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Let g be the nilpotent Lie algebra underlying the nilmanifold N = Γ G, and let ( g∗, d) be the \ Chevalley-Eilenberg complex seen as a commutative differential graded algebra (CDGA).V Hasegawa proved in [10] that this CDGA provides not only the R-minimal model of N but also its Q-minimal model. A result by Bazzoni and Mu˜noz asserts that, in six dimensions, there are infinitely many rational homotopy types of nilmanifolds, but only 34 different real homotopy types (see [3, Theorem 2]). Hence, there only exits a finite number of real homotopy types of 6-dimensional nilmanifolds admitting a generalized complex structure. In this paper we prove that the situation is completely different in eight dimensions: Theorem 1.1. There are infinitely many real homotopy types of 8-dimensional nilmanifolds admitting generalized complex structures of every type k, for 0 k 4. ≤ ≤ It is worth remarking that the existence of infinitely many real homotopy types of 8-dimensional nilmanifolds with complex structure (having special Hermitian metrics) is proved in [13]. However, such nilmanifolds do not admit any symplectic form.
This paper is structured as follows. In Section 2 we review some general results about minimal models and homotopy theory, and we define a family of nilmanifolds Nα in eight dimensions depending on a rational parameter α > 0. Section 3 is devoted to the construction of generalized complex structures on the nilmanifolds Nα. More concretely, we prove the following: + Proposition 1.2. For each α Q , the 8-dimensional nilmanifold Nα has generalized complex struc- tures of every type k, for 0 k∈ 4. ≤ ≤ In Section 4 we study the real homotopy types of the nilmanifolds in the family Nα α∈Q+ . More precisely, the result below is attained: { } ′ Proposition 1.3. If α = α , then the nilmanifolds Nα and Nα′ have non-isomorphic R-minimal models, so they have different real6 homotopy type. Note that Theorem 1.1 is a direct consequence of Propositions 1.2 and 1.3. Moreover, by taking products with even dimensional tori the result holds in any dimension 2n 8 and for every 0 k n. Furthermore, our result in Theorem 1.1 can be extended to the complex≥ homotopy setting≤ (see≤ Remark 4.4 for details). Since the nilmanifolds Nα cannot admit any K¨ahler metric [10], one has the following Corollary 1.4. Let n 4. There are infinitely many complex homotopy types of 2n-dimensional compact non-K¨ahler manifolds≥ admitting generalized complex structures of every type k, for 0 k n. ≤ ≤ 2. The family of nilmanifolds Nα Let us start recalling some general results about homotopy theory and minimal models, with special attention to the class of nilmanifolds. In [17] Sullivan shows that it is possible to associate a minimal model to any nilpotent CW-complex X, i.e. a space X whose fundamental group π1(X) is a nilpotent group that acts in a nilpotent way on the higher homotopy group πk(X) of X for every k > 1. Recall that a minimal model is a commutative differential graded algebra, CDGA for short, ( VX , d) defined over the rational numbers Q and satisfying a certain minimality condition, that encodesV the rational homotopy type of X [8]. More generally, let K be the field Q or R. A CDGA ( V, d) defined over K is said to be minimal if the following conditions hold: V (i) V is the free commutative algebra generated by the graded vector space V = V l; ⊕ V 3
(ii) there exists a basis xj j∈J , for some well-ordered index set J, such that deg(xk) deg(xj ) { } ≤ for k < j, and each dxj is expressed in terms of the preceding xk (k < j).
A K-minimal model of a differentiable manifold M is a minimal CDGA ( V, d) over K together with a quasi-isomorphism φ from ( V, d) to the K-de Rham complex of M, i.e.V a morphism φ inducing an isomorphism in cohomology. Here,V the K-de Rham complex of M is the usual de Rham complex of differential forms (Ω∗(M), d) when K = R, whereas for K = Q one considers Q-polynomial forms instead. Notice that the K-minimal model is unique up to isomorphism, since char (K) = 0. By [5, 17], two nilpotent manifolds M1 and M2 have the same K-homotopy type if and only if their K-minimal models are isomorphic. It is clear that if M1 and M2 have different real homotopy type, then M1 and M2 also have different rational homotopy type. Let N be a nilmanifold, i.e. N = Γ G is a compact quotient of a connected and simply connected \ nilpotent Lie group G by a lattice Γ of maximal rank. For any nilmanifold N one has π1(N) = Γ, which is nilpotent, and πk(N) = 0 for every k 2. Therefore, nilmanifolds are nilpotent spaces. Let n be the dimension of the nilmanifold≥ N = Γ G, and let g be the Lie algebra of G. It is well known that the minimal model of N is given by the Chevalley-Eilenberg\ complex ( g∗, d) of g. Recall that by [14] the existence of a lattice Γ of maximal rank in G is equivalent to the nilpotentV Lie algebra g being rational, i.e. there exists a basis e1,...,en for the dual g∗ such that the structure constants are rational numbers. Thus, the rational and{ the nilpotency} conditions of the Lie algebra g allow to take a basis e1,...,en for g∗ satisfying { } (4) de1 = de2 =0, dej = aj ei ek for j =3, . . . , n, ik ∧ i,k