Formality and the Lefschetz property in symplectic and cosymplectic geometry

We review topological properties of K\"ahler and symplectic manifolds, and of their odd-dimensional counterparts, coK\"ahler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the K\"ahler/symplectic situation) and the $b_1=1$ case (in the coK\"ahler/cosymplectic situation).


Introduction
Symplectic geometry is the study of symplectic manifolds, that is, smooth manifolds endowed with a 2-form which is closed and non-degenerate. Important examples of symplectic manifolds are R 2n " R nˆp R n q˚and, more generally, the cotangent bundle T˚M of any smooth manifold M , which is endowed with a canonical symplectic structure. Cotangent bundles are especially important in classical mechanics, where they arise ase phase spaces of a classical physical system (see [5]). In case the physical system has symmetries, one can perform a reduction procedure on the phase space; this often produces compact examples of symplectic manifolds (see [74]). Therefore, the study of compact symplectic manifolds is also relevant. In the last decades, starting with the first questions on the global nature of symplectic manifolds and the foundational work of Gromov (see [48]), symplectic geometry has become a branch of geometry which is interesting per se. Standard references for symplectic geometry include [7,24,78]. For more advanced applications, including connections with Mirror Symmetry, we refer to [79,88].
Kähler manifolds are special examples of symplectic manifolds. Recall that a Hermitian manifold is Kähler if its Kähler form is closed (see Section 2.1 for all the relevant definition). In particular, symplectic geometry receives many imputs from algebraic geometry. Indeed, the complex projective space CP n is endowed with a Kähler structure, which is inherited by all projective varieties.
For a long time, the only known examples of symplectic manifolds came in fact from Kähler and algebraic geometry. It was only in 1976 that Thurston gave the first example of a symplectic manifold which carries no Kähler metric (see [90]). Since then, the quest for examples of symplectic manifolds which do not carry Kähler metrics has been a very active area of research in symplectic geometry, whose main contribution has been the introduction of new techniques for constructing symplectic manifolds (see [9,31,33,44,46,48,49,77] as well as Section 3.5 for an account).
Cosymplectic geometry was first studied by Libermann in [68]. Blair and the Japanese school tracking back to Sasaki have studied cosymplectic geometry in the setting of almost contact metric structures (see [21] and the references therein). To some extent, cosymplectic geometry can be viewed as the odd-dimensional counterpart of symplectic geometry. Indeed, cosymplectic manifolds are the right geometric framework for time-dependent Hamiltonian mechanics. The main example of a cosymplectic manifold is the product of a symplectic manifold with the real line, or a circle. There exist however cosymplectic manifolds which are not products (see Example 4.18). Very recently, cosymplectic manifolds have appeared in the study of a special type of Poisson structures, namely b-symplectic structures (see [52,53]), as well as a special class of foliated manifolds (see [75]). We refer to [25] for a recent survey on cosymplectic geometry.
Examples of cosymplectic manifolds come from coKähler geometry. CoKähler structures are normal almost contact metric structures pφ, η, ξ, gq for which the tensor φ is parallel with respect to the Levi-Civita connection (see Section 2.2 for the relevant definitions). The main example is provided by the product of a Kähler manifold and the real line (or the circle). We point out here as well that there exist compact coKähler manifolds which are not the product of a compact Kähler manifold and a circle (see [18]).
Compact Kähler and coKähler manifolds satisfy quite stringent topological properties, which we have collected in Theorems 3.1 and 4.1 below. One way to produce an example of a symplectic manifold which does not carry Kähler metrics (resp. of a cosymplectic manifold which does not carry coKähler metrics) consists in constructing a symplectic manifold (resp. a cosymplectic manifold) which violates some of these topological properties.
In this paper we study two of these topological properties, which manifest both in the Kähler and in the coKähler case, namely the Lefschetz property and the formality of the rational homotopy type. The interplay between them has been extensively investigated in the symplectic case, yet nothing has been said in the cosymplectic case.
In the symplectic framework, our starting point is the paper [58], in which the authors collected the known (at the time) examples of symplectic manifolds which are, for instance, formal, but do not satisfy the Lefschetz property. Some questions remained open in [58]. Due to recent work of many authors (among them we quote Bock [23] and Cavalcanti [26]), we are able to answer such questions, completing the picture started in [58].
2.3. Formality. The de Rham theorem asserts that the real cohomological information of a smooth manifold M can be recovered from the analysis of the smooth forms Ω˚pM q. Apart from the additive structure, Ω˚pM q is endowed with two other operators: ‚ the de Rham differential d : Ω k pM q Ñ Ω k`1 pM q; ‚ the wedge product^: Ω k pM q b Ω ℓ pM q Ñ Ω k`ℓ pM q; these interact through the Leibnitz rule, which says that d is a (graded) derivation with respect to the wedge product, i.e. dpα^βq " dα^β`p´1q deg α α^dβ.
In the seminal paper [89], and following previous work of Quillen, Sullivan showed that the infinitesimal nature of the wedge product could be used to extract a great deal of homotopical information of a space X from the analysis of piecewise linear differential forms ΩP L pXq on X. More precisely, from ΩP L pXq one can recover not only the rational cohomology H˚pX; Qq of X, but also its rational homotopy groups π k pXq b Q. For this to work, X has to be a nilpotent space. This means that π 1 pXq is a nilpotent group whose action on higher homotopy groups is nilpotent. Here and in the sequel, by space we mean a CW-complex of finite type. We refer to [35,36,47] for all the results we quote in rational homotopy theory.
Definition 2.2. Let k be a field of zero characteristic. A commutative differential graded algebra over k (k-cdga for short) is a graded vector space A " ' ně0 A n together with a product¨which is commutative in the graded sense, i.e. x¨y " p´1q |x||y| y¨x, where |x| is the degree of a homogeneous element x, and a k-linear map d : A n Ñ A n`1 such that d 2 " 0, which is a graded derivation with respect to¨, i.e. dpx¨yq " dx¨y`p´1q |x| x¨dy. (1) Given two cdga's pA, dq and pA 1 , d 1 q, a morphism is a map ϕ : A Ñ A 1 which preserves the degree and such that d 1˝ϕ " ϕ˝d. A morphism of cgda's which induces an isomorphism in cohomology is a quasi isomorphism. (a) Given a cdga pA, dq, its cohomology is a cgda with trivial differential. (b) The de Rham algebra of a smooth manifold, endowed with the wedge product and the exterior differential, is an R-cdga. (c) Piecewise linear differential forms pΩP L pXq, dq on a space X are a Q-cdga. (d) Given a Lie algebra g defined over k, the Chevalley-Eilenberg complex p Ź g˚, dq is a k-cdga. Given a basis te 1 , . . . , e n u of g and its dual basis te 1 , . . . , e n u of g˚, the differential of e k P g˚is defined by de k pe i , e j q "´e k pre i , e j sq, then extended to Ź gb y imposing (1). That d squares to zero is equivalent to the Jacobi identity in g.
