ON THE FUNDAMENTAL GROUPS OF
SYMPLECTICALLY ASPHERICAL MANIFOLDS
R. IB'A~NEZ, YU. RUDYAK, AND A. TRALLE
Abstract. In this paper we are interested in the fundamental
groups of closed symplectically aspherical manifolds. Motivated
by some results of Gompf, we introduce two classes of fundamen-
tal groups ß1(M) of symplectically aspherical manifolds M with
ß2(M) = 0 and ß2(M) 6= 0. Relations between these classes are
discussed. We show that several important classes of groups can
be realized in both classes. Also, we notice that there are some
dimensional phenomena in the realization problem.
Introduction
Throughout the paper the term "symplectic manifold" means a closed
symplectic manifold (M, !) such that the cohomology class [!] 2 H2(M; R)
lies in the integral lattice H2(M)=tors. We say that the symplectic form
! is symplectically aspherical if
Z
f*! = 0,
S2
for every map f : S2 ! M. In cohomological terms, it means that
<[!], h(a)> = 0
for every a 2 ß2(M), where h : ß2(M) ! H2(M) is the Hurewicz
homomorphism. Frequently one writes the last equality as [!]|i2(M) =
0.
By the definition, a symplectically aspherical manifold is a symplectic
manifold whose symplectic form is symplectically aspherical. The im-
portance of symplectically aspherical manifolds in symplectic geometry
and topology is well-known, see e.g. [F , H , LO , R2 , RO , RT ].
____________
2000 Mathematics Subject Classification. 53D05, 14F35.
The first author was partially supported by the project UPV/EHU 00127.310-
EA-7781/2000. The second author was partially supported by Max-Planck Institute
of Mathematics, Bonn, Germany. The third author acknowledges the support of
the Polish Committee for the Scientific Reseach (KBN).
1
2 R. IB'A~NEZ, YU. RUDYAK, AND A. TRALLE
Clearly, every symplectic manifold M with ß2(M) = 0 is symplecti-
cally aspherical. On the other hand there are some reasons to know
whether there are symplectically aspherical manifolds with ß2(M) 6= 0,
see e.g. [G2 ]. Examples of such manifolds were given in [G2 ] as some
4-dimensional closed manifolds obtained as branched coverings. Here
we use a theorem of Donaldson [D ] on hyperplane sections of sym-
plectic manifolds in order to give another construction and get other
examples of 4-dimensional symplectically aspherical manifolds M with
non-trivial ß2(M).
In fact, we are interesting in searching for fundamental groups of sym-
plectically aspherical manifolds. It is well known that every finitely
presented group can be realized as the fundamental group of a closed
symplectic manifold, [G1 ]. However, not every such group can be re-
alized as the fundamental group of a closed symplectically aspherical
manifold. For example, the trivial (or, more generally, any finite) group
cannot. In the sequel we call a group symplectically aspherical if it can
be realized as a fundamental group of a closed symplectically aspherical
manifold. According to what we have said above, it is interesting to
compare the fundamental groups of symplectically aspherical manifolds
having ß2(M) = 0 with these ones having ß2(M) 6= 0.
We always identify de Rham cohomology of a manifold M with H*(M; R).
As usual, we call a closed connected manifolod M2n cohomologically
symplectic or, brielfy, c-symplectic if there exists a class a 2 H2(M; R)
with an 6= 0. Finally, we call a group ß c-symplectic if there exists a
(closed) c-symplectic manifold M which is K(ß, 1).
1. Preliminaries
1.1. Theorem (Hopf). Let X be a connected CW -space with ß1(X) =
ß and ßi(X) = 0 for i < n. Then there is an exact sequence
ßn(X) -- h-! Hn(X) -- - ! Hn(ß) -- - ! 0.
Proof. See [B , Theorem II.5.2].
The following theorem is a symplectic analog of the Lefschetz Theorem
on Hyperplane Sections.
1.2. Theorem. Let (M2n, !) be a symplectic manifold, and let h 2
H2(M) be an integral lift of [!]. Then for N large enough the Poincar'e
dual of Nh, in H2n-2(M), can be realized by a symplectic submanifold
V 2n-2 of M2n. Moreover, we can choose V such that the inclusion
SYMPLECTICALLY ASPHERICAL MANIFOLDS 3
i : V ,! M is an (n - 2)-equivalence, i.e. the homomorphism i* :
ßk(V ) ! ßk(M) is an isomorphism for k n - 2 and an epimorphism
for k = n - 1.
