivial.
Finally, all the higher (i.e, quadruple, etc.) Massey products are trivial f*
*or the
dimensional reasons.
HOMOTOPY PROPERTIES OF SYMPLECTIC MANIFOLDS 7
3. Relation between homotopy properties
of closed symplectic manifolds
3.1. Theorem. The relations between the Hard Lefschetz property, evenness of
odd-degree Betti numbers and vanishing of the Massey products for closed symple*
*ctic
manifolds are given by the following tables:
TABLE 1: symplectically aspherical case;
TABLE 2: simply-connected case.
The word Impossible in the table means that there is no closed symplectic
manifold (aspherical or simply connected) that realizes the combination in the
corresponding line.
The sign ? means that we (the authors) do not know whether a manifold with
corresponding properties exists.
8 RA'UL IB'A~NEZ, YULI RUDYAK, ALEKSY TRALLE AND LUIS UGARTE
Table 1: Symplectically Aspherical Symplectic Manifolds
______________________________________________________________________________|*
*||||
| Triviality of |Hard Lefschetz | Evenness of | |
|_Massey_Products_____|___Property______|____b2i+1_____|______________________|*
*|||||
2n
||_______yes__________||_____yes________||____yes______||___Kähler_(T__)______||
| | | | |
||_______yes__________||_____yes________||____no_______||____Impossible_______||
| | | | |
||_______yes__________||_____no_________||____yes______||_________?___________||
| | | | |
||_______yes__________||_____no_________||____no_______||_________?___________||
| | | | |
||_______no___________||_____yes________||____yes______||_________?___________||
| | | | |
||_______no___________||_____yes________||____no_______||____Impossible_______||
| | | | |
||_______no___________||_____no_________||____yes______||______K_x_K__________||
| | | | |
||_______no___________||_____no_________||____no_______||________K____________||
Table 2: Simply-Connected Symplectic Manifolds
______________________________________________________________________________|*
*||||
| Triviality of |Hard Lefschetz |Evenness of | |
|_Massey_Products_____|___Property_______|___b2i+1_____|______________________|*
*|||||
n
||_______yes__________||_____yes_________||___yes______||___Kähler_(CP_)______||
| | | | |
||_______yes__________||_____yes_________||___no_______||____Impossible_______||
| | | | |
||_______yes__________||_____no__________||___yes______||____M(6,{e},0)_______||
| | | | |
||_______yes__________||_____no__________||___no_______||________V____________||
| | | | |
||_______no___________||_____yes_________||___yes______||________?____________||
| | | | |
||_______no___________||_____yes_________||___no_______||____Impossible_______||
|| || || || ||
||_______no___________||_____no__________||___yes______||___CgP5_x_gCP_5______||
|| || || || ||
||_______no___________||_____no__________||___no_______||_______gCP5__________||
HOMOTOPY PROPERTIES OF SYMPLECTIC MANIFOLDS 9
Proof. We prove the theorem via line-by-line analysis of Tables 1 and 2.
Line 1 in Tables 1 and 2. For closed Kähler manifolds, the Hard Lefschetz
property is proved in [GH], the evenness of b2i+1 follows from the Hodge theory
[W], the triviality of Massey products follows from the formality of any closed
Kähler manifold [DGMS].
One can ask if there are non-Kähler manifolds having the properties from line
1. In the symplectically aspherical case the answer is affirmative. Let G = R*
* xOE
R2 be the semidirect product determined by the one-parameter subgroup OE(t) =
diag (ekt, e-kt), t 2 R, ek + e-k 6= 2. One can check that G contains a lattice*
*, say
. Then the compact solvmanifold
M = G= x S1
is symplectic and has the same minimal model as the Kähler manifold S2 x T 2.
Hence such manifold fits into line 1. It cannot be Kähler, since it admits no c*
*om-
plex structure. The latter follows from the Kodaira-Yau classification of compa*
*ct
complex surfaces (see [TO] for details).
Line 2 in Tables 1 and 2. Any manifold satisfying the Hard Lefschetz prop-
erty must have even b2i+1. Indeed, consider the usual non-singular pairing p :
H2k+1 (M) H2m-2k-1 (M) ! R of the form
Z
p ([ff], [fi])= ff ^ fi.
M
Define a skew-symmetric bilinear form <-, -> : H2k+1 (M) H2k+1 (M) ! R via
the formula
<[ff], [fl]> = p [ff], Lm-2k-1 [fl],
for [ff], [fl] 2 H2k+1 (M). Since this form is non-degenerate and skew-symmetri*
*c, its
domain H2k+1 (M) must be even-dimensional, i.e. b2k+1 is even.
Line 3 in Table 1. We do not know any non-simply-connected (and, in particular,
symplectically aspherical) examples to fill in this line.