A very important class of examples of cdga's is provided by Sullivan minimal algebras: V is the free commutative algebra generated by a graded vector space V " ' ně0 V n ; ‚ there exists a basis tx τ u τ PI of generators of V , for some well-ordered index set I, such that |x ν | ď |x µ | for ν ă µ and dx µ is expressed in term of the x ν with ν ă µ.
In particular, d has no linear part.
The reason why Sullivan minimal algebras are important is given by the following result: Theorem 2.5. Let pA, dq be a k-cdga, where charpkq ‰ 0. Then there exist a Sullivan minimal algebra p Ź V, dq and a morphism ϕ : p Ź V, dq Ñ pA, dq which induces an isomorphism on cohomology. p Ź V, dq, which is unique up to automorphisms, is the minimal model of pA, dq. Definition 2.6. Let X be a nilpotent space. The minimal model of X is the minimal model of the cdga pΩP L pXq, dq. It is a Q-cdga, usually denoted by p Ź V X , dq.
Remark 2.7. When M is a smooth manifold, its real minimal model is the minimal model of the de Rham algebra pΩ˚pM q, dq.
Recall that the rationalization of a space X is a rational space X Q (i.e. a space whose homotopy groups are rational vector spaces) together with a map f : X Ñ X Q inducing isomorphisms π k pXq b Qπ k pX Q q. Two spaces X and Y have the same rational homotopy type if their rationalizations X Q and Y Q have the same homotopy type. Sullivan constructed a 1-1 correspondence between nilpotent rational spaces and isomorphism classes of Sullivan minimal algebras over Q, given by In this sense, one can study rational homotopy types algebraically.
Definition 2.8. Let pA, dq be a cdga and let p Ź V, dq be its minimal model. pA, dq is formal if there exists a quasi isomorphism p Since, by definition, the cohomology of p Ź V X , dq is precisely pH˚pX; Qq, 0q, a space X is formal if its minimal model (hence its rational homotopy type) is determined by its rational cohomology. There are many examples of formal spaces, among them Kähler and coKähler manifolds, symmetric spaces, H-spaces.
Remark 2.9. In [64], Kotschick called a manifold geometrically formal if it carries a Riemannian metric for which all wedge products of harmonic forms are harmonic. One sees easily that a geometrically formal manifold is formal. Indeed, consider the map pH˚pM q, 0q Ñ pΩ˚pM q, dq which assigns to each cohomology class its unique harmonic representative (harmonic with respect to the metric which makes M geometrically formal). Such map is then a morphism of cdga's, and is clearly a quasi isomorphism. By general theory of minimal models (see for instance [84,Chapter 1]), one gets a quasi isomorphism p Ź V M , dq Ñ pH˚pM q, 0q, hence M is formal. There are, however, examples of formal manifolds which are not geometrically formal (see [65]). Such examples are generalised symmetric spaces of compact simple Lie groups. It is unclear whether there exist examples of formal, not geometrically formal compact homogeneous non-symmetric spaces. Geometric formality influences the topology of the underlying manifold (see [85]).
Massey products are an obstruction to formality. We describe here triple Massey products and refer to [42,84] for their higher order analogue. Let M be a manifold and let a i P H p i pM ; Rq, 1 ď i ď 3, be three cohomology classes such that a 1 Y a 2 " 0 " a 2 Y a 3 . Take forms α i on M with rα i s " a i and write α 1^α2 " dσ, α 2^α3 " dτ . The Massey product of these classes is xa 1 , a 2 , a 3 y " rα 1^τ`p´1 q p 1`1 σ^α 3 s P It was proven in [32] that all Massey products vanish on a formal manifold.
In the present paper we will also deal with spaces which are not nilpotent, for instance solvmanifolds (see Section 2.5 below). When it comes to them, we shall also address the question of whether they are formal spaces or not. When we ask such a question, we simply mean to ask whether the minimal model is a formal cdga, in the sense of Definition 2.8.
2.4. s-formality. In [42], the second and third authors introduced the notion of s-formality, which is a suitable weakening of the notion of formality, and prove that for compact oriented manifolds the weaker notion implies the stronger one. First note the following result of [32] which gives a characterization of formality. Ź V and the space V decomposes as a direct sum V " C ' N with dpCq " 0, d is injective on N and such that every closed element in the ideal IpN q generated by N in Ź V is exact.
In [42] we weaken the condition of formality as follows.
Definition 2.11. We say that a minimal model pA, dq is s-formal if we can write A " Ź V such that for each i ď s the space V i of generators of degree i decomposes as a direct sum V i " C i ' N i , where the spaces C i and N i satisfy the following three conditions: (1) dpC i q " 0, (2) the differential map d : any closed element in the ideal Ip À iďs N i q, generated by À A connected manifold is s-formal if its minimal model is s-formal. The main result of [42] is: Theorem 2.12. Let M be a connected and orientable compact differentiable manifold of dimension 2n, or p2n´1q. Then M is formal if and only if it is pn´1q-formal.
Definition 2.13. A (compact) nilmanifold is the quotient of a simply connected nilpotent Lie group G by a lattice Γ.
Definition 2.14. A (compact) solvmanifold is the quotient of a simply connected solvable Lie group G by a lattice Γ.
Notice that every nilpotent group is solvable, hence every nilmanifold is a solvmanifold, but the converse is not true. A solvable group G is completely solvable if the adjoint representation on g has only real eigenvalues. Every nilpotent Lie group is completely solvable.
A connected, simply connected nilpotent Lie group is diffeomorphic to R n for some n; the diffeomorphism is given by the exponential map. Also, a simply connected solvable Lie group is homeomorphic to R n for some n. Hence a solvmanifold S " ΓzG (in particular, a nilmanifold) is an aspherical space with π 1 pSq " Γ.