Proof. See [D , Theorem 1 and Proposition 39].
We need also the following homotopic characterization of symplectically
aspherical closed manifolds.
1.3. Proposition. Let (M, !) be a symplectic manifold, and let K
denote the Eilenberg-Mac Lane space K(ß1(M), 1). The following three
conditions are equivalent:
(i) (M, !) is symplectically aspherical;
(ii) there exists a map f : M ! K which induces isomorphism on the
fundamental groups and such that
[!] 2 Im {f* : H2(K, R) ! H2(M, R)};
(iii) there exists a map f : M ! K such that
[!] 2 Im {f* : H2(K; R) ! H2(M, R)}.
Proof. See [RT , Corollary 2.2], cf. also [LO , Lemma 4.2].
1.4. Proposition. If a group ß is symplectically aspherical and ø is
a subgroup of a finite index of ß, then ø is symplectically aspherical.
Proof. This holds, because a finite covering space over a closed symplec-
tically aspherical manifold is a closed symplectically aspherical mani-
fold.
1.5. Corollary. No finite groups are symplectically aspherical.
Proof. The trivial group is not symplectically aspherical because the
Hurewicz homomorphism ß2(M) ! H2(M) is an isomorphism for every
simply-connected space M. Now the result follows from Proposition
1.4.
2. Dimension phenomena
2.1. Proposition. Suppose that a group ß can be realized as the fun-
damental group of a symplectically aspherical manifold (M2n, !) with
n 3. Then ß can be realized as the fundamental group of a (2n - 2)-
dimensional symplectically aspherical manifold.
4 R. IB'A~NEZ, YU. RUDYAK, AND A. TRALLE
Proof. Without loss of generality we can assume that the symplectic
form ! is an integral form. (Indeed, we can a find a rational form j
which is C1 -closed to !, and therefore j is a symplectic form. Now
take a suitable multiple of j.) According to the Donaldson Theorem
1.2, there exists a symplectic submanifold V 2n-2 of M such that the
inclusion i : V ,! M is an (n - 2)-equivalence. In particular, ß1(V ) =
ß1(M). Clearly, V is symplectically aspherical since M is, and the
result follows.
2.2. Corollary. Suppose that a group ß can be realized as the fun-
damental group of a symplectically aspherical manifold (M2n, !) with
n 3. Then ß can be realized as the fundamental group of a 4-
dimensional symplectically aspherical manifold.
So, we can decrease the dimension of the symplecticaly aspherical man-
ifold with a prescribed fundamental group. However, we are not always
able to increase the dimension, as the following proposition shows.
2.3. Proposition. Let ß be a group such that Hi(ß; R) = 0 for i > k.
Suppose that ß can be realized as the fundamental group of a symplec-
tically aspherical manifold M2n. Then 2n k.
Proof. Because of 1.3, there exists a map f : M ! K(ß, 1) such that
[!] 2 Im {f* : H2(ß; R) = H*(K(ß, 1); R) ! H2(M; R)}.
Since [!]2n 6= 0, we conclude that 2n k.
2.4. Corollary. The group Zm cannot be realized as the fundamental
group of a symplectically aspherical manifold of dimension 2k with 2k >
m.
Notice that Z2n is the fundamental group of the torus T 2n. Since
ß2(T 2n) = 0, Z2n can be realized as the fundamental group of a sym-
plectically aspherical manifold of dimension 2k with 2 k n.
2.5. Remark. Because of Propositions 2.2 and 2.3, it makes sense to
introduce the following invariant of symplectically aspherical groups.
Namely, given a symplectically aspherical group ß, we define (ß) to be
the largest n such that ß can be realized as the fundamental group of a
closed symplectically aspherical manifold M2n. For example, (Z2n) =
n. Furthermore, if ß is the fundamental group of a closed orientable
surface then (ß) = 1, and if G is the direct product of n such groups
then (G) = n.
SYMPLECTICALLY ASPHERICAL MANIFOLDS 5
3. Two classes of symplectically aspherical groups
Let A be the class of symplectically aspherical groups which can be
realized as the fundamental groups of symplectically aspherical mani-
folds with trivial ß2, and let B be the class of symplectically aspherical
groups which can be realized as the fundamental groups of symplecti-
cally aspherical manifolds with non-trivial ß2. In this section we want
to investigate the relation between the classes A and B. First, some
trivial remarks.
