Simply connected K-contact and Sasakian manifolds of dimension 7

Abstract

We construct a compact simply connected 7-dimensional manifold admitting a K-contact structure but not a Sasakian structure. We also study rational homotopy properties of such manifolds, proving in particular that a simply connected 7-dimensional Sasakian manifold has vanishing cup product $H^2 \times H^2 \rightarrow H^4$ and that it is formal if and only if all its triple Massey products vanish.

1 Introduction

Sasakian geometry has become an important and active subject, especially after the appearance of the fundamental treatise of Boyer and Galicki [3]. Chapter 7 of this book contains an extended discussion of the topological problems in the theory of Sasakian, and, more generally, K-contact manifolds. These are odd-dimensional analogues to Kähler and symplectic manifolds, respectively.

The precise definition is as follows. Let $(M,\eta )$ be a co-oriented contact manifold with a contact form $\eta \in \Omega ^1(M)$, that is $\eta \wedge (d\eta )^n>0$ everywhere, with $\dim M=2n+1$. We say that $(M,\eta )$ is K-contact if there is an endomorphism $\Phi$ of TM such that:

• $\Phi ^2=-{{\mathrm{Id}}}+ \xi \otimes \eta$, where $\xi$ is the Reeb vector field of $\eta$ (that is $i_\xi \eta =1$, $i_\xi (d\eta )=0$),

• the contact form $\eta$ is compatible with $\Phi$ in the sense that $d\eta (\Phi X,\Phi Y)\,=\,d\eta (X,Y)$, for all vector fields X, Y,

• $d\eta (\Phi X,X)>0$ for all nonzero $X\in \ker \eta$, and

• the Reeb field $\xi$ is Killing with respect to the Riemannian metric defined by the formula $g(X,Y)\,=\,d\eta (\Phi X,Y)+\eta (X)\eta (Y)$.

In other words, the endomorphism $\Phi$ defines a complex structure on ${\mathcal {D}}=\ker \eta$ compatible with $d\eta$, hence $\Phi$ is orthogonal with respect to the metric $g|_{\mathcal {D}}$. By definition, the Reeb vector field $\xi$ is orthogonal to $\ker \eta$, and it is a Killing vector field.

Let $(M,\eta ,g,\Phi )$ be a K-contact manifold. Consider the contact cone as the Riemannian manifold $C(M)=(M\times {\mathbb {R}}^{>0},t^2g+dt^2)$. One defines the almost complex structure I on C(M) by:

• $I(X)=\Phi (X)$ on $\ker \eta$,

• $I(\xi )=t{\partial \over \partial t},\,I\left( t{\partial \over \partial t}\right) =-\xi$, for the Killing vector field $\xi$ of $\eta$.

We say that $(M,\eta ,\Phi ,g,I)$ is Sasakian if I is integrable. Thus, by definition, any Sasakian manifold is K-contact.

There are several topological obstructions to the existence of the aforementioned structures on a compact manifold M of dimension $2n+1$, for example:

• (1) the evenness of the pth Betti number for p odd with $1\, \le \, p \, \le \, n$, of a Sasakian manifold,

• (2) some torsion obstructions in dimension 5 discovered by Kollár [17],

• (3) the fundamental group of Sasakian manifolds are special,

• (4) the cohomology algebra of a Sasakian manifold satisfies the hard Lefschetz property,

• (5) formality properties of the rational homotopy type.

An early result [13] establishes that the odd Betti numbers up to the middle dimension of Sasakian manifolds must be even. The parity of $b_1$ was used to produce the first examples of K-contact manifolds with no Sasakian structure [3, example 7.4.16]. More refined tools are needed in the case of even Betti numbers. The cohomology algebra of a Sasakian manifold satisfies a hard Lefschetz property [4]. Using it examples of K-contact non-Sasakian manifolds are produced in [5] in dimensions 5 and 7. These examples are nilmanifolds with even Betti numbers, so in particular they are not simply connected.

The fundamental group can also be used to construct K-contact non-Sasakian manifolds. Fundamental groups of Sasakian manifolds are called Sasaki groups, and satisfy strong restrictions. Using this it is possible to construct (non-simply connected) compact manifolds which are K-contact but not Sasakian [8].

When one moves to the case of simply connected manifolds, K-contact non-Sasakian examples of any dimension $\ge$9 were constructed in [16] using the evenness of the third Betti number of a compact Sasakian manifold. Alternatively, using the hard Lefschetz property for Sasakian manifolds there are examples [19] of simply connected K-contact non-Sasakian manifolds of any dimension $\ge$9.