According to a theorem of Mal'čev (see [73]), a simply connected nilpotent Lie group G admits a lattice Γ if and only if there exists a basis of g such that the structure constants are rational numbers. So far, a statement of this flavor is not known for simply connected solvable Lie groups. A necessary condition for a Lie group to admit a compact quotient is unimodularity. The construction of lattices in solvable Lie groups is an active area of research (see for instance [23,30] , .
and the lattice Γ " tA P H | x, y, z P Zu. Then N " ΓzH is a 3-dimensional nilmanifold. The Chevalley-Eilenberg complex of h is Since b 1 pN q " 2, N is not diffeomorphic to a torus.
3. Formality and the Lefschetz property in symplectic geometry (ii) the Lefschetz map L n´k : H k pM ; Rq Ñ H 2n´k pM ; Rq, rαs Þ Ñ rωs n´k^r αs is an isomorphism, 0 ď k ď n; (iii) the rational homotopy type of M is formal.
piq and piiq follow from Hodge-de Rham theory on a compact Kähler manifold. Notice that piiq implies piq. This is clear since, for 1 ď k ď t n 2 u, the bilinear map H 2k´1 pM ; RqˆH 2k´1 pM ; Rq Ñ H 2n pM ; Rq -R, prαs, rβsq Þ Ñ rαs^rβs^rωs n´2k`1 is a symplectic form on H 2k´1 pM ; Rq. Then one simply applies Poincaré duality. For a proof of these facts we refer to [57].
These properties have proven to be extremely useful in the task of constructing examples of compact symplectic manifolds with no Kähler metric (see for instance [14,26,44,46,77]). It is natural to ask whether these three properties are related on a compact symplectic manifold. Such a question was first tackled in [58]. The authors collected the examples, known at the time, of compact symplectic manifolds which violate some of the properties of Theorem 3.1. The kind of question we want to answer is Is there a compact symplectic manifold M which satisfies piq and piiiq above but not piiq?
The fundamental group plays a crucial role in setting this question. Indeed, while it is relatively easy to come up with non-simply connected compact symplectic manifolds which are non-formal or do not satisfy the Lefschetz property, the same question is harder in the simply connected case. We shall therefore subdivide our examples into the simply connected and the non-simply connected case.
Remark 3.2. There is a fourth topological constraint on the topology of compact Kähler manifolds: their fundamental groups are not arbitrary (see [2]). On the other hand, Gompf showed in [46] that every finitely presented group is the fundamental group of a symplectic 4-manifold.
3.2. The Lefschetz property on compact symplectic manifolds. Let pM 2n , ωq be a compact symplectic manifold and consider, for 0 ď k ď n, the map Clearly (3) sends closed (resp. exact) forms to closed (resp. exact) forms, hence it descends to a well defined map Definition 3.3. We say that a symplectic manifold pM 2n , ωq satisfies the Lefschetz property if the Lefschetz map (3) is an isomorphism for 0 ď k ď n. pM, ωq is of Lefschetz type if (3) is an isomorphism for k " 1.
The Lefschetz map is related to some other important objects which can defined on a compact symplectic manifold. On the one hand, following work of Koszul [63], Brylinski (see [22]) and Libermann (see [69]) showed that, on a compact symplectic manifold pM, ωq, one can define a symplectic ‹-operator, ‹ : Ω k pM q Ñ Ω 2n´k pM q; from this, one gets a symplectic codifferential δ : Ω k pM q Ñ Ω k´1 pM q defined by δ -p´1q k p‹d‹q. A form α P Ω k pM q is symplectically harmonic if dα " 0 and δα " 0. Denote by Ω˚pM, ωq the space of sympletically harmonic forms on pM, ωq. Clearly, The Lefschetz property appears indirectly in another feature of the study of cohomological properties of symplectic manifolds. Given a symplectic manifold pM, ωq, we consider the differential d and the symplectic codifferential δ. The following property is known as dδlemma: Im d X ker δ " Im δ X ker d " Im dδ.
Merkulov (see [80]) related the dδ-lemma with the symplectically harmonic cohomology. More precisely, he proved: Theorem 3.5. A symplectic manifold pM, ωq satisfies the dδ-lemma if and only if the inclusion H k pM, ωq ãÑ H k pM ; Rq is an isomorphism.
We mention here that the Lefschetz property has been studied also in the context of almost Kähler manifolds. Let pM, g, Jq be an almost Kähler manifold and let ω be the Kähler form. Motivated by the Donaldson "tamed to compatible" conjecture (see [34]), Li and Zhang considered in [67] the following subspaces of H 2 pM ; Rq: One has HJ pM q`HJ pM q Ă H 2 pM ; Rq, but the sum is in general neither direct nor equal to H 2 pM ; Rq. The almost complex structure J is said to be ‚ C 8 -pure if HJ pM q X HJ pM q " 0; ‚ C 8 -full if HJ pM q`HJ pM q " H 2 pM ; Rq; ‚ C 8 -pure-and-full if HJ pM q ' HJ pM q " H 2 pM ; Rq. Theorem 3.6. Let pM 2n , g, Jq be an almost Kähler manifold and assume that the Lefschetz map L n´2 : Ω 2 pM q Ñ Ω 2n´2 pM q takes harmonic forms to harmonic forms. Then, if J is To conclude this section, we quote a result of Benson and Gordon, see [19, Proof of Theorem A].
Theorem 3.7. Let pN 2n , ωq be a compact nilmanifold endowed with a symplectic structure. Assume that pN, ωq is of Lefschetz type. Then N is diffeomorphic to a torus T 2n .
Hence every symplectic non-toral nilmanifold violates condition piiq in Theorem 3.1 and is therefore non Kähler. By Nomizu's theorem 2.15, a symplectic form on a nilmanifold is cohomologous to a left-invariant one. Symplectic structures on nilpotent Lie algebras have been studied in [51]. A complete classification of symplectic nilmanifolds up to dimension 6 is available in [17]. As we observed, nilmanifolds are never simply connected.
3.3. Formality of compact symplectic manifolds. As we already pointed out, formality is a property of the rational homotopy type of a space, or manifold. Since every compact Kähler manifold is formal according to Theorem 3.1 (see also [32]), it is reasonable to investigate whether formality also holds for arbitrary symplectic manifolds. In the context of nilmanifolds, a formal symplectic nilmanifold is diffeomorphic to a torus by Hasegawa's theorem 2.17. Hence, a symplectic non-toral nilmanifold violates condition piiiq in Theorem 3.1. Using nilmanifolds, we obtain many examples of non formal symplectic manifolds, albeit non simply connected.
