Formality and symplectic structures of almost abelian solvmanifolds
aa r X i v : . [ m a t h . DG ] F e b FORMALITY AND SYMPLECTIC STRUCTURES OFALMOST ABELIAN SOLVMANIFOLDS
MAURA MACR`I
Abstract.
In this paper we study some properties of almost abeliansolvmanifolds using minimal models associated to a fibration. In par-ticular we state a necessary and sufficient condition to formality and amethod for finding symplectic strucures of this kind of solvmanifolds. Introduction
Nilmanifolds and solvmanifolds are compact quotients of (respectively)nilpotent and solvable Lie groups by a lattice. They have been intensivelystudied from many points of view (geometry, topology, group theory) sincethey are, on the one hand, spaces for which the computation of some of theirinvariants is tractable and, on the other hand, they are involved enough toshow all sort of different behaviors.Typical examples in this context are the Nomizu Theorem for nilmani-folds, which states that their de Rham cohomology agrees with the one ofthe Lie algebra [12], and Benson and Gordon result on the non existenceof any K¨ahler structure on a nilmanifold (unless it is a torus) [1]. K¨ahlermanifolds are quite relevant within rational homotopy theory: in [5] it wasshown that a compact K¨ahler manifold is formal. Hasegawa, using an ex-plicit description of the minimal model of a nilmanifold, proved that a nontoral nilmanifold cannot be formal [10], yielding an alternate proof of theabove mentioned result by Benson and Gordon. In the case of solvmanifolds,Hasegawa proved in [11] that a solvmanifold carries a K¨ahler metric if andonly if it is covered by a finite quotient of a complex torus, which has thestructure of a complex torus bundle over a complex torus.In this paper we study the formality and the symplectic structures ofalmost abelian solvmanifolds, i.e. compact homogeneous spaces S = G/ Γ,where the solvable Lie group G is a semidirect product R⋉R n and Γ = Z ⋉ Z n . Our idea starts from a work of Oprea and Tralle in which the theory ofminimal models is used to compute the cohomology of some solvmanifolds[14]. Indeed the above mentioned result of Nomizu does not apply in gene-ral to the cohomology of solvmanifolds. Almost abelian solvmanifolds areprobably the most tractable class of solvmanifolds whose cohomology doesnot in general agree with the one of the Lie algebra.In particular, in [14], the authors use a theorem of Felix and Thomas(Theorem 1) in which is described the model of a fibration and they apply it to the
Mostow fibration (1) N/ Γ N = ( N Γ) / Γ ֒ → G/ Γ −→ G/ ( N Γ) = T k , where N is the nilradical of G , associated to every solvmanifold.This construction is related to a submodule U of the cohomology algebraof the abelian Lie algebra R n H ∗ ( R n ) computed using (1) and defined in[13, 14] (cf. Section 2).Rather than using this theory to compute the cohomology of the solv-manifold (cf. [4]), we find some of its properties. Indeed, while in generalthe submodule U is difficult to compute, its construction is quite simple foralmost abelian solvmanifolds. Hence we are able to find some properties of U and relate them to those of the solvmanifold.Let ( M U , d ) be a minimal commutative differential graded algebra (cdga),such that its cohomology algebra is isomorphic to U , then this algebra is asubspace of the minimal model ( M S , D ) of the solvmanifold S , (Theorem1).In particular in section 3 we find a necessary and sufficient condition forthe formality of S : Main Theorem. If M S is of finite type, then S is formal if and only if ker D | M U = ker d . In section 4 we give a method to find symplectic forms on an almostabelian solvmanifold and in the last section we give two examples of appli-cation of the Main Theorem.2.
Preliminaries
Definition 1.
