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Contemporary Mathematics
Cohomology of Nilmanifolds and Gontcharova's Theorem
Dmitri V. Millionschikov
Abstract. Let Mn be a nilmanifold with a corresponding nilpotent Lie al­
gebra Vn = f! e1 ; : : : ; en ?; [e i ; e j ] = (j \Gamma i)e i+j ; i + j Ÿ ng: We study the
cohomology H \Lambda (Mn ; R): we prove that a stable Betti number bq (Mn ); n ?? q;
is equal to the (q+2)­th Fibonacci number F q+2 :
1. Introduction.
Nilmanifolds are a fruitful source of examples of symplectic manifolds with no
K¨ahler structure or non formal symplectic manifolds (see [TO], [CFG] for refer­
ences). I. Babenko and I. Taimanov described in [BaT1], [BaT2] examples of non
formal simply connected symplectic manifolds. They considered an interesting fam­
ily of nilmanifolds Mn that we'll discuss below. I. Babenko and I. Taimanov used
only the existence of a non­trivial triple Massey product in H \Lambda (Mn ; R) in their proof
and they didn't study the cohomology H \Lambda (Mn ; R): The question of calculation of
H \Lambda (Mn ; R) appears to be very interesting and important.
2. Nilmanifolds and symplectic structures.
Definition 2.1. A nilmanifold M is a compact homogeneous space of the form
G=\Gamma; where G is a simply connected nilpotent Lie group and \Gamma is a lattice in G.
Example 2.2. A n\Gammadimensional torus T n = R n =Z n .
Example 2.3. The Heisenberg manifold M 3 = H 3 =\Gamma 3 ; where H 3 is the group
of all matrices of the form 0
@ 1 x z
0 1 y
0 0 1
1
A ; x; y; z 2 R;
and a lattice \Gamma 3 is a subgroup of matrices with x; y; z 2 Z.
Now we introduce the main example of this article.
2000 Mathematics Subject Classification. Primary 17B30, 17B56; Secondary 55P62, 55P20.
Key words and phrases. Nilpotent Lie algebras, cohomology, nilmanifolds.
Partially supported by the Russian Foundation for Fundamental Research, grant no. 99­01­
00090.
c
fl2001 American Mathematical Society
1

2 DMITRI V. MILLIONSCHIKOV
Example 2.4. Let Vn be a nilpotent Lie algebra with a basis e 1 ; e 2 ; : : : ; e n and
a Lie bracket:
[e i ; e j ] =
ae
(j \Gamma i)e i+j ; i + j Ÿ n;
0; i + j ? n:
One can also define a group structure \Lambda in the vector space Vn using the Campbell­
Hausdorff formula:
x \Lambda y = x + y + 1
2 [x; y] + 1
12 ([x; [x; y]] \Gamma [y; [y; x]]) + : : : :
This nilpotent group Vn = (Vn ; \Lambda) has a discrete subgroup (lattice) \Gamma n and ac­
cording to A.Maltsev's theorem ([Mal]) quotient spaces Mn = Vn =\Gamma n are compact
manifolds (nilmanifolds).