In fact, the construction of a simply connected symplectic non formal manifold is a much harder problem. As very often happens in symplectic geometry, one of the main problems is that there are relatively few techniques to construct symplectic manifolds. Due to this lack of examples, Lupton and Oprea (see [70]) asked whether a simply connected compact symplectic manifold is formal (this is what they called the formalising tendency of a symplectic structure). Nowadays, however, we know many simply connected symplectic non formal manifolds: Theorem 3.8. For every n ě 4 there exists a compact, simply connected non formal symplectic manifold of dimension 2n. Remark 3.9. We point out that, due to a result of Miller (see [42,81]), a simply connected manifold of dimension ď 6 is automatically formal. Hence Theorem 3.8 covers all possible cases in which such a phenomenon can occur.

3.4.
Examples: the non-simply connected case. Thanks to recent contributions of many authors, we can fill up Table 1 of [58].
Let us describe the manifolds which appear in Table 1.
‚ T 2n is the 2n-dimensional torus, which carries a Kähler structure. ‚ M 4 is a 4-dimensional compact symplectic completely solvable solvmanifold. According to [40] (see also [84,  is a lattice; we refer to [23,Theorem 9.4] for a complete description of this example. ‚ E 4 is the compact nilmanifold defined by the equations de 1 " de 2 " 0, de 3 " e 1^e2 , de 4 " e 1^e3 considered in [38]. It was the first example of compact symplectic manifold of dimension 4 which does not admit complex structures. In fact, E 4 is non-formal and does not satisfy the Lefschetz property, but its odd Betti numbers are even. Thus, by the Enriques-Kodaira classification [62], E 4 does not have complex structures. Note that in dimension 6, Iwasawa manifold was the first example of compact symplectic and complex manifold whose odd Betti numbers are even, but not admitting Kähler metrics, as it is non-formal [39]. Non-Kähler compact symplectic manifolds in higher dimension are given for example in [31]. ‚ KT is the so-called Kodaira-Thurston manifold. As a symplectic manifold, it was first described by Thurston in [90]; there, he showed that b 1 pKT q " 3, hence KT is not Kähler. KT also has the structure of a compact complex surface (a primary Kodaira surface, see [11,Page 197]) and was known to Kodaira. Abbena described KT as a nilmanifold in [1]. It is the product of the Heisenberg manifold (see Example 2.19) and a circle. ‚ Cp12q is a 12-dimensional simply connected symplectic manifold. We shall describe it in Section 3.5.
Remark 3.10. In [58], the authors distinguish between the simply connected and the aspherical case, rather than non simply connected. All the examples in Table 1 are aspherical, except for Cp12qˆT 2 . We should point out here that a symplectic manifold pM, ωq is called for every map f : S 2 Ñ M . An aspherical symplectic manifold is clearly symplectically aspherical. Symplectically aspherical manifolds play an important role in symplectic geometry, see [61] for a survey on this topic.
Remark 3.11. In [60], Kasuya   Before describing the content of Table 2, we would like to recall a construction of McDuff (see [77,84]) which allows to obtain new symplectic manifolds: the symplectic blow-up. Together with the fibre connected sum (see [46] and [78,Chapter 7]), symplectic fibrations (see [78,Chapter 6] and the references therein), approximately holomorphic techniques of Donaldson (see [9,33,82]) and symplectic resolutions (see [27,44]), it is the most effective technique when it comes to constructing new symplectic manifolds.
Using ideas of Gromov, and generalizing a known construction in algebraic geometry, McDuff defined the notion of symplectic blow-up of a symplectic manifold pX, σq along a symplectic submanifold pY, τ q. Since Y Ă X is a symplectic submanifold, the normal bundle N Y of Y in X has the structure of a complex vector bundle. The blow-up of X along Y , replaces a point y P Y with the projectivization of N y Y . This produces a new manifold X " Bl Y X and a map p :X Ñ X, with the following properties: ‚ [77, Proposition 2.4] π 1 pXq " π 1 pXq and H˚pX; Rq fits into a short exact sequence of R-modules 0 Ñ H˚pX; Rq Ñ H˚pX; Rq Ñ A˚Ñ 0, where A˚is a free module over H˚pY ; Rq with one generator in each dimension 2i, 1 ď i ď k´1, where 2k is the codimension of Y in X. ‚ [77, Proposition 3.7] If Y is compact,X carries a symplectic formσ which agrees with p˚σ outside a neighborhood of p´1pY q.
We also recall the following result of Gompf, which allows to construct compact symplectic manifolds which do not satisfy the Lefschetz property. We begin now the description of the manifolds in Table 2.
‚ CP n is the complex projective space, which is known to have a Kähler structure. ‚ M p6, 0, 0q is the manifold constructed by taking n " 6, G the trivial group and b " 0 in Theorem 3.12. Since it is a simply connected 6-manifold, it is formal by the result of Miller, see Remark 3.9. ‚ N is a 6-dimensional simply connected (hence formal) symplectic and complex manifold which does not satisfy the Lefschetz property. Such a manifold was constructed by the authors in [14]. ‚ We shall construct Ą CP 7 in Proposition 3.13 below. ‚ Cp12q is a simply connected, symplectic 12-manifold which satisfies the Lefschetz property but has a non-zero triple Massey product. Hence it is non-formal. Such example was constructed by Cavalcanti in [26, Example 4.4] using Donaldson's techniques together with the symplectic blow-up. ‚ Ą CP 5 is the symplectic blow-up of CP 5 along a symplectic embedding of the Kodaira-Thurston manifold KT . Historically, this was the first example of a simply connected symplectic manifold with no Kähler structures. It was constructed by McDuff in [77].
We come now to the description of Ą CP 7 . Let us consider the symplectic 6-manifold S " ΛzG 6.78 which appeared in Section 3.4. Recall that S is a formal symplectic manifold which does not satisfy the Lefschetz property and has b 1 pSq " 1. According to Tischler's result, we find a symplectic embedding of pS, ωq in pCP 7 , ω 0 q. Now set Ą CP 7 -Bl S CP 7 .