1. If ß 2 B and ø is simplectically aspherical, then ß x ø 2 B.
2. Let G be the fundamental group of a closed orientable surface.
Then G =2B (by Corollary 2.2).
3.1. Theorem. Let (M4, !) be a 4-dimensional closed symplectically
aspherical manifold and let ß1(M) = ß. If ß2(M) = 0 then b1(ß)
b3(ß).
Proof. Because of the Hopf Theorem 1.1, there is an epimorphism
H3(M) ! H3(ß). So, b1(ß) = b1(M) = b3(M) b3(ß).
3.2. Corollary. If ß is a symplectically aspherical group and b1(ß) <
b3(ß), then ß 2 B.
Proof. Because of Corollary 2.2, ß can be realized as the fundamental
group of a 4-dimensional symplectically aspherical manifold. Now the
result follows from Theorem 3.1
3.3. Corollary. Suppose that ß can be realized as the fundamental
group of a symplectically aspherical manifold M with ß3(M) = 0 and
b1(M) < b3(M). Then ß 2 B.
Proof. If ß2(M) 6= 0 then we are done. So, suppose that ß2(M) = 0.
Since ß3(M) = 0, the Hopf exact sequence from Theorem 1.1 for n = 3
yields an isomorphism H3(M) ~=H3(ß). So,
b1(ß) = b1(M) < b3(M) = b3(ß)
and the result follows from 3.2.
3.4. Corollary. Let ß and ø be two symplectically aspherical groups.
(i) If max {b3(ß), b3(ø )} 1, then ß x ø 2 B.
(ii) If max {b2(ß), b2(ø )} 2 and max {b1(ß), b1(ø )} 1, then ß x ø 2
B.
6 R. IB'A~NEZ, YU. RUDYAK, AND A. TRALLE
Proof. Notice that b2(G) > 0 for every symplectically aspherical group
G. Now, b1(ß x ø ) = b1(ß) + b1(ø ), while (by the Künneth formula)
b3(ß x ø ) = b3(ø ) + b1(ß)b2(ø ) + b2(ß)b1(ø ) + b3(ß).
Now, each of the conditions in (i), (ii) implies that
b3(ß x ø ) > b1(ß x ø ),
and the result follows from 3.3.
3.5. Corollary. If ß is a symplectically aspherical group, then ßxZ4 2
B.
According to the results of Sections 2 and 3, it seems reasonable to
introduce the classes A2n an B2n as follows. The group ß belongs to
A2n if ß can be realized as the fundamental group of a symplectically
aspherical 2n-dimensional manifold with ß2(M) = 0. Similarly, the
group ß belongs to B2n if ß can be realized as the fundamental group of
a symplectically aspherical 2n-dimensional manifold with ß2(M) 6= 0.
Because of what is done above, we have the following Proposition.
3.6. Proposition.
A2n+2 A2n . . .A6 A4 [ B4, B2n+2 B2n . . .B4.
Proof. Consider the (n - 2) equivalence i : V 2n-2 ! M2n from the
proof of Theorem 2.1. The homomorphism i* : ß2(V ) ! ß2(M) is an
isomorphism for n 44 and an epimorphism for n = 3. Therefore the
result holds.
3.7. Remarks and Questions. 1. Notice that B2 = ; 6= A2.
2. We have A6 \ B6 6= ; since Z8 2 A6 \ B6. Indeed, Z8 2 A8 A6.
On the other hand, Z6 = ß1(T 6) and therefore Z6 2 B4 by Theorem
3.1. Thus, Z8 = Z6 x Z2 2 B6.
3. Similalry, A2n \ B2n 6= ; for n 3.
4. We don't know whether A4 \ B4 6= ;. In particular, is it true that
Z4 2 B4?
3. Is Z2n+1 symplectically aspherical if n > 1? (Z and Z3 are not by
Proposition 2.3.) If the answer is negative, the proof should be delicate
because the answer is positive at c-cymplectical leveL, see Proposition
3.8 below.
5. Generally, is it true that B A?
SYMPLECTICALLY ASPHERICAL MANIFOLDS 7
3.8. Proposition. For every n > 3 there exists a manifold N2n and a
cohomology class a 2 H2(N; R) such that ß1(N) = Z2n+1 and a|i2(N)=
0.