Remark 2.5. The manifolds Mn are Eilenberg­MacLane spaces K (\Gamma n ; 1):
It was Victor Buchstaber who introduced these manifolds in [Bu1], [Bu2] for
the study of the Landweber­Novikov algebra. V. Buchstaber proposed also a very
interesting interpretation of Vn as a group of polynomial real line infinitesimal
transformations
ft ! Px (t) = t + x 1 t 2 + \Delta \Delta \Delta + xn t n+1 g
with a nilpotent group structure defined by
(P x \Lambda Qy )(t) := Qy (Px (t)) mod t n+2 ;
where we denote by x = (x 1 ; x 2 ; : : : ; xn ) a vector in the Vn :
The cohomology H \Lambda (Mn ; R) is isomorphic to the cohomology H \Lambda (Vn ) of the
corresponding Lie algebra Vn (Nomizu theorem [Nz]). One of the most important
Babenko­Taimanov's observations was
Remark 2.6. ( [BaT1]) M 2k is a symplectic manifold with a left­invariant
symplectic form
\Omega 2k+1 = (2k\Gamma1)! 1 “ ! 2k + (2k\Gamma3)! 2 “ ! 2k\Gamma1 + : : : +! k “ ! k+1 ;
where ! 1 ; : : : ; ! 2k is a dual basis of left­invariant 1­forms to the e 1 ; : : : ; e 2k :
Remark 2.7. A rational minimal model MMn for Mn has generators x 1 ; : : : ; xn
of degree 1, with a differential
dx 1 = 0; dx 2 = 0; dx k =
X
i!j;i+j=k
(j \Gamma i)x i “ x j ; k = 3; : : : n:
In fact I. Babenko, I. Taimanov in [BaT1], [BaT2] applied the symplectic
blow­up procedure to the symplectic embedding M 2k ! C P 2k+1 (as it was done
by D. McDuff in [McD] where she described the first example of simply connected
symplectic non K¨ahler manifold).
3. Cohomology of infinite dimensional Lie algebras and Gontcharova's
theorem.
The problem of calculation of the cohomology of the Vn is closely related to
Gontcharova's theorem in the infinite dimensional Lie algebras theory.
Definition 3.1. L k is an infinite dimensional Lie algebra of polynomial vector
fields on R with a trivial k­jet at the origin. L k has a basis
e i = x i+1 d
dx
; i = k; k + 1; : : : ; [e i ; e j ] = (j \Gamma i)e i+j :

COHOMOLOGY OF NILMANIFOLDS AND GONTCHAROVA'S THEOREM 3
So the algebra Vn is a quotient algebra Vn = L 1 =Ln+1 :
The problem of calculation of H \Lambda (L k ) was solved by Lidiya Gontcharova in
1973. In [G] Gontcharova used the second grading p of a complex of continuous
cochains C \Lambda (L k ) (p(! i 1 “ : : : “! i q
) = i 1 +\Delta \Delta \Delta+i q ) for her computation of the bigraded
cohomology H q
p (L k ).
Theorem 3.2. (L.Gontcharova,[G])
1) dimH q
k (L 1 ) =
ae
1; if k = 3q 2 \Sigmaq
2 ;
0; otherwise:
Thus dimH q (L 1 ) = 2 for all q – 1.
2) dimH q (L k ) =
` q + k \Gamma 1
k \Gamma 1
'
+
` q + k \Gamma 2
k \Gamma 1
'
:
Remark 3.3. One can consider a formula for Euler characteristic üN of the
complex C \Lambda
N (L 1 ) of cochains of the second grading p = N and then define a corre­
sponding generating function:
X
N
üN t N =
1
Y
n=1
(1 \Gamma t n ) = 1 \Gamma t \Gamma t 2 + t 5 + t 7 + ::: + (\Gamma1) q (t 3q 2 \Gammaq
2 + t 3q 2 +q
2 ) + : : :
This is the well­known Euler identity in combinatorics. From topological point
of view this identity is an equality of two expressions of Euler characteristic üN
(the right part of the formula is an alternated sum of Betti numbers, see [Fu] for
details).
Numbers 3q 2 \Sigmaq
2 are so called Euler pentagonal numbers. From their properties
it follows that the algebra H \Lambda (L 1 ) has a trivial multiplication (see [FFR]).