An almost abelian solvmanifold is a quotient S = G/ Γ wherethe solvable Lie group G and its lattice Γ are semidirect products of kind G = R ⋉ ϕ R n , Γ =
Z ⋉ ϕ | Z Z n . In particular if g is the Lie algebra of G , then g = R ⋉ ad Xn +1 R n , where R = h X n +1 i and R n = h X , · · · , X n i , and ϕ ( t ) := e t ad Xn +1 .In this case the Mostow fibration is R n / Z n ֒ → S −→ R / Z and we want to apply to this fibration the following theorem of Felix andThomas Theorem 1. [13, 14]
Let F → E → B be a fibration and let U be the largest π ( B ) -submodule of H ∗ ( F, Q ) on which π ( B ) acts nilpotently. Suppose that H ∗ ( F, Q ) is a vector space of finite type and that B is a nilpotent space, thenin the Sullivan model of the fibration ORMALITY OF ALMOST ABELIAN SOLVMANIFOLDS 3 A ( B ) / / A ( E ) / / A ( F )( V X, d X ) / / σ O O ( V ( X ⊕ Y ) , D ) τ O O / / ( V Y, d Y ) ρ O O the cdga homomorphism ρ : ( V Y, d Y ) → A ( F ) induces an isomorphism ρ ∗ : H ∗ ( V Y, d Y ) → U . We give here only basic definitions of cdga and minimal models and werefer to [8] for a depth study of these topics.
Definition 2.
Let K ba a field of characteristic . A graded K -vector spaceis a family of K -vector spaces A = {A p } p ≥ . An element of A has degree p if it belongs to A p .A commutative differential graded K -algebra, cdga, ( A , d ) is a graded K -vector space A together with a multiplication A p ⊗ A q → A p + q that is associative, with unit ∈ A and commutative in the graded sense,i.e. ∀ a ∈ A p , b ∈ A q a · b = ( − pq b · a , and with a differential d : A p → A p +1 such that d = 0 and ∀ a ∈ A p , b ∈ A q d ( a · b ) = da · b + ( − p a · db . Given a K -cdga ( A , d ) its cohomology algebra H ∗ ( A , K ) is well definedand it is a K -cdga with d ≡ Definition 3.
A cdga homomorphism f : ( A , d A ) → ( B , d B ) is a family ofhomomorphisms f p : A p → B p such that f p + q ( a · b ) = f p ( a ) · f q ( b ) and d B f p = f p d A . Definition 4.
A cdga ( M , d ) is minimal if it is free commutative,i.e. M = V V with V graded vector space, and there exist a ordered basis { x α } of V such that V = K , dV ⊂ V ≥ V and dx α ∈ V ( x β ) β<α , wherewith V ≥ V we mean V i V with i ≥ .A minimal model of the cdga ( A , d ) is a minimal cdga ( M , d ) together witha cdga quasi isomorphism ψ : M → A , i.e. a morphism that induces anisomorphism on cohomology.
For every topological space T Sullivan defined a Q -cdga A ( T ) associatedto T . We refer to [8] for its definition, we only need to know that itscohomology is the cohomology of the space T over the constant sheaf Q ,then we can apply Theorem 1 to differential manifolds.In particular by definition of Sullivan model of a fibration [8], we havethat ORMALITY OF ALMOST ABELIAN SOLVMANIFOLDS 4 • ( V X, d X ) and ( V Y, d Y ) are minimal cdga, • σ and τ are quasi isomorphisms, • ∀ x ∈ X Dx = d X x and ∀ y ∈ Y Dy = d Y y + cx ∧ y ′ with c ∈ Q , x ∈ V X + and y ′ ∈ V Y For every α, β ∈ H ∗ ( R n ) , where R n is the n-dimensionalabelian Lie algebra, if α and β ∈ U then also α ∧ β ∈ U .Proof. Due to Proposition 1 this proof is very simple: α and β ∈ U isequivalent to ϕ s ( α ) = α and ϕ s ( β ) = β , then ϕ s ( α ∧ β ) = ϕ s ( α ) ∧ ϕ s ( β ) = α ∧ β. (cid:3) Remark 1. U is a submodule of H ∗ ( R n ) , then also in U the zero class isrepresented only by the zero form in V ∗ ( R n ) . Formality We begin stating two