In [24] and in [2] the rational homotopy type of Sasakian manifolds is studied. In [2] it is proved that all higher order Massey products for simply connected Sasakian manifolds vanish, although there are Sasakian manifolds with non-vanishing triple Massey products. This yields examples of simply connected K-contact non-Sasakian manifolds in dimensions $\ge$17. However, Massey products are not suitable for the analysis of lower dimensional manifolds.

Hence, the problem of the existence of simply connected K-contact non-Sasakian compact manifolds (open problem 7.4.1 in [3]) is still open in dimensions 5 and 7. Dimension 5 is the most difficult one, and it is treated in [3] separately. Here one has to use the obstructions of [17] which are very subtle torsion obstructions associated to the classification of Kähler surfaces. By definition, a simply connected compact oriented 5-manifold is called a Smale–Barden manifold. These manifolds are classified topologically by $H_2(M,{\mathbb {Z}})$ and the second Stiefel–Whitney class. Chapter 10 of the book by Boyer and Galicki is devoted to a description of some Smale–Barden manifolds which carry Sasakian structures. The following problem is still open (open problem 10.2.1 in [3]).

Do there exist Smale–Barden manifolds which carry K-contact but do not carry Sasakian structures?

In this note we solve the described problem in the easier case of dimension 7 (the solution is still possible by means of homotopy theory combined with symplectic surgery).

Theorem 1

There exist 7-dimensional compact simply connected K-contact manifolds which do not admit a Sasakian structure.

We then turn around to the study of the rational homotopy type of K-contact and Sasakian simply connected manifolds of dimension 7. In particular, we prove:

Corollary 2

Let M be a simply connected compact K-contact 7-dimensional manifold. Suppose that the cup product map $H^2(M)\times H^2(M)\longrightarrow H^4(M)$ is non-zero. Then M does not admit a Sasakian structure.

Formality is a very useful rational homotopy property that has been widely used to distinguish between symplectic and Kähler manifolds [21] (see Sect. 6 for definitions and details). Simply connected compact manifolds of dimension $\le$6 are always formal, so formality becomes interesting in dimension 7. We study this property in detail giving a precise characterisation for Sasakian manifolds (see Theorem 15). In particular, we have the following:

Corollary 3

Let M be a simply connected compact Sasakian 7-dimensional manifold. Then M is formal if and only if all triple Massey products are zero.

2 Gompf–Cavalcanti manifold

Let $(M,\omega )$ be a symplectic manifold of dimension 2n. For every $0\le k \le n$, we define the Lefschetz map as $L_\omega :H^{n-k}(M) \rightarrow H^{n+k}(M)$, $L_\omega ([\beta ])=[\beta \wedge \omega ^{n-k}]$. We say that M satisfies the hard Lefschetz property if $L_\omega$ is an isomorphism for every $0 \le k \le n$.

Proposition 4

There exists a simply connected 6-dimensional symplectic manifold $(M,\omega )$ such that $\dim \ker \,(L_\omega : H^2(M)\rightarrow H^4(M))$ is odd.

Proof

Gompf constructs in [14, Theorem 7.1] an example of a simply connected 6-dimensional symplectic manifold $(M,\omega )$ which does not satisfy the hard Lefschetz property, that is, the Lefschetz map $L_\omega : H^2(M)\rightarrow H^4(M)$ is not an isomorphism. If $\dim \ker L_\omega$ is already odd then we have finished.

So let us suppose that $\dim \ker L_\omega$ is even. Take a cohomology class $a\in H^2(M)$ which belongs to the kernel of $L_\omega$. In [7, Lemma 2.4] Cavalcanti proves that given a symplectic manifold $(M,\omega )$ as above satisfying that there exists a symplectic surface $S\hookrightarrow M$ with $\langle a,[S]\rangle \ne 0$, then there is another 6-dimensional symplectic manifold $(M',\omega ')$ (the symplectic blow-up of M along S) satisfying

$$\begin{aligned} \dim \ker \left( L_{\omega '}: H^2\left( M'\right) \rightarrow H^2\left( M'\right) \right) = \dim \ker \left( L_\omega : H^2\left( M\right) \rightarrow H^2\left( M\right) \right) -1. \end{aligned}$$

The symplectic blow-up of M along S is constructed in [20], where it is proved that the fundamental groups $\pi _1(M')\cong \pi _1(M)$, hence $M'$ is simply connected. This means that the simply connected 6-dimensional symplectic manifold $M'$ satisfies that $\dim \ker (L_{\omega '} : H^2(M')\rightarrow H^4(M'))$ is odd, as required.