Proof. Take the torus T 2n+2and consider its hyperplane section as in
Theorem 1.2. Then we get a 2n-dimensional symplectically aspher-
ical manifold (M, !) with the fundamental group Z2n+2. Then, by
proposition 1.3, there exists a map f : M ! T 2n+2which induces an
isomorphism of fundamental groups and such that f*(un) = [!]n 6= 0
for some u 2 H2(T 2n+2; R). So, there are cohomology classes xi 2
H1(T 2n+2; R), i = 1, 2, . .2.n such that f*(x1 . .x.2n) 6= 0. This im-
plies, in turn, that there exists a map g : M ! T 2nwith g*[!T ]n 6= 0.
Here !T is the symplectic form on T 2n. In particular, the degree of g
is non-zero.
Consider the induced homomorphism
g* : Z2n+2 = ß1(M) ! ß1(T 2n) = Z2n
and take any a 2 Ker g*. Let A be the subgroup generated by a. Then
Z2n+2=A ~=Z2n+1 F
where F is a finite abelian group.
Now, we represent a by an embedded circle S and perform the surgery
of g along S. Then we get a map h : N ! T 2n which is bordant
to g, and therefore h has non-zero degree. So, h*[!]n 6= 0, and thus
N is c-symplectic. Furthermore, ß1(N) = Z2n+1 F . Now, passing
to a finite cover of of N, we obtain a c-symplectic manifold with the
fundamental group Z2n+1. finally, the class h*[!] vanishes on the image
of the Hurewicz map since [!] does.
4. Some results about realization
Now we describe some classes of symplectically aspherical groups. For
this purpose, we recall several notions.
A lattice in a Lie group G is a discrete subgroup ß G. A lattice ß in
G is called uniform if G=ß is compact.
4.1. Definition. A Lie group G is called completely solvable, if any
adjoint linear operator ad V : g ! g of the Lie algebra g of G has only
real eigenvalues.
It is well known that every completely solvable Lie group is solvable,
[VGS ] .
8 R. IB'A~NEZ, YU. RUDYAK, AND A. TRALLE
4.2. Lemma. If ß is a uniform lattice in a simply-connected com-
pletely solvable Lie group G of dimension 2n and ß is c-symplectic,
then ß 2 A. In particular, ß is symplectically aspherical.
Proof. Consider the closed manifold M := G=ß. Since G is solvable,
and, hence, diffeomorphic to euclidean space, we conclude that M =
K(ß, 1) and so H*(ß) ~=H*(M). Now, since ß is c-symplectic, then M
is c-symplectic, so there exists a cohomology class ff 2 H2(M, R) such
that ffn 6= 0 2 H2n(M; R). By the Hattori theorem [Ha ], there is an
isomorphism
H*(M; R) ~=H*( g*, ffi),
where ( g*, ffi) denotes the standard Chevalley-Eilenberg complex for
the Lie algebra g. Therefore ff can be represented by a closed differ-
ential 2-form ! whose pullback e! to G is a left-invariant form. Fur-
thermore, e!is non-degenerate since it is left-invariant and [e!]n 6= 0 on
H2n( g*, ffi). Hence, ! is non-degenerate, and so (M, !) is a symplectic
manifold.
Let ß be a polycylic group. Let ff 2 Aut (ß). There exists a subnormal
series ß = ßn ßn-1 ... ß0 such that ff(ßi) ßi, [Gb ]. (Here
subnormality means that ßi is normal in ßi+1 and Fi = ßi+1=ßi are
finitely generated abelian groups.) Hence ff induces automorphisms
ffi 2 Aut (Fi C) = GL(ki, C). One can easily check that the set
of eigenvalues of all operators ffi does not depend on the choice of a
subnormal series. We call the elements of this set eigenvalues of ff.
4.3. Definition. A polycyclic group ß is called a group of type (R), if
for all fl 2 ß all eigenvalues of the inner automorphism Int(fl) are real
and positive.
4.4. Theorem. A group ß is isomorphic to a uniform lattice in a
completely solvable simply-connected Lie group if and only if ß is of
type (R).
Proof. See Gorbatsevich [Gb ].
4.5. Corollary. Every c-symplectic group ß of type (R) belongs to A.
Furthermore, ß 2 B if b1(ß) < b3(ß).