Theorem 3.4. A stable Betti number b q (Mn ) = dim(H q (Vn )); n ?? q; is equal
to the (q+2)­th Fibonacci number F q+2 :
b 1 (Mn ) = dim(H 1 (Vn )) = 2; b 2 (Mn ) = dim(H 2 (Vn )) = 3;
b 3 (Mn ) = dim(H 3 (Vn )) = 5; : : : ; b q (Mn ) = dim(H q (Vn )) = F q+2 ; n ?? q:
Proof. (Sketch of the proof). Let us consider the Serre­Hochschild spectral
sequence of a pair of Lie algebras (L 1 ; Ln+1 ) (see [HS]) that converges to the
H \Lambda (L 1 ):
In fact the first row of E 2 is our answer. We compute here H q (L 1 =Ln+1 ) only
for n ?? q:
Lemma 3.5. H q (L 1 ) = E q;0
1 = E q;0
3 : In other words: Gontcharova's cocycles of
small degrees q !! n are in the first row of E 2 and survive till E1 :
Lemma 3.6. H q (L 1 =Ln+1 ) = Imd 2 H q\Gamma2 (L 1 =Ln+1 ; H 1 (Ln+1 )) \Phi H q (L 1 ): In
other words: to get a basis of H q (L 1 =Ln+1 ) for large values n ?? q we have to add
to two Gontcharova's cocycles of degree q a basis of Imd 2 H q\Gamma2 (L 1 =Ln+1 ; H 1 (Ln+1 )):
Example 3.7. Let us consider dimensions q = 1; 2; 3 and n ?? q :
1) H 1 (Vn ) is generated by two classes [! 1 ] and [! 2 ] -- Gontcharova's classes
g 1 = [! 1 ]; g 2 = [! 2 ].

4 DMITRI V. MILLIONSCHIKOV
0 ! 1 ; ! 2 g 5 ; g 7 g 12 ; g 15 : : :
0 0 0 0 : : :
0 0 0 0 : : :
E 3 = E 4 = \Delta \Delta \Delta = E1 :
Figure 1. Gontcharova's cocycles and E1 :
0
!n+1
! 1 ; ! 2
!n+1“!n+4
!2“!n+1
!2“!n+2+::
!2“!n+3+::
\Omega n+1
g 5 ; g 7
g5“!2n+::
g7“!n+1 : : :
g7“!n+5+::
\Omega n+1“!n+1
!2“\Omega n+1
!2“\Omega n+2+::
!2“\Omega n+3+::
g12 ; g15
g7“\Omega n+1
: : :
g7“\Omega n+5+::
\Omega 2
n+1 ; g22 ; g26
P P P P P P P P Pq
P P P P P P Pq
P P P P P P P Pq
P P P
P Pq
P P P P P P P Pq
d 2
d 2
Figure 2. The term E 2 of the spectral sequence.
2) The basis of H 2 (Vn ) consists of two Gontcharova's classes g 5 = [! 1 “! 4 ];
g 7 = [! 2 “! 5 + 3
2 ! 1 “! 6 ] and the third generator is the ''symplectic
class''[\Omega n+1 ];
where\Omega n+1 = (n\Gamma1)! 1 “ !n + (n\Gamma3)! 2 “ !n\Gamma1 + : : : :
3) In H 3 (Vn ) to the classes g 12 = [! 1 “! 4 “! 7 + : : : ]; g 15 = [! 2 “! 5 “! 8 + : : : ]
(degree 3 with second gradings p=12,15 respectively, see [G] for details) we add
three new ones
[! 2
]“[\Omega n+1 ]; [! 2
“\Omega n+2 \Gamma n! 3
“\Omega n+1 ];
[! 2
“\Omega n+3 \Gamma (n+1)! 3
“\Omega n+2 + n(n + 1)
2 ! 4
“\Omega n+1 ];
where\Omega n+2
;\Omega n+3 are projections of ''non­existing'' d!n+2 ; d!n+3 to the C 2 (Vn ) :
\Omega n+2 = (n\Gamma2)! 2 “!n + (n\Gamma4)! 3 “!n\Gamma1 + : : :
;\Omega n+3 = (n\Gamma3)! 3 “!n + : : : :
Remark 3.8. An elementary calculation shows that the cohomology algebra
H \Lambda (Vn ) does not satisfy the hard Lefschetz condition:
[! 1
]“[\Omega n+1 ] = 0; ! 1
“\Omega n+1 = 1
n
d\Omega n+2 :
Lemma 3.9.