Line 3 in Table 2. We use Theorem 2.2 with n = 6 and G = {e}. Then, for every
b, the corresponding manifold M(6, {e}, b) has even odd-degree Betti numbers and
does not have the Hard Lefschetz property. Furthermore, all the Massey products
in M vanish by 2.1.
Line 4 in Table 2. Consider the manifold V from Theorem 2.6. Since b3(V ) = 1,
V does not have the Hard Lefschetz property.
Line 5 in Tables 1 and 2 and Line 4 in Table 2. We do not know any examples
to fill in these lines.
Line 6 in Tables 1 and 2. This is impossible, see the argument concerning line
2.
Lines 7 and 8 in Table 1. Consider the Kodaira-Thurston manifold K [Th].
Recall that this manifold is defined as a nilmanifold
K = N3= x S1,
10 RA'UL IB'A~NEZ, YULI RUDYAK, ALEKSY TRALLE AND LUIS UGARTE
where N3 denotes the 3-dimensional nilpotent Lie group of triangular unipotent
matrices and denotes the lattice of such matrices with integer entries. One c*
*an
check that the Chevalley-Eilenberg complex of the Lie algebra n3 is of the form
( (e1, e2, e3), d), de1 = de2 = 0, de3 = e1e2.
with |ei| = 1. We have already mentioned that the minimal model of any nilman-
ifold N= is isomorphic to the Chevalley-Eilenberg complex of the Lie algebra n.
In particular, one can get the minimal model of the Kodaira-Thurston manifold in
the form
( (x, e1, e2, e3), d), dx = de1 = de2 = 0, de3 = e1e2
with degrees of all generators equal 1. One can check that the vector space H1(*
*K)
has the basis {[x], [e1], [e2]}. Hence, b1(K) = 3, which also shows that K does
not have the Hard Lefschetz property. Furthermore, K possesses a symplectic
form ! with [!] = [e1e3 + e2x], and one can prove that the Massey triple product
<[e1], [e1], [!]> is non-trivial. Thus, K realizes Line 8 of Table 1.
Finally, K x K realizes Line 7 of Table 1.
Lines 7 and 8 in Table 2. We use Theorem 2.5. Consider a symplectic embedding
i : K ! CP m, m 5, and perform the blow-up along i. Then, by 2.5(i), C]P m is
simply-connected. Furthermore, it realizes Line 8 of Table 2 by 2.5(ii) and 2.5*
*(iii).
Finally, ]CP m x ]CP m realizes Line 7 of Table 2.
3.2. Remark. The result of Lupton [L] shows that the problem of constructing
of a non-formal manifold with the Hard Lefschetz property turns our to be very
delicate. In [L] there is an example of a DGA, whose cohomology has the Hard
Lefschetz property, but which is not intrinsically formal. This means that the*
*re
is also a non-formal minimal algebra with the same cohomology ring. Sometimes,
using Browder-Novikov theory, one can construct a smooth closed manifold M with
such non-formal Sullivan minimal model. However, there is no way in sight to get
a symplectic structure on M.
4. Flexible symplectic manifolds
Let (M2m , !) be a symplectic manifold. It is known that there exists a uniq*
*ue
non-degenerate Poisson structure associated with the symplectic structure (se*
*e,
for example [LM, TO]). Recall that is a skew symmetric tensor field of order 2
such that [ , ] = 0, where [-, -] is the Schouten-Nijenhuis bracket.
The Koszul differential ffi : k(M) ! k-1 (M) is defined for Poisson, in pa*
*rtic-
ular symplectic, manifolds as
ffi = [i( ), d].
Brylinski has proved in [B] that the Koszul differential is a symplectic codiff*
*erential
of the exterior differential with respect to the symplectic star operator. We c*
*hoose
the volume form associated to the symplectic form, say vM = !m =m!. Then we
define the symplectic star operator
* : k(M) ! 2m-k (M)
HOMOTOPY PROPERTIES OF SYMPLECTIC MANIFOLDS 11
by the condition fi ^ (*ff) = k( )(fi, ff)vM , for all ff, fi 2 k(M). It turn*
*s out to
be that
ffi = (-1)k+1 (* O d O *).
4.1. Definition. A k-form ff on the symplectic manifold M is called symplectica*
*lly
harmonic, if dff = 0 = ffiff.
We denote by khr(M) the space of symplectically harmonic k-forms on M. We
set
Hkhr(M, !) = khr(M)=(Im d \ khr(M)), hk(M) = hk(M, !) = dim Hkhr(M, !).
We say that a de Rham cohomology class is symplectically harmonic if it contains
a symplectically harmonic representative, i.e. if it belongs to the subgroup H**
*hr(M)
of H*(M).