It remains to find $S\hookrightarrow M$ as required. The cohomology class a is non-zero, so there is some $b\in H^4(M,{\mathbb {Z}})$ such that $a\cup b\ne 0$. It is easy to see that there is a rank 2 complex vector bundle $E\rightarrow M$ with $c_1(E)=0,\,c_2(E)=2b$. This corresponds to the fact that the map $[M,B{\text {SU}}(2)]\rightarrow H^4(M,{\mathbb {Z}})$ given by the second Chern class exhausts $2\, H^4(M,{\mathbb {Z}})$. A short proof runs as follows: $B{\text {SU}}(2)$ has trivial 3-skeleton and it has $\pi _4(B{\text {SU}}(2))={\mathbb {Z}}$ and $\pi _5(B{\text {SU}}(2))={\mathbb {Z}}_2$. Represent the cohomology class b by a cocycle $\varphi _b:C_4(M)\rightarrow {\mathbb {Z}}$, where $C_4(M)$ is the space of cellular chains. Given b, we define $f:M\rightarrow B{\text {SU}}(2)$ inductively on the skeleta (in what follows we denote by X[k] the k-skeleton of a space X). It is trivial on the 3-skeleton of M. For every 4-cell c, we define $f:c\rightarrow B{\text {SU}}(2)[4]=S^4$ to have degree $\varphi _b(c)\in {\mathbb {Z}}$. As M is simply connected there are no 5-cells, so it only remains to attach the 6-cell $c_6$ to the 4-skeleton M[4]. The attaching map is given by some $g:S^5\rightarrow M[4]$. When composed with f, we have a map $f\circ g: S^5 \rightarrow B{\text {SU}}(2)$, which gives an obstruction element $o_f\in \pi _5(B{\text {SU}}(2))={\mathbb {Z}}_2$. If we multiply b by two, then the map $\varphi _b$ gets multiplied by 2. The corresponding f is given by composing f with a double cover of $S^4$, hence the obstruction element is $2o_f=0$. This means that the map f associated to 2b can be extended to $M\rightarrow B{\text {SU}}(2)$.

Now take the rank 2 bundle $E \rightarrow M$ just constructed. Assume that $[\omega ]$ is a an integral cohomology class (which can always be done by perturbing $\omega$ slightly to make it rational and multiplying it by a large integer). Let $L\rightarrow M$ be the line bundle with first Chern class $c_1(L)=[\omega ]$. We now use the asymptotically holomorphic techniques introduced by Donaldson [10]. Specifically, the result of [1] guarantees the existence of a suitable large $k\gg 0$ and a section of $E\otimes L^{\otimes k}$ whose zero locus is a symplectic manifold (an asymptotically holomorphic manifold in fact). This zero locus $S\subset M$ is a symplectic surface, and the cohomology class defined by S is $c_2(E\otimes L^{\otimes k})= c_2(E)+2k c_1(L)=2b+2k[\omega ]$. Therefore $\langle a,[S]\rangle =\langle a, 2b+2k[\omega ] \rangle =2\langle a,b\rangle \ne 0$, as required. $\square$

We will call the manifold produced in Proposition 4 the Gompf–Cavalcanti manifold, because it is constructed by the surgery technique of Gompf [14] together with the symplectic blow-up of Cavalcanti [7]. Note however that this is not a unique one but a family of manifolds.

3 Simply-connected K-contact non-Sasakian manifolds in dimension 7

We show the existence of simply connected compact K-contact non-Sasakian manifolds in dimension 7 by proving that the Boothby–Wang fibration over the Gompf–Cavalcanti manifold is K-contact but non-Sasakian. The existence of a K-contact structure on such fibration is shown in [2] and [16]. For the convenience of the reader we briefly recall these constructions.

Let $(B,\omega )$ be a symplectic manifold such that the cohomology class $[\omega ]$ is integral. Consider the principal $S^1$-bundle $\pi : M\rightarrow B$ given by the cohomology class $[\omega ]\,\in \, H^2(B,\,{\mathbb {Z}})$. Fibrations of this kind were first considered by Boothby and Wang and are called Boothby–Wang fibrations. By [25], the total space M carries an $S^1$-invariant contact form $\eta$ such that $\eta$ is a connection form whose curvature is $d\eta =\pi ^*\omega$. We have the following result (which is known, compare Theorem 6.1.26 and Proposition 7.1.2 in [3]).