Obviously, the class of completely solvable Lie groups contains all nilpo-
tent Lie groups. Furthermore, it is well known that every finitely gen-
erated torsion free nilpotent group is of type (R). Now we show that
some of these groups really belong to B.
SYMPLECTICALLY ASPHERICAL MANIFOLDS 9
4.6. Corollary. The fundamental group of any 6-dimensional sym-
plectic nilmanifold is a symplectically aspherical group of class B.
Proof. All 6-dimensional symplectic nilmanifolds are classified (see [Sa ,
IRTU ]). In particular, the first and second Betti numbers of each of
34 such manifolds can be found in the corresponding tables in these
papers. Note that since the Euler characteristic of any nilmanifold is
zero, we get the following relation for the Betti numbers: 2 - 2b1 +
2b2 - b3 = 0. Hence b1 < b3 is the same as
2 + 2b2 > 3b1.
One can check that each of the symplectic nilmanifolds from the tables
[Sa , IRTU ] satisfies this inequality.
Notice that one can also get groups of type (R) which are solvable but
non-nilpotent. For example, consider the following simply-connected
completely solvable Lie group G consisting of matrices
0 t t 1
e 0 xe 0 0 y1
B 0 e-t 0 xe-t 0 y2C
B t C
B 0 0 e 0 0 z1C
B -t C .
B 0 0 0 e 0 z2C
@ 0 0 0 0 1 tA
0 0 0 0 0 1
It is shown in [FLS ], that this group contains a uniform lattice ß, and
that the compact solvmanifold M := G=ß has b1(M) = 2 < b3(M) = 4.
Thus, ß 2 B.
4.7. Example. Tori, products of complex curves and hyperplane sec-
tions of these manifolds give us examples of symplectically aspherical
algebraic (and therefore Kähler) manifolds. Here we show how to con-
struct symplectically aspherical closed Kähler manifolds. Let G be a
semisimple simply-connected Lie group of non-compact type, and let
K be a maximal compact connected subgroup of G. If the homoge-
neous space G=K is a symmetric Hermitian space then G=K turns out
to be a Kähler manifold with the invariant Kähler metric. All such
pairs (G, K) are listed in [He , Ch. IX]. Moreover, every such group G
contains a uniform lattice ß, [VGS ]. Thus, M := ß\G=K is a symplec-
tically aspherical Kähler manifold with the fundamental group ß (since
G=K is diffeomorphic to Euclidean space). In particular, ß 2 A.
Now we give an example of ß as above with ß 2 B. Let Ø(ß) denote
the Euler characteristic of ß. It was shown in [VGS , Theorem 7.9] that
Ø(ß) 6= 0 if and only if rank(G) = rank(K), and in the latter case one
10 R. IB'A~NEZ, YU. RUDYAK, AND A. TRALLE
has also the sign of Ø(ß) equal to (-1)n, where n = 1=2 dim G=K. Now,
consider G = Sp(2, R) and K = U(2). Then G=K is a 6-dimensional
Hermitian symmetric space of non-compact type, and therefore Ø(ß) <
0. Furthermore, b1(ß) = 0, see [VGS , Theorem 7.1]. On the other hand,
Ø(ß) = 2 - 2b1 + 2b2 - b3 = 2 + 2b2 - b3 < 0, which implies b3(ß) > 0.
Thus, ß 2 B.
5. Nilpotent groups in A4
In this section we describe the nilpotent groups which can be realized
as the fundamental groups of symplectic manifolds with ß2(M) = 0.
Here we use some ideas from [R1 ].
Let ß be a finitely presented group, and let X be a CW -space with
ß1(X) = ß and finite 2-skeleton. Let eX be the universal covering space
of X, and let H1c(Xe) be the 1-dimensional cohomology with compact
supports of Xe.
5.1. Proposition-Definition. The group H1c(Xe) depends on the group
ß only. We denote it by H1c(ß) and call the 1-dimensional cohomology
with compact supports of ß.
Proof. Consider two spaces X1 and X2 as the above described space X.
First, assume that both X1 and X2 are K(ß, 1)'s. Consider homotopy
equivalences f : X1 ! X2 and g : X2 ! X1 with gf ' 1X1 and
fg ' 1X2. We can assume that f(X(2)1) X(2)2and g(X(2)2) X(2)1.
Moreover, the homotopies H : gf ' 1 and H0 : fg ' 1 can be chosen
so that H(X(1)1x I) X(2)1and H0(X(1)2x I) X(2)2.