Proposition 3.13. Ą CP 7 is a simply connected, formal symplectic manifold which does not satisfy the Lefschetz property and has b 3 p Ą CP 7 q " 1.
Proof. That π 1 p Ą CP 7 q " 0 follows immediately from McDuff result we discussed above. b 3 p Ą CP 7 q " 1 follows from (5) and the fact that b 1 pSq " 1. This immediately implies that Ą CP 7 does not have the Lefschetz property.
By [41, Theorem 1.1], Ą CP 7 is formal. However, let us see this explicitly by computing its minimal model. The 6-manifold S " ΛzG 6.78 is the quotient of the simply connected completely solvable symplectic Lie group G 6.78 of [23, Theorem 9.4] by a lattice Λ Ă G 6.78 .
We check that Ą CP 7 is 6-formal (see Definition 2.10). Take an element β P IpN 6 q which is closed, and we have to check that rϕpβqs " 0 in H˚p Ą CP 7 ; Rq. As β can have degree at most 14, it has to be β " β 1¨w¨u , with β 1 P Ź px, y, zq. But clearly rϕpw¨uqs " 0 since rη^γs " 0 P H 5 pS; Rq. By Theorem 2.12, Ą CP 7 is formal.
Remark 3.14. It seems quite hard to find a compact simply connected symplectic manifold which is formal and satisfies the Lefschetz property but is not Kähler.
Remark 3.15. The manifold N which appears in Table 2, line 3, carries a symplectic structure and a complex structure but no Kähler metric. For a discussion on manifolds which are simultaneously complex and symplectic but not Kähler, we refer to [14, Section 1].

Formality and the Lefschetz property in cosymplectic geometry
A compact cosymplectic manifold pM 2n`1 , η, ωq is never simply connected. Indeed, consider the 1-form η; being closed, it defines a cohomology class rηs P H 1 pM ; Rq. If η " df , with f P C 8 pM q, by compactness of M there exists p P M so that η p " 0. But this contradicts the condition ηpξq " 1. Alternatively, one can argue using the fact that η^ω n is a volume form, hence, if η were exact, the same would be true for the volume form, which is absurd. In this way, one sees that b 1 pM q ě 1 on a cosymplectic manifold. This argument allows actually to conclude that η^ω k is a closed, non exact form, for 0 ď k ď n. Hence, b k pM q ě 1 for 0 ď k ď 2n`1 on a cosymplectic manifold.
CoKähler manifolds are the odd-dimensional counterpart of Kähler manifolds. It is not a big surprise, therefore, that they satisfy as well strong topological conditions. Theorem 4.1. Let pM 2n`1 , φ, η, ξ, gq be a compact coKähler manifold and let ω be the Kähler form. Then is an isomorphism, 0 ď k ď n; (iii) the rational homotopy type of M is formal.
A proof of this result can be found in [28]; see also [16] for a different perspective on how such properties can be deduced from the corresponding properties of compact Kähler manifolds. Here H˚pM q denotes harmonic forms on the Riemannian manifold pM, gq; clearly H˚pM q -H˚pM ; Rq. The map (6) really sends harmonic forms to harmonic forms, as it is proved in [28]; recall that both η and ω are parallel on a coKähler manifold, hence harmonic.
Here as well, we want to answer questions such as Is there a compact cosymplectic manifold M which satisfies piq and piiiq above but not piiq?
Unfortunately, as we shall see in the next Section, the Lefschetz map can not be defined, in general, on arbitrary cosymplectic manifolds. We will identify a certain property, morally equivalent to the Lefschetz type condition of Definition 3.3, and address the Lefschetz question in this setting.
4.1. The Lefschetz property in cosymplectic geometry. Let pM, ωq be a symplectic manifold. As we remarked above, the Lefschetz map (2) sends closed forms to closed forms, hence descends to cohomology, giving (3). In particular, the Lefschetz map can be defined on any symplectic manifold. Of course, one needs then to use Kähler identities to prove that (3) is an isomorphism in the Kähler case and we have seen that there are symplectic manifolds for which (3) is not an isomorphism.
The cosymplectic case is much subtler. A first instance is that, differently from what happens in the Kähler case, the Lefschetz map on coKähler manifolds is defined only on harmonic forms. On a cosymplectic manifold, however, there is no metric. Let pM, η, ωq be a compact cosymplectic manifold; if one tries to define a map L n´k : Ω k pM q Ñ Ω 2n`1´k pM q, α Þ Ñ ω n´k`1^ı ξ α`ω n´k^η^α (7) one sees that it does not send closed forms to closed forms! Indeed, for α a closed k-form, dpω n´k`1^ı ξ α`ω n´k^η^α q " ω n´k`1^d pı ξ αq, which is not zero in general. where e ij is a short-hand for e i^ej . Set η " e 5 and ω " e 13´e24 . Then pg, η, ωq is a cosymplectic Lie algebra; the Reeb field is ξ " e 5 . Let G denote the simply connected nilpotent Lie group with Lie algebra g. Since the Lie algebra g is defined over Q, G contains a lattice Γ; hence N -ΓzG is a compact nilmanifold. The cosymplectic structure on g gives a left-invariant cosymplectic structure on G, which descends to N . Hence pN, η, ωq is a cosymplectic nilmanifold. By Nomizu's theorem, p Ź g˚, dq ãÑ pΩ˚pN q, dq is a quasi isomorphism. We study the Lefschetz map on p Ź g˚, dq, i.e.
L 5´k : ľ k g˚ÝÑ ľ 5´k g˚, 0 ď k ď 2. A computation shows that L 4 sends closed forms to closed forms. However, the closed 2-form α " e 35 is sent to β "´e 234 , which is not closed, since dβ " e 1245 .
A way to bypass this difficulty would be to work with forms on M which are preserved by the flow of the Reeb field ξ, that is, to consider the differential subalgebra Ωξ pM q of Ω˚pM q, defined by Ω k ξ pM q " tα P Ω k pM q | L ξ α " 0u. Lemma 4.3. The Lefschetz map (7) restricts to a map L n´k : Ω k ξ pM q Ñ Ω 2n`1´k ξ pM q (9) which sends closed forms to closed forms.
Everything works just fine, but there is of course a problem: the differential graded algebra pΩξ pM q, dq Ă pΩ˚pM q, dq does not compute, in general, the de Rham cohomology of M . The question arises, whether there exists a class of almost coKähler structures for which the inclusion pΩξ pM q, dq ãÑ pΩ˚pM q, dq is a quasi isomorphism, i.e. for which Hξ pM q -H˚pΩξ pM q, dq satisfies Hξ pM q -H˚pM ; Rq.