equivalent definitions of s -formality and formality,[6], [7], [8]: Definition 5. A cdga ( V V, d ) is s -formal if there is a cdga homomorphism ψ : V V ≤ s → H ∗ ( V V ) , such that the map ψ ∗ : H ∗ ( V V ≤ s ) → H ∗ ( V V ) induced on cohomology is equal to the map i ∗ : H ∗ ( V V ≤ s ) → H ∗ ( V V ) induced by the inclusion i : V V ≤ s → V V . Definition 6. A minimal cdga ( V V, d ) is s -formal if for every i ≤ sV i = C i ⊕ N i such that • d ( C i ) = 0 ORMALITY OF ALMOST ABELIAN SOLVMANIFOLDS 7 • d is injective on N i • ∀ n ∈ I s := V V ≤ s · N ≤ s such that dn = 0 , then n is exact in V V . We say that ( V V, d ) is formal if it is s -formal ∀ s ≥ 0, in particular thismeans Definition 7. A cdga ( V V, d ) is formal if there exists a cgda homomorphism ψ : V V → H ∗ ( V V ) that induces the identity in cohomology. Definition 8. A minimal cdga ( V V, d ) is formal if V = C ⊕ N such that • d ( C ) = 0 • d is injective on N • ∀ n ∈ I := V V · N such that dn = 0 , then n is exact in V V . We denote by ( M U , d ) the minimal cdga ( V Y, d Y ) and by ( M S , D ) theminimal model ( V ( X ⊕ Y ) , D ) of S .Proposition 2 implies the following Proposition 3. ( M U , d ) is always formal.Proof. Consider U as a vector space and define A as the subspace of U spanned by generators of U that are wedge of generators of lower degree,and B as the subspace of U spanned by generators of U that are not wedgeof generators of lower degree. Then U = A ⊕ B .Using the notation of Definition 8 we have that if Y = C ⊕ N ,then C ∼ = B as vector spaces. Then by Proposition 2, the cohomology of M U is given bythe elements of B . This means that for every b ∈ B exist c b ∈ C such that[ c b ] ∼ = b by the isomorphism ρ ∗ , then dc b = 0 and c b is not exact. Moreover,every n ∈ N is not closed.Suppose that there exists a closed element in M U which is not a generatorand that it lies in I . This means that it is a product of two elements andat least one of them is not closed. By Proposition 2 the cohomology of M U is given only by the elements of B , so this element must be also exact.Otherwise H ∗ ( M U ) ≇ U .Then by Definition 8 ( M U , d ) is formal. (cid:3) Now consider the minimal model ( M S , D ) of the solvmanifold S .By definition DA = 0 and(2) ∀ x ∈ Y Dx = h dx or dx + yA with y ∈ Λ Y A cdga A is of k -finite type if ∀ i ≤ k A i is a finite dimensionalvector space. Remark 2. Obviously M S is of k -finite type if and only if M U is of k -finitetype. We can now prove the main result: Theorem 2. If M S is of k -finite type, then S is k -formal if and only if ker D i | M U = ker d i ∀ i ≤ k , where with d i we mean d | M iU .Proof. Suppose that for some i ≤ k ker D i | M U ( ker d i , then there exists x ∈ M iU such that dx = 0, but Dx = 0. This means for (2) that Dx = yA with 0 = y ∈ M ORMALITY OF ALMOST ABELIAN SOLVMANIFOLDS 9 Symplectic structures Suppose that S = R ⋉ R n − has dimension 2 n . Recall that a symplecticform on S is ω ∈ V S such that dω = 0 and ω n = 0. We denote with { α , · · · , α n − } the basis of V R n − and with { α n } the basis of V R Definition 10. If M is a (2 n − -dimensional manifold a co-symplecticstructure on M is a couple ( F, η ) where F is a -form, η is a -form on M ,both are closed and F n − ∧ η = 0 . In particular we call a co-symplectic structure on U a co-symplectic struc-ture ( F, η ) on R n − such that [ F ] , [ η ] ∈ U . Observe