Passing to the universal coverings, we get the homotopy equivalences
ef: eX1! eX2and homotopies eH : eX1x I ! eX1and eH0: eX2x I ! eX2.
Clearly, the maps
ef| (2): X(2)! X , eg| (2): X(2)! X
X1 1 2 X2 1 1
and the homotopies
eH| (1) : X(1)x I ! X , eH0| (1) : X(1)x I ! X
X1 xI 1 1 X2 xI 2 2
are proper maps. Therefore fe induces an isomorphism H1c(Xe2) !
H1c(Xe1).
Now we consider an arbitrary space X as above. We attach to X cells
of dimension 3 and get an embedding X Y where Y = K(ß, 1).
SYMPLECTICALLY ASPHERICAL MANIFOLDS 11
Since X(2)= Y (2), we conclude that H1c(Xe) = H1c(Ye). This completes
the proof.
5.2. Remark. Certainly, the group H1c(ß) admits a purely algebraic
description in terms of the group ß, cf [R1 , N]. However, the description
from 5.1 is enough for our goals.
5.3. Theorem (cf. [R1 ]). Let Mn be a closed manifold with ß1(M) =
ß and ßi(M) = 0 for 2 i n - 2. Then ßn-1(M) = H1c(ß).
Proof. Let fM be the universal covering space for M. Because of the
Poincar'e duality we have
H1c(ß) = H1c(fM ) = Hn-1(fM ).
But, by the Hurewicz Theorem, Hn-1(fM ) = ßn-1(fM ) = ßn-1(M).
5.4. Lemma. If ß is finitely generated nilpotent group with rank ß >
1, then H1c(ß) = 0.
Proof. First, we assume that ß is torsion free. We embed ß as a uniform
lattice in a contractible nilpotent Lie group G with dim G = rank ß =
n, [M ]. Since n > 1, we conclude that ßn-1(G) = 0. Thus, since
ß1(G=ß) = ß, we deduce from Theorem 5.3 that H1c(ß) = 0.
Now, if ß is not torsion free then it contains a torsion free subgroup
ß0 of finite index, [Ku ]. Then K(ß0, 1) can be regarded as a finite
covering over K(ß, 1). So, K(ß0, 1) and K(ß, 1) have the same universal
covering, and thus, H1c(ß) = H1c(ß0) = 0.
5.5. Corollary. Let M be a closed n-dimensional manifold, n > 1
with ßi(M) = 0 for i = 2, . .,.n - 2. If ß1(M) is a nilpotent group ß
with rank ß > 1, then ß is torsion free and rank ß = n.
Proof. By Lemma 5.4, H1c(ß) = 0. Therefore, by 5.3, ßn-1(M) = 0.
Furthermore, Hn(fM ) = 0 because ß is infinite. So, ßn(fM ) = 0, i.e. Mf
is contractible, i.e. M = K(ß, 1). So, ß is torsion free since M is finite
dimensional. Finally, M is homotopy equivalent to a closed nilmanifold
G=ß of dimension n, and therefore rank ß = n.
5.6. Theorem. Let M be a closed 4-dimensional symplectic manifold
M with ß2(M) = 0. If ß1(M) is a nilpotent, then ß1(M) is a tor-
sion free nilpotent group of rank 4. Conversely, every finitely presented
torsion free group can be realized as the fundamental group of closed
4-dimensional symplectic manifold with ß2(M) = 0.
12 R. IB'A~NEZ, YU. RUDYAK, AND A. TRALLE
Proof. First, notice that rank ß1(M) > 1. Indeed, if rank ß1(M) = 1
then ß1(M) contains Z as a subgroup of finite index. Considering the
finite covering with respect to the inclusion Z ß1(M), we get a 4-
dimensional closed symplectic manifold with the fundamental group Z.
But this is impossible by Proposition 2.3.
Now, by Lemma 5.5, ß1(M) must be a torsion free nilpotent group of
the rank 4.