Recall that an almost coKähler structure pφ, η, ξ, gq is K-cosymplectic if the Reeb field is Killing. K-cosymplectic structures have been extensively studied in [15]; the inspiration there came from the contact (metric) case, where one defines a K-contact structure as a contact metric structure whose Reeb field is Killing.
On a K-cosymplectic manifold M , the 1-dimensional distribution defined by ξ integrates to a Riemannian foliation F ξ , whose leaf through x P M is the flowline of ξ.
We denote by H˚pM ; F ξ q the basic cohomology. One can think of it as the cohomology of the space of leaves M {F ξ . We collect in the following Proposition the relevant features of K-cosymplectic structures (compare [15,Theorem 4
Here we interpret η as a harmonic 1-form. As a consequence, the Lefschetz map (9) on a K-cosymplectic manifold pM 2n`1 , φ, η, ξ, gq descends to a map which can, or not, be an isomorphism.
From now on, we restrict to K-cosymplectic manifolds in order to study the Lefschetz map.
Definition 4.6. Let M 2n`1 be a compact manifold endowed with a K-cosymplectic structure pφ, η, ξ, gq. We say that M has the Lefschetz property if (11) is an isomorphism for 0 ď k ď n. We say that M is of Lefschetz type if (11) is an isomorphism for k " 1.
Proposition 4.7. Let M 2n`1 be a compact manifold endowed with a K-cosymplectic structure pφ, η, ξ, gq. Assume that M satisfies is of Lefschetz type. Then b 1 pM q is odd.
Proof. The splitting (10) tells us that H 1 ξ pM q " H 1 pM ; F ξ q ' η and that H 2n ξ pM q " H 2n pM ; F ξ q ' η^H 2n´1 pM ; F ξ q. By assumption, L n´1 : H 1 ξ pM q Ñ H 2n ξ pM q is an isomorphism. Its restriction to H 1 pM ; F ξ q sends α to ω n´1^η^α P η^H 2n´1 pM ; F ξ q. In particular, ω n´1^α ‰ 0. Now consider the bilinear map Ψ is clearly skew-symmetric and non-degenerate. Hence dim H 1 pM ; F ξ q is even. Since b 1 pM q " dim H 1 pM ; F ξ q`1, the thesis follows.
We have some kind of converse to this result: Proposition 4.8. Suppose M is a compact manifold endowed with a K-cosymplectic structure pφ, η, ξ, gq. If b 1 pM q " 1, then M is of Lefschetz type.
Proof. Since M is compact and b 1 pM q " 1, H 1 ξ pM q " η. The Lefschetz map (11) sends η to ω n P H 2n ξ pM q, which is non-zero. Remark 4.9. It is known that b 1 is odd on a compact manifold endowed with a coKähler structure; however, nothing can be said on the higher odd-degree Betti numbers, even up to middle dimension. Consider, for instance, the manifold MˆMˆS 1 , where M is the K3 surface. This is clearly coKähler and has b 3 " 44. This is why, in Table 4 below, the column about Betti numbers has been removed.
We show next that K-cosymplectic manifolds abund. Indeed, the following holds (see [15,Proposition 2.12]): Proposition 4.10. Let pK, h, ωq be a compact almost Kähler manifold and let ϕ : K Ñ K be a diffeomorphism such that ϕ˚h " h and ϕ˚ω " ω. Then the mapping torus 1 K ϕ has a natural K-cosymplectic structure.
In particular, the product KˆS 1 of an almost Kähler manifold pK, h, ωq and a circle has a natural K-cosymplectic structure, with the product metric g " h`dθ 2 . The cosymplectic structure is given by taking ξ to be the vector tangent to S 1 and η to be the dual 1-form. For a product K-cosymplectic manifold, M " KˆS 1 , the space of leaves of F ξ is simply K, hence (10) becomes H k ξ pM q " H k pK; Rq ' η^H k´1 pK; Rq.
Under this splitting, α P H k ξ pM q can be written as α 0`η^α1 with α 0 P H k pK; Rq and α 1 P H k´1 pK; Rq. We have ı ξ α j " 0, j " 0, 1, and η^η^α 1 " 0. As a consequence, the Lefschetz map (11) sends α 0 P H k pK; Rq Ă H k ξ pM q to ω n´k^η^α 0 and η^α 1 to ω n´k`1^α 1 . We easily obtain the following result: We focus on a situation in which one could think of studying some analogue of the Lefschetz property on arbitrary cosymplectic manifolds. We make the following definition: The usual terminology for a morphism of cdga's, inducing an isomorphism on cohomology, is quasi-isomorphism. It follows from this definition that if M is an n-dimensional manifold, then dimpV M q " n, because a generator of the top power of V M must be sent to the volume form on M . Suppose dim V M " n; then Ź n V M is one-dimensional, hence a generator vol P Ź n V M is sent to the volume form on M by ϕ. By compactness, and using the fact that ϕ commutes with the differential, this implies that vol can not be exact and, as a consequence, that every element w in Ź n´1 V M must be closed; otherwise one would necessarily have dw " cvol, with c P R˚, making vol exact. Proof. This follows from Nomizu Theorem [83] when S is a nilmanifold and from Hattori Theorem [56] when S is a completely solvable solvmanifold. In both cases, the algebraic model is the Chevalley-Eilenberg complex p Ź g˚, dq of the Lie algebra g of G. For the non-completely solvable case, one uses a result of Guan ([50], see also [29]) to obtain a modification of the Lie group G,G, such that S " ΓzG and the Lie algebra cohomology of g computes H˚pS; Rq.
Let pM 2n`1 , η, ωq be a cosymplectic manifold satisfying the hypotheses of Definition 4.12. Let p Ź V M , dq be the algebraic model and let ϕ : p Ź V M , dq Ñ pΩ˚pM q, dq. Since ϕ induces an isomorphism in cohomology, and both η and ω are closed non-exact forms, one can choose cocycles v P V M and w P Ź 2 V M such that rϕpvqs " rηs and rϕpwqs " rωs. Notice that v^w n is a generator of is such that rws " rws in H 2 p Ź V M q, then there exists u P V M such thatw " w`du. Then v^w n " v^w n , since they differ by an exact form and there are no exact forms in Ź 2n`1 V M . The (algebraic) Reeb field θ P VM is determined by the equation ı θ pv^w n q " w n .
