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On Low-Dimensional Solvmanifolds

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On Low-Dimensional Solvmanifolds On Low-Dimensional Solvmanifolds Christoph Bock MSC 2000: Primary 53C30, 57T15; Secondary 57R17 1 Introduction In this note we want to study compact homogeneous spaces G/Γ, where G is a connected and simply-connected Lie group and Γ a discrete subgroup in G. It is well known that the existence of such a Γ implies the unimodularity of the Lie group G. Recall that a Lie group G is called unimodular if for all X g holds trad X = 0, where g denotes the Lie algebra of G. ∈ If we further demand G/Γ to be symplectic (when G is even-dimensional), a result of Chu [11] shows that G has to be solvable. Therefore, we regard compact quotients of connected and simply-connected solvable Lie groups by discrete subgroups, so called solvmanifolds. First, we recall the definition of nilpotent and solvable groups resp. Lie alge- bras. (i) Let G be group and denote its neutral element by e. We define the de- (k) (k) rived series (D G)k N, descending series (G )k N and ascending series ∈ ∈ (G(k))k N of subgroups in G inductively as follows: ∈ D(0)G := G(0) := G, arXiv:0903.2926v4 [math.DG] 31 Aug 2009 (k) (k 1) (k 1) (k) (k 1) D G := [D − G, D − G], G := [G,G − ], G(0) = e , G(k) := g G [g,G] G(k 1) . { } { ∈ | ⊂ − } (k0) G is called nilpotent if there exists k0 N such that G = e . ∈ { } (k0) G is called solvable if there exists k0 N such that D G = e . ∈ { } (ii) Given a Lie algebra g, one defines the derived, descending and ascending series of subalgebras in g via D(0)g := g(0) := g, (k) (k 1) (k 1) (k) (k 1) D g := [D − g,D − g], g := [g, g − ], g(0) = 0 , g(k) := X g [X, g] g(k 1) { } { ∈ | ⊂ − } 1 and calls g nilpotent resp. solvable if its ascending resp. derived series be- comes trivial for k0 large enough. We collect some properties in the following proposition. The parts which are not obvious can be found in [60, Section 3.18]. Proposition 1.0.1. (i) The subgroups arising in the derived, descending and ascending series of a group are normal. Moreover, they are closed and simply-connected Lie subgroups in the case of a connected and simply-connected Lie group. (ii) The subalgebras arising in the derived, descending and ascending series of a Lie algebra are ideals. (iii) A Lie group is nilpotent resp. solvable if and only if its Lie algebra is nilpo- tent resp. solvable. We shall examine whether certain solvmanifolds are formal, symplectic, K¨ahler or Lefschetz. Therefore, we give a brief introduction of these notions. 1.1 Formality i A differential graded algebra (DGA) is a graded R-algebra A = i N A together with an R-linear map d: A A such that d(Ai) Ai+1 and∈ the following conditions are satisfied: → ⊂ L .i) The R-algebra structure of A is given by an inclusion R ֒ A0) → (ii) The multiplication is graded commutative, i.e. for a Ai and b Aj one i j i+j ∈ ∈ has a b =( 1) · b a A . · − · ∈ i (iii) The Leibniz rule holds: a Ai b A d(a b)= d(a) b +( 1) a d(b) ∀ ∈ ∀ ∈ · · − · (iv) The map d is a differential, i.e. d2 = 0. Further, we define a := i for a Ai. The i-th cohomology| | of a DGA∈ (A, d) is the algebra ker(d: Ai Ai+1) Hi(A, d) := → . im(d: Ai 1 Ai) − → If (B,dB) is another DGA, then a R-linear map f : A B is called mor- → phism if f(Ai) Bi, f is multiplicative, and d f = f d . Obviously, any ⊂ B ◦ ◦ A such f induces a homomorphism f ∗ : H∗(A, dA) H∗(B,dB). A morphism of differential graded algebras inducing an isomorphism→ on cohomology is called quasi-isomorphism. 2 Definition 1.1.1. A DGA ( ,d) is said to be minimal if M i (i) there is a graded vector space V = i N+ V = Span ak k I with ∈ { | ∈ } homogeneous elements ak, which we call the generators, L (ii) = V , M (iii) the indexV set I is well ordered, such that k<l a a and the ⇒ | k|≤| l| expression for dak contains only generators al with l<k. We shall say that ( ,d) is a minimal model for a differential graded al- M gebra (A, d ) if ( ,d) is minimal and there is a quasi-isomorphism of DGAs A M ρ: ( ,d) (A, dA), i.e. it induces an isomorphism ρ∗ : H∗( ,d) H∗(A, dA) on cohomology.M → M → The importance of minimal models is reflected by the following theorem, which is taken from Sullivan’s work [56, Section 5]. 