Finally, consider a torsion free finitely presented nilpotent group ß,
rank ß = 4. It is easy to see that H2(ß; R) 6= 0. (You can use
classification of such groups, [VGS ], or notice that b1(ß) > 1 while
Ø(ß) = 0.) We embed ß as a uniform lattice in a 4-dimensional con-
tractible nilpotent group G, [M ]. Consider the closed oriented mani-
fold M := G=ß, dim M = 4. Then H2(M; R) = H2(ß; R) 6= 0. Take
any a 2 H2(M; R), a 6= 0. Then, by Poincar'e duality, there exists
b 2 H2(M; R) with ab 6= 0. Since ab = ba, we have
(a + b)2 = a2 + 2ab + b2,
and so at least one of elements a2, b2 or (a+b)2 must be non-zero. Thus,
M is a c-symplectic manifold. Now, asserting as in 4.2, we conclude
that the nilmanifold M is symplectic and symplectically aspherical.
5.7. Corollary. Let ß be a torsion free finitely generated c-symplectic
nilpotent group. If rank ß > 4 then ß 2 A6, ß 2 B4 and ß =2A4.
This Corollary strength Corollary 4.6.
Proof. Recall that ß is a uniform lattice in a certain simply connected
group G, dim G = rank ß. Asserting as in Lemma 4.2, we conclude that
ß 2 A2n. Therefore ß 2 A6, see Proposition 3.6. Furthermore, ß =2A4
by Theorem 5.6. Thus, ß 2 B4 since, by proposition 3.6, A6 A4[ B4.
6. Gompf symplectic sum and symplectic asphericity
Here we mention briefly how to built symplectically aspherical manifold
from other ones. Certainly, this yields to other examples of symplecti-
cally aspherical groups. We do not dwell these things here, but hope
to do it somewhere later.
First, recall the construction of the connected sum of two manifolds
along a submanifold, with the aim to emphasize the symplectic version
of this construction, [G1 ] .
SYMPLECTICALLY ASPHERICAL MANIFOLDS 13
Let Mn1, Mn2and Nn-2 be smooth closed oriented manifolds (not nec-
essarily connected), of dimensions n and n - 2, respectively. Assume
that we are giventwo embeddings j1 : N ! M1 and j2 : N ! M2,
with the normal bundles 1 and 2, respectively, such that their Euler
classes differ only by sign: e( 1) = -e( 2). It turns out to be that there
exists an orientation-reversing bundle isomorphism ff : 1 ! 2. Let
Vi denote a tubular neighborhood of ji(N), which we identify with the
total space of i. Then ff yields a diffeomorphism _ : V1 ! V2, which
maps j1(N) to j2(N). Then _ determines an orientation-preserving
diffeomorphism
' : (V1 - j1(N)) ! (V2 - j2(N)), ' = ` O _,
where v
`(p, v) = (p, _____)
||v||2
is a diffeomorphism which turns each punctured normal fiber inside
out.
6.1. Definition. Let M1 [_ M2 denote the smooth, closed oriented
manifold obtained from the disjoint union M1- (j1(N)) t M2- (j2(N))
via gluing V1 - j1(N) and V2 - j2(N) by ':
M1 [_ M2 = M - (j1(N) [ j2(N))= '
where a ' b if and only if b = '(a), a 2 V1 - j1(N), b 2 V2 - j2(N).
It was noted in [G1 ] that there exists a cobordism X between M1 t M2
and M1[_ M2. It will be important for us to notice that the cobordism
X is obtained from (M1 t M2) x I (I = [0, 1]) by identifying closed
tubular neighborhoods of j1(N) x 1 and j2(N) x 1 by _ and rounding
corners.
Now, we need the following observation. Every closed k-form !M on
M for which j*1!M = j*2!M induces a cohomology class [ ] 2 Hk(X; R)
and, hence, by restriction, a class [!] 2 Hk(M1 [_ M2; R). Note that
[!] = i*[ ], where i : M1 [_ M2 ! X is the canonical embedding.
In the sequel we will need the following result.
6.2. Theorem. Let (M1, M2, N and ji : N ! Mi, i = 1, 2 be as in
Definition 6.1. Suppose in addition that M1, M2 and N are symplectic
manifolds and both embeddings ji : N ! Mi are symplectic. Then,
for any choice of (orientation reversing) _ : V1 ~= V2, the manifold
M1 [_ M2 admits a canonical symplectic structure !, which is induced
by the symplectic form on M1 t M2 after a perturbation near j2(N).
More precisely, there is a unique isotopy class of symplectic forms on
14 R. IB'A~NEZ, YU. RUDYAK, AND A. TRALLE
M1 [_ M2 (independent of fiber isotopies of _) that contains forms !
with the following characterization: the class [!] 2 H2(MM1[_ M2; R)
is the restriction of the class [ ] 2 H2(X; R) canonically induced on
the cobordism X by the symplectic on M1 t M2.