4.2. Definition. We say that a closed smooth manifold M is flexible, if M pos-
sesses a continuous family of symplectic forms !t, t 2 [a, b], such that hk(M, *
*!a) 6=
hk(M, !b) for some k.
So, the McDuff-Khesin Question (see the introduction) asks about existence of
flexible manifolds.
In order to prove our result on the existence of flexible 6-dimensional nilm*
*ani-
folds, we need some preliminaries. The following lemma is proved in [IRTU1] and
generalizes an observation of Yan [Y].
4.3. Lemma. For any symplectic manifold (M2m , !) and k = 0, 1, 2 we have
H2m-khr(M) = Im {Lm-k : Hk (M) ! H2m-k (M)} H2m-k (M).
In other words,
h2m-k (M, !) = dim Im {Lm-k : Hk (M) ! H2m-k (M)}.
The following fact can be deduced from 4.3 using standard arguments from lin*
*ear
algebra, see [IRTU1].
4.4. Lemma. Let !1 and !2 be two symplectic forms on a closed manifold M2m .
Suppose that, for k = 1 or k = 2, we have
h2m-k (M, !1) 6= h2m-k (M, !2).
Then M is flexible.
4.5. Proposition. Let G be a simply connected 6-dimensional nilpotent Lie group
such that its Lie algebra g has the basis {Xi}6i=1and the following structure r*
*ela-
tions:
[X1, X2] = -X4, [X1, X4] = -X5, [X1, X5] = [X2, X3] = [X2, X4] = -X6
(all the other brackets [Xi, Xj] are assumed to be zero). Then G admits a latti*
*ce ,
and the corresponding compact nilmanifold N := G= admits two symplectic forms
!1 and !2 such that
dim Im L2[!1]= 0, dim Im L2[!2]= 2.
12 RA'UL IB'A~NEZ, YULI RUDYAK, ALEKSY TRALLE AND LUIS UGARTE
Proof. First, G has a lattice by 2.4(ii). Furthermore, by 2.4(iii), in the Chev*
*alley-
Eilenberg complex ( *g*, d) we have
dff1= dff2 = dff3 = 0,
dff4= ff1ff2,
dff5= ff1ff4,
dff6= ff1ff5 + ff2ff3 + ff2ff4,
where we write ffiffj instead of ffi^ ffj. One can check that the following ele*
*ments
represent closed homogeneous 2-forms on N:
!1 = ff1ff6 + ff2ff5 - ff3ff4,
!2 = ff1ff3 + ff2ff6 - ff4ff5.
Since [!31] 6= 0 6= [!32], these homogeneous forms are symplectic. Indeed, by 2*
*.4(iii)
the cohomology classes [!0] and [!1] have homogeneous representatives whose thi*
*rd
powers are non-zero. Then the same is valid for their pull-backs to invariant 2*
*-forms
on the Lie group G. But for invariant 2-forms this condition implies non-degene*
*racy.
Since G ! N is a covering, the homogeneous forms !1 and !2 on N are also non-
degenerate.
Obviously, the R-vector space H1(N) has the basis {[ff1], [ff2], [ff3]}. On*
*e can
check by direct calculation that
[!1]2[ffi] = 0, i = 1, 2, 3
and that
[!2]2[ff1] = -2[ff1ff2ff4ff5ff6], [!2]2[ff2] = 0, [!2]2[ff3] = 2[ff2ff3ff*
*4ff5ff6].
Finally, it is straightforward that the above cohomology classes span 2-dimensi*
*onal
subspace in H5(N).
4.6. Theorem. There exists a flexible 6-dimensional nilmanifold.
Proof. Consider the nilmanifold N as in 4.5. Because of 4.3 and 4.5, we conclude
that
h5(N, !1) = 0 6= 2 = h5(N, !2),
and the result follows from 4.4.
Acknowledgment. The first and the fourth authors were partially supported
by the project UPV 127.310-EA147/98. This work was partially done in Oberwol-
fach and financed by Volkswagen-Stiftung. The second and third authors were also
partially supported by Max-Planck Institut für Mathematik, Bonn.
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Departamento de Matem'aticas, Facultad de Ciencias, Universidad del Pais Vas*
*co,
Apdo. 644, 48080 Bilbao, Spain,
E-mail address: : mtpibtor@lg.ehu.es
Yu. B. Rudyak, Department of Mathematics, University of Florida, 358 Little
Hall, PO Box 118105 Gainesville, FL 32611-8105, USA,
E-mail address: rudyak@math.ufl.edu, july@mathi.uni-heidelberg.de
University of Warmia and Masuria, 10561 Olsztyn, Poland
E-mail address: tralle@tufi.wsp.olsztyn.pl
Departamento de Matem'aticas, Facultad de Ciencias, Universidad de Zaragoza,
50009 Zaragoza, Spain
E-mail address: : ugarte@posta.unizar.es