H q (Vn ) = H q (L 1 ) \Phi Imd 2 = H q (L 1 ) \Phi H q\Gamma2 (L 2 ) \Phi H q\Gamma4 (L 3 ) \Phi : : : :
The last proposition together with Gontcharova's theorem concludes the proof,
namely, it gives us a representation of Fibonacci numbers by means of binomial
coefficients:

COHOMOLOGY OF NILMANIFOLDS AND GONTCHAROVA'S THEOREM 5
F q+2 =
` q
0
'
+
` q \Gamma 1
0
'
+
` q \Gamma 1
1
'
+
` q \Gamma 2
1
'
+
` q \Gamma 2
2
'
+
` q \Gamma 3
2
'
+ : : : :
It is a finite sum that can be illustrated by means of the Pascal triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
)
\Lambda
\Lambda \Lambda
\Lambda 2 1
1 3 \Lambda \Lambda
1 \Lambda \Lambda \Lambda \Lambda
:
Example 3.10. Let us consider a sum of the fourth and the third diagonals of
the Pascal triangle
F 6 = f1 + 3 + 1g + f1 + 2g = 8;
or in terms of binomial coefficients
F 6 = F 4+2 =
ae` 4
0
'
+
` 3
1
'
+
` 2
2
'oe
+
ae` 3
0
'
+
` 2
1
'oe
= 8:
References
[BaT1] I.K. Babenko and I.A. Taimanov. On the existence of nonformal simply connected sym­
plectic manifolds, Russian Math. Surveys 53:4 (1998), 1082­1083.
[BaT2] I.K. Babenko and I.A. Taimanov. On nonformal simply connected symplectic manifolds,
SFB 2888, Preprint 358, 1998.
[Bu1] V.M. Buchstaber, A.V. Shokurov, The Landweber--Novikov algebra and formal vector fields
on the line, Funct. Anal. and Appl. 12:3 (1978), 1--11.
[Bu2] V.M. Buchstaber, The groups of polynomial transformations of the line, nonformal sym­
plectic manifolds, and the Landweber­Novikov algebra, Russian Math. Surveys 54 (1999).
[CFG] L.A. Cordero, M. Fernandez, A. Gray, Symplectic manifolds with no K¨ahler structure,
Topology 25 (1986), 375--380.
[DGMS] P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of K¨ahler struc­
ture, Invent. Math. 19 (1975), 245--274.
[FFR] B.L. Feigin, D.B. Fuchs, V.S. Retakh, Massey operations in the cohomology of the infinite
dimensional Lie algebra L1 , Rokhlin Seminar, Lecture Notes in Math. 1346 (1988), 13--31.
[Fu] D.B. Fuchs, Cohomology of the infinitedimensional Lie algebras, Consultant Bureau, New
York, 1987.
[G] L.V. Gontcharova, Cohomology of Lie algebras of formal vector fields on the line, Funct.
Anal. and Appl. 7:2 (1973), 6--14.
[HS] G.P. Hochschild, J.­P. Serre, Cohomology of Lie algebras, Ann. Math. 57:3 (1953), 367--401.
[Mal] A. Malcev, On a class of homogeneous spaces, Izvestia Akad. Nauk SSSR Ser. Mat. 3 (1949),
9­32 (Russian); English translation: Amer. Math. Soc. Transl. (1) 9 (1962), 276--307.
[McD] D. McDuff, Examples of symplectic simply­connected manifolds with no K¨ahler structure,
J. Diff. Geom. 20 (1984), 267--277.
[Mill] D.V. Millionschikov, Cohomology of nilmanifolds and Gontcharova's theorem, Russian
Math. Surveys, to appear.
[Nz] K. Nomizu, On the cohomology of homogeneous spaces of nilpotent Lie groups, Ann. of Math.
59 (1954), 531--538.
[TO] A. Tralle, J. Oprea, Symplectic manifolds with no K¨ahler structure, Lecture Notes in Math.
1661 (1997).
Department of Mathematics and Mechanics, Moscow State University, 119899 Moscow,
RUSSIA
E­mail address: million@mech.math.msu.su