that every form on R n − is closed, so the only necessary condition to get this structure is thenon-degeneracy.Let ( F, η ) be a co-symplectic structure on U . This means that F := X ≤ i We have then proved the following proposition: Proposition 4. If D | M U = d and there exists a co-symplectic structureon U , then there exists a symplectic structure on S . Examples We conclude giving two examples of computation:5.1. An example in dimension 6. Consider the almost abelian solvmani-fold S defined by the action ofad X = − with lattice generated by t = 2 π .The Lie algebra associated to this solvmanifold in [2] is called g a =06 . .According to the method developed in [3] and [9], this solvmanifold is dif-feomorphic to the 6-dimensional, almost abelian, completely solvable solv-manifold ˜ G/ Γ π with ˜ G = R ⋉ ˜ ϕ R and˜ ϕ = . Then its cohomology groups are isomorphic to those of the Lie algebra ˜ g given by [ X , X ] = X , [ X , X ] = X , [4].In particular we have H ( S ) = h α , α , α , α i H ( S ) = h α , α , α , α , α , α , α i H ( S ) = h α , α , α , α , α , α , α , α i Now we compute U : ϕ = e π ad X , then ϕ ( α ) = α + 2 πα + 2 π α ,ϕ ( α ) = α + 2 πα ,ϕ ( α ) = α ,ϕ ( α ) = α ,ϕ ( α ) = α . In this case ∀ i = 1 , · · · , α i ∈ U , then U ≡ H ∗ ( R ) and M U ≡ M U = ( ^ ( e, f, z, p, q ) , . ORMALITY OF ALMOST ABELIAN SOLVMANIFOLDS 11 Knowing the cohomology groups of the solvmanifold we can compute itsminimal model: M S = ( ^ ( A, e, f, z, p, q ) , D ) , DA = De = Df = Dz = 0 , Dp = eA, Dq = pA with the map τ : M S → V ∗ S given by τ ( A ) = α , τ ( e ) = α , τ ( f ) = α , τ ( z ) = α , τ ( p ) = α , τ ( q ) = α . Then for Theorem 2 S is not 1-formal.Now consider the symplectic forms on S .In this case the generic co-symplectic structure on U is given by F = X ≤ i Now consider the almost abelian solv-manifold S defined by the action ofad X = b − b − b 10 0 0 0 0 − − b b = 0 , e πb + e − πb ∈ Z with lattice generated by t = 2 π . ORMALITY OF ALMOST ABELIAN SOLVMANIFOLDS 12 Again this solvmanifold is diffeomorphic to the 8-dimensional, almostabelian, completely solvable solvmanifold ˜ G/ Γ π with ˜ G = R ⋉ ˜ ϕ R and˜ ϕ = b b − b 00 0 0 0 0 0 − b . Then its cohomology groups are isomorphic to those of the Lie algebra ˜ g given by [ X , X ] = X , [ X , X ] = X , [ X , X ] = bX , [ X , X ] = bX , [ X , X ] = − bX , [ X , X ] = − bX . In particular we have H ( S ) = h α , α i .Now we compute U : ϕ ( α ) = α + 2 πα + 2 π α ,ϕ ( α ) = α + 2 πα ,ϕ ( α ) = α ,ϕ ( α ) = e πb α ,ϕ ( α ) = e πb α ,ϕ ( α ) = e − πb α ,ϕ ( α ) = e − πb α , then U = h α , α , α i ,U = h α , α , α , α , α , α , α i U = h α , α , α , α , α , α , α , α , α ,α , α , α , α i U = h α , α , α , α , α , α , α , α ,α , α , α , α , α i U = h α , α , α , α , α , α , α i U = h α , α , α i U = h α i . The minimal cdga M U is quite difficult to compute, indeed the big di-mension of U implies a need of many generators in degree 2, and then manyrelations to check to get the cohomology isomorphism.Fortunately we do not need to construct all M U and M S to understand ifthe solvmanifold is formal: we can simply find out that M U = ( V ( x, y, z ) , H ( S ) = h α , α i , then M S = ( ^ ( A, x, y, z ) , D ) with DA = Dx = 0 , Dy = xA, Dz = yA and so for Theorem 2 S is not 1-formal. ORMALITY OF ALMOST ABELIAN SOLVMANIFOLDS 13 Acknowledgments. 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