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 ]. 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