Proof. See Gompf [G1 ].
Let (X; A, B) be a CW -triad. We set C = A \ B and denote by
j1 : A ! X, j2 : B ! X, i1 : C ! A, i2 : C ! B the obvious
inclusions.
6.3. Proposition. Fix any k and any coefficient group. If the homo-
morphism i*1: Hk(A) ! Hk(C) is injective and the homomorphism
i*1: Hk-1 (A) ! Hk-1 (C) is surjective, then the j*2: Hk(X) ! Hk(B)
is injective and the homomorphism j*2: Hk-1 (X) ! Hk-1 (B) is sur-
jective.
Proof. The exactness of the sequence
i*1 k-1 k i*1 k k
Hk-1 (A) --- ! H (C) ! H (A, C) -- - ! H (A) ! H (C)
implies that Hk(A, C) = 0. So, because of the excision property,
Hk(X, B) = Hk(A, C) = 0. Now, the exactness of the sequence
j*2 k-1 k i*2 k k
Hk-1 (X) --- ! H (B) ! H (X, B) --- ! H (X) ! H (B)
implies the required claim on j*2.
We say that a 2-dimensionalPcohomology class a is decomposable if it
can be represented as a = i aia0iwhere ai and a0i are 1-dimensional
classes. Notice that a symplectic form ! on a symplectic manifold is
aspherical if its cohomology class [!] is decomposable.
6.4. Theorem. Let M1, M2, N and ji : N ! Mi, i = 1, 2 be as in
Theorem 6.2, and suppose that j1 induces a surjection on the first
cohomology group and an injection on the second cohomology group.
Assume that H2(M2; R) consists on decomposable elements. Then the
symplectic manifold (M1 [_ M2) is symplectically aspherical.
Proof. Let X be the cobordism described in after Definition 6.1. It
suffices to prove that the cohomology class [ ] of the form on X
is decomposable. Notice that X is homotopy equivalent to the space
Y = M1[N M2. So, we have the triad (Y ; M1, M2) with M1\ M2 = N.
Since smooth manifolds are triangulable, we can regard the above triad
SYMPLECTICALLY ASPHERICAL MANIFOLDS 15
as a CW -triad. Let j : M2 ! Y be the inclusion. Because of the
conditions of the Theorem,
X
j*[ ] = bib0i, bi, b0i2 H1(M2).
By the Proposition 6.3, the map j* : H1(Y ) ! H1(M2) is an epi-
morphism. So,Pthere are ai, ai 2 H1(Y ) with j*(ai) = bi, j*(ai) = bi.
So, j*([ ] - aiai) = 0. But, again by the PropositionP6.3, the map
j* : H2(Y ) ! H2(M2) is a monomorphism, and thus [ ] - aiai.
The Proposition below gives us one more source of symplectically as-
pherical manifolds.
6.5. Proposition. Let M1, M2, N and ji : N ! Mi, i = 1, 2 be as in
Theorem 6.2. Suppose in addition that ßk(N) = 0 = ßk(Mi\ji(N)), i =
1, 2 for k > 1 and the induced homomorphisms
(ji)* : ß1(@Ni) ! ß1(Mi\ Ni), i = 1, 2
are monomorphisms Then ßk(M1 [_ M2) = 0 for k > 1. In particular
the group
ß1(M1 \ N1) *i1(N1\f1(S))ß1(M2 \ N2)
is symplectically aspherical.
Proof. See [Proposition 3.1][K ].
One more sourse of symplectically aspherical groups comes from the
observation of Gompf [G2 , Lemma 1] who proved that a branched cov-
ering over a 4-dimensional symplectically aspherical manifold is sym-
plectically aspherical.
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E-mail address: mtpibtor@lg.ehu.es
SYMPLECTICALLY ASPHERICAL MANIFOLDS 17
Yu. Rudyak, Department of Mathematics, Universoty of Florida, 358
Little Hall, Gainesville, FL 32601, USA
E-mail address: rudyak@math.ufl.edu
E-mail address: rudyak@mathi.uni-heidelberg.de
A. Tralle, Department of Mathematics, University of Warmia and
Mazura, 10561 Olsztyn, Poland
E-mail address: tralle@matman.uwm.edu.pl