Take a closed form γ P Ω 1 pM q. We find a cocycle z P V M such that rϕpzqs " rγs. Define an algebraic Lefschetz map V M Ñ Ź 2n V M by z Þ Ñ yw n´1^v^z`wn^ı θ z.
By what we said so far, y is a closed element in Ź 2n V M . Hence the algebraic Lefschetz map sends 1-cocycles to 2n-cocycles. Therefore, it descends to cohomology and it is reasonable to ask whether it is an isomorphism there.
Definition 4.14. Let pM, η, ωq be a compact cosymplectic manifold and assume that the cohomology of M can be computed from an algebraic model. We say that M is 1-Lefschetz if (13) is an isomorphism.
In the case of nilmanifolds, we have the following result, which can be seen as the cosymplectic analogue of the Benson-Gordon Theorem 3.7.
Theorem 4.15. Let N " ΓzG be a compact nilmanifold of dimension 2n`1 endowed with a cosymplectic structure pη, ωq. Assume that N is 1-Lefschetz. Then N is diffeomorphic to a torus.
Proof. The Chevalley-Eilenberg complex p Ź g˚, dq of the Lie algebra g is an algebraic model of the cohomology of N , as follows from Proposition 4.13; notice that dim g " 2n`1. Let ϕ : p Ź g˚, dq Ñ pΩ˚pN q, dq be the quasi-isomorphism. In particular, the cohomology of N can be computed from this algebraic model. In order to prove that N is diffeomorphic to a torus, it is enough to prove that d " 0 in p Ź g˚, dq (see the argument in [84, Theorem 2.2, Chapter 2]). Arguing as above, we can assume that there are cocycles v P g˚and w P Ź 2 gm apping to η and ω respectively under ϕ. Assume that d is non-zero. Then we can choose a Mal'cev basis of g˚, g˚" xv, x 1 , . . . , x s , x s`1 , . . . , x 2n y, with v " x 0 , dx i " 0, 0 ď i ď s, and dx j , s`1 ď j ď 2n, is a non-zero linear combination of products x kℓ :" x k^xℓ with k, ℓ ă j, for some s ă 2n`1. Then w can be written as for some coefficients a ij P R; we have collected all the summands that contain x 2n into z P g˚, where z does not contain x 2n . Also, notice that z must be a cocycle. Indeed, when one computes dw, which must be zero, the term dz¨x 2n pops up. But since we have chosen a Mal'cev basis for g˚, the generator x 2n can not appear in the differential of any other x k , and this forces z to be a cocycle. We define a derivation λ of degree´1 on g˚by the rule λpx i q " 0, 0 ď i ď 2n, λpx 2n q " 1, and extend it to p Ź g˚, dq by forcing the Leibnitz rule. Assume that the algebraic Lefschetz map (13) is an isomorphism; the cocycle z is sent to the cocycle w n´1^v^z . For degree reasons we have y^v^w n " 0 for every 1-cocycle. Applying λ we get 0 " λpy^v^w n q " λpyq^η^w n´y^λ pvq^w n`y^v^λ pω n q " " n y^v^λpwq^w n´1 " n y^v^z^w n´1 .
This must be true for every 1-cocycle y, and v^z^w n´1 is non-zero by the Lefschetz-type hypothesis. But this violates Poincaré duality. Thus we obtain a contradiction with the existence of a non-closed generator of g˚, since λ was defined as 0 on such generators. Hence d " 0.
Remark 4.16. Recall that a manifold M 2n is cohomologically symplectic is there is a class ω P H 2 pM ; Rq such that ω n ‰ 0. Every symplectic manifold is cohomologically symplectic, but the converse is not true. Indeed, CP 2 #CP 2 is cohomologically symplectic but admits no almost complex structures, as shown by Audin [6]. Hence, it can not be symplectic. The nice interplay between geometry and topology on cohomologically symplectic manifolds has been unveiled by Lupton and Oprea (see [71]). It was proven recently by Kasuya (see [59]) that cohomologically symplectic solvmanifolds are genuinely symplectic. Following these ideas, one can call a manifold M 2n`1 cohomologically cosymplectic if there exist classes η P H 1 pM ; Rq and ω P H 2 pM ; Rq such that η^ω n ‰ 0. By arguing as in [59], one can show that a cohomologically cosymplectic solvmanifold is cosymplectic.
Example 4.2 above shows that, even in the case of cosymplectic manifolds whose cohomology can be computed from an algebraic model, the Lefschetz map does not necessarily send cocycles of degree k ě 2 to cocycles. 4.2. Formality in cosymplectic geometry. As a consequence of formality of compact Kähler manifolds, it was proved in [28] that a compact coKähler manifold is formal. In [13], we constructed examples of non-formal cosymplectic manifolds with arbitrary Betti numbers. More precisely, we proved the following result: It is known that an orientable compact manifold of dimension ď 4 with first Betti number equal to 1 is formal, see [43]; in other words, the exception p3, 1q is not due to the fact that we require the manifold to have a cosymplectic structure.

Examples.
We let now Sections 4.1 and 4.2 come together, and fill up two tables similar to those given in Section 3 in the symplectic case. We have already observed that a compact manifold endowed with a cosymplectic structure always has b 1 ě 1. Therefore, one can never have simply connected compact cosymplectic manifolds. We will therefore distinguish two cases: ‚ compact K-cosymplectic manifolds with arbitrary first Betti number; ‚ compact K-cosymplectic manifolds with first Betti number equal to 1.
Since the product of a compact symplectic manifold and a circle always admits a Kcosymplectic structure, it is enough to take products of the manifolds which appear in Tables 1 and 2    We end this section with two further examples: a compact cosymplectic non-formal 5manifold with b 1 " 1 which is not the product of a 4-manifold and a circle and a compact K-cosymplectic 7-manifold which is not coKähler and is not the product of a 6-manifold and a circle. Example 4.18. In [13], we constructed a cosymplectic, non-formal 5-dimensional solvmanifold S with b 1 pSq " 1. S is the quotient of the completely solvable Lie group H, whose elements are matrices with px 1 , . . . , x 5 q P R 5 , by a lattice Γ. A global system of coordinates on H is given by tX 1 , . . . , X 5 u with X i pAq " x i . A basis of left-invariant 1-forms on H (which we identify with h˚) is given by A straightforward computation shows that The manifold S has a cosymplectic structure, defined by taking η " α 5 and ω " α 1^α4`α2^α3 .