0 Theorem 1.1.2. A differential graded algebra (A, dA) with H (A, dA) = R pos- sesses a minimal model. It is unique up to isomorphism of differential graded algebras. We quote the existence-part of Sullivan’s proof, which gives an explicit con- struction of the minimal model. Whenever we are going to construct such a model for a given algebra in this article, we will do it as we do it in this proof. Proof of the existence. We need the following algebraic operations to “add” resp. “kill” cohomology. Let ( ,d) be a DGA. We “add” cohomology by choosing a new generator x and settingM := (x), d˜ = d, d˜(x)=0, M M ⊗ |M ^ k and “kill” a cohomologyf class [z] H ( ,d) by choosing a new generator y of degree k 1 and setting ∈ M − := (y), d˜ = d, d˜(y)= z. M M ⊗ |M ^ Note that z is a polynomialf in the generators of . 0 M Now, let (A, dA) a DGA with H (A, dA)= R. We set 0 := R, d0 := 0 and M ρ0(x)= x. Suppose now ρk : ( k,dk) (A, dA) has been constructed so that ρk induces isomorphisms on cohomologyM → in degrees k and a monomorphism in degree ≤ (k + 1). “Add” cohomology in degree (k + 1) to get morphism of differential graded algebras ρ : ( ,d ) (A, d ) which induces an isomorphism (k+1),0 M(k+1),0 (k+1),0 → A ρ∗ on cohomology in degrees (k + 1). Now, we want to make the induced (k+1),0 ≤ map ρ(∗k+1),0 injective on cohomology in degree (k + 2) . 3 We “kill” the kernel on cohomology in degree (k+2) (by non-closed generators of degree (k+1)) and define ρ : ( ,d ) (A, d ) accordingly. (k+1),1 M(k+1),1 (k+1),1 → A If there are generators of degree one in ( (k+1),0,d(k+1),0) it is possible that this killing process generates new kernel on cohomologyM in degree (k + 2). Therefore, we may have to “kill” the kernel in degree (k + 2) repeatedly. We end up with a morphism ρ(k+1), : ( (k+1), ,d(k+1), ) (A, dA) which ∞ ∞ ∞ induces isomorphisms on cohomology in degreesM (k + 1) and a→ monomorphism ≤ in degree (k +2). Set ρk+1 := ρ(k+1), and ( k+1,dk+1) := ( (k+1), ,d(k+1), ). ∞ M M ∞ ∞ Inductively we get the minimal model ρ: ( ,d) (A, d ). M → A A minimal model ( ,d) of a connected smooth manifold M is a minimal MM model for the de Rahm complex (Ω(M),d) of differential forms on M. The last theorem implies that every connected smooth manifold possesses a minimal model which is unique up to isomorphism of differential graded algebras. Now, we are able to introduce the notion of formality. Endowed with the triv- ial differential, the cohomology of a minimal DGA is a DGA itself, and therefore it also possesses a minimal model. In general, this two minimal models need not to be isomorphic. A minimal differential graded algebra ( ,d) is called formal if there is a morphism of differential graded algebras M ψ : ( ,d) (H∗( ,d),d = 0) M → M H that induces the identity on cohomology. This means that ( ,d) and (H∗( ,d),d = 0) share their minimal model. M M H The following theorem gives an equivalent characterisation. Theorem 1.1.3 ([49, Theorem 1.3.1]). A minimal model ( ,d) is formal if and only if we can write = V and the space V decomposesM as a direct sum M V = C N with d(C)=0, d is injective on N, and such that every closed element in the ideal⊕ I(N) generated by NV in V is exact. This allows us to give a weaker versionV of the notion of formality. Definition 1.1.4. A minimal model ( ,d) is called s-formal, s N, if we can write = V and for each i s Mthe space V i generated by∈ generators of M ≤ degree i decomposes as a direct sum V i = Ci N i with d(Ci) = 0, d is injective i V ⊕ i on N and such that every closed element in the ideal I( i s N ) generated by i i ≤ i s N in i s V is exact in V . ≤ ≤ L L Obviously,V formalityL implies s-formalityV for every s. The following theorem is an immediate consequence of the last definition. 4 Theorem 1.1.5. Let ( ,d) be a minimal model, where = V , V = C N with d(C)=0 and d isM injective on N. M ⊕ r V i Assume that there exist r, s N+, n N and x i s V such that holds ∈ ∈ ∈ ≤ V L c Cr (n + c) x is closed and not exact.
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