Notice that S is 1-Lefschetz, according to Definition 4.14. We prove that S is not the product of a 4-manifold and a circle. Assume this is the case and write S " PˆS 1 . We use the product structure of H˚pS; Rq -H˚pP ; Rq b H˚pS 1 ; Rq. The generator rηs of H 1 pS 1 ; Rq is a zero divisor, since rηs^rα 1^α2 s " 0 for rα 1^α2 s P H 2 pP ; Rq, and this is absurd.
Example 4.19. Finally, we give an example of a 7-dimensional K-cosymplectic solvmanifold without coKähler structures, which is not a product of a 6-dimensional manifold and a circle. This shows that, although we considered product K-cosymplectic manifolds, there are more. This example is inspired by the discussion in [84, Chapter 3, Section 3]. We start with the 6-dimensional nilpotent Lie algebra h with basis te 1 , . . . , e 6 u and non-zero brackets re 1 , e 2 s "´e 4 , re 1 , e 3 s "´e 5 and re 2 , e 3 s "´e 6 .
Notice that h is the Lie algebra h 7 in the notation of [86] and the Lie algebra L 6,4 in that of [17]. The Chevalley-Eilenberg complex is p Ź h˚, dq with non-zero differential given, in terms of the dual basis te 1 , . . . , e 6 u, by de 4 " e 12 , de 5 " e 13 and de 6 " e 23 .
Consider the derivation D : h Ñ h given, with respect to te 1 , . . . , e 6 u, by the matrix Since D is skew-symmetric, it is an infinitesimal isometry of ph, hq; furthermore, one can check that D is an infinitesimal symplectic derivation of ph, ωq, i.e., D t ω`ωD " 0. We denote by g the semi-direct product h ' D R, which is a 7-dimensional solvable (but not completely solvable) Lie algebra with brackets: ‚ re i , e j s g " re i , e j s h , for 1 ď i ă j ď 6; ‚ re 7 , e i s g " Dpe i q for 1 ď i ď 6; here e 7 generates the R-factor.
We see that g sits in a short exact sequence 0 Ñ h Ñ g Ñ R Ñ 0 of Lie algebras and that h Ă g is an ideal. Define ϕ t -expptDq P Autphq. Then ϕt ω " ω and ϕt h " h. Let H be the unique simply connected nilpotent Lie group with Lie algebra h and consider the following diagram: where Φ t -exp˝ϕ t˝e xp´1 is a well-defined Lie group automorphism; indeed, H is nilpotent, hence exp is a global diffeomorphism. We consider h and ω as left-invariant objects on H. Then Φ t is, by construction, an isometry of pH, hq and a symplectomorphism of pH, ωq.
Let Λ be the lattice in h spanned over Z by the basis te 1 , . . . , e 6 u and set Λ -exppΛq. Then Λ Ă H is a lattice and N -ΛzH is a compact nilmanifold. Notice that ϕ π 2 preserves Λ, hence Φ " Φ π 2 preserves Λ and descends to a diffeomorphism of N . Since h and ω are left-invariant, they define a metric and a symplectic structure on N . Clearly, Φ is both an isometry and a symplectomorphism of pN, h, ωq. Now consider the solvable Lie group G " H¸Φ R and the lattice Γ " Λ¸Φ Z. Notice that Γ is a solvable group. We have two short exact sequnces of groups, namely 0 Ñ Λ Ñ Γ Ñ Z Ñ 0 and 0 Ñ H Ñ G Ñ R Ñ 0.
Then S -ΓzG is a solvmanifold which can be identified with the mapping torus of the diffeomorphism Φ : N Ñ N . Indeed, the mapping torus bundle coincides with the Mostow bundle of S, which is N Ñ S Ñ S 1 .
By Proposition 4.10, S has a natural K-cosymplectic structure. We can describe as a leftinvariant K-cosymplectic structure on G, i.e., a K-cosymplectic structure on g. Set η " e 7 , ξ " e 7 , ω "´e 16`e25`2 e 34 and let g be the scalar product which makes te 1 , . . . , e 7 u orthonormal. From this we recover φ. Hence pφ, η, ξ, gq is a K-cosymplectic structure on S. We proceed to show (1) b 1 pSq " 2, hence S is not coKähler; (2) S is not the product of a 6-dimensional manifold and a circle.
Notice that since G is not completely solvable, we can not use Hattori's theorem to compute the cohomology of S. We will instead regard S as the mapping torus N Φ and use the following result (see [13,Lemma 12] for a proof): Lemma 4.20. Let M be a smooth, n-dimensional manifold, let ϕ : M Ñ M be a diffeomorphism and let M ϕ be the corresponding mapping torus. For every 0 ď k ď n, set V k " kerpϕ˚´Id : H k pM ; Rq Ñ H k pM ; Rqq and C k " cokerpϕ˚´Id : H k pM ; Rq Ñ H k pM ; Rqq, where ϕ˚is the map induced by ϕ on H k pM ; Rq. Then where η is the pullback of the generator of H 1 pS 1 ; Rq under the mapping torus projection M ϕ Ñ S 1 .
According to [84, Chapter 3, Theorem 3.8], the action of Φ " Φ π 2 on H k pN ; Rq is obtained by taking the action of ϕ π 2 on Ź k h˚and considering the action induced on the cohomology H k ph; Rq. We are implicitly identifying the cohomology of N with the cohomology of h. We can do this thanks to Nomizu's theorem. Applying (15) with k " 1, one computes that H 1 pS; Rq " xre 3 s, re 7 sy.
Hence b 1 pSq " 2; this proves p1q above. Using again (15), one computes the remaining Betti numbers of S to be b 2 pSq " 3 and b 3 pSq " 7. Assume now that S is a product, S " PˆS 1 . Notice that P is an aspherical manifold and that π 1 pP q is solvable, being a subgroup of Γ " π 1 pSq. Also, by applying the Künneth formula, we obtain that χpP q "´1. But this contradicts Proposition 4.21 below. This proves p2q.
Proposition 4.21. Let M be a compact aspherical manifold whose fundamental group is solvable. Then χpM q " 0.