Theorem 5

Any Boothby–Wang fibration admits a K-contact structure on the total space.

Proof

To prove this theorem we need to introduce a certain tool, called the universal contact moment map in the sense of Lerman [18]. Recall that by our assumption the given contact distribution ${\mathcal {D}}$ is determined by the contact form $\eta$, that is ${\mathcal {D}}=\ker \eta$. Consider its annihilator ${\mathcal {D}}^0\subset T^*M$. Clearly, ${\mathcal {D}}^0$ is a line bundle, and, therefore, it has two components after the removal of the zero section,

$$\begin{aligned} {\mathcal {D}}^0{\setminus } M={\mathcal {D}}^0_+\sqcup {\mathcal {D}}^0_{-}. \end{aligned}$$

Single out one of these components, say ${\mathcal {D}}^0_{+}$. Consider the Lie algebra of contact vector fields $\chi (M,\eta )$ on M. It is known that this Lie algebra can be identified with a space of sections of the vector bundle $TM/{\mathcal {D}}$, that is $\chi (M,\eta )\cong \Gamma (M,TM/{\mathcal {D}})$. Because of that there is a natural pairing between points of the line bundle ${\mathcal {D}}^0$ and contact vector fields given by the formula

$$\begin{aligned} {\mathcal {D}}^0\times \chi (M,\eta )\rightarrow {\mathbb {R}},\quad ((p,\beta ),X)\mapsto \langle \beta ,X_p\rangle \end{aligned}$$

where $\beta \in {\mathcal {D}}^0,X_p\in T_pM, p\in M$. Suppose that a Lie algebra ${\mathfrak {g}}$ acts on M by contact vector fields, that is, there exists a representation $\rho :{\mathfrak {g}}\rightarrow \chi (M,\eta )$. Define the universal moment map as the map

$$\begin{aligned} \psi :{\mathcal {D}}_{+}^0\rightarrow {\mathfrak {g}}^* \end{aligned}$$

by the formula

$$\begin{aligned} \langle \psi (p,\beta ),X\rangle =\langle (p,\beta ),\rho (X)\rangle =\langle \beta ,\rho (X)_p\rangle , \end{aligned}$$

where $(p,\beta )\in ({\mathcal {D}}_{+}^0)_p\subset T^*_pM,X\in {\mathfrak {g}}$. Now the proof becomes a consequence of the following criterion proved by Lerman [18].

Proposition 6

A compact co-orientable contact manifold $(M,\eta )$ admits a K-contact metric g if and only if there exists an action of a torus T on M preserving the contact structure ${\mathcal {D}}$ and a vector $X\in {\mathfrak {t}}=L(T)$ so that the function $\langle \psi ,X\rangle :{\mathcal {D}}^0_{+}\rightarrow {\mathbb {R}}$ is strictly positive. $\square$

We continue with the proof of Theorem 5. Consider the $S^1$-action on M given by the Reeb vector field. Let ${\mathfrak {g}}=L(S^1)$, and $\rho :{\mathfrak {g}}\rightarrow \chi (M,\eta )$ be the homomorphism of Lie algebras determined by this action (thus, ${\mathfrak {g}}={\mathfrak {t}}=L(S^1)$ in this particular situation). Since the $S^1$-action is free, $\rho (X)_p\not =0$ for any $p\in M$. Now,

$$\begin{aligned} \langle \psi ,X\rangle (p,\beta )=\langle \psi (p,\beta ),X\rangle =\langle \beta ,\rho (X)_p\rangle . \end{aligned}$$

Note that in the considered case $\beta \in ({\mathcal {D}}^0_{+})_p\subset T_p^*M$, and, therefore, $\beta \not =0$. Also $(p,\beta )$ belongs to the annihilator of the distribution ${\mathcal {D}}$, while $\rho (X)$ is transversal to ${\mathcal {D}}$, since it is given by the Reeb vector field. Thus, for any point p, $\langle (p,\beta ),\rho (X)_p\rangle \not =0$. Hence, X may be chosen to yield positive sign everywhere, and we complete the proof by applying Proposition 6. $\square$

Remark 7

Proposition 7.1.2 from [3] is due to Rukimbira. In this work we give a different proof based on Lerman’s criterion given by Proposition 6.

The following gives a proof of Theorem 1.

Theorem 8

The total space of the Boothby–Wang fibration over the Gompf–Cavalcanti manifold is a simply connected K-contact non-Sasakian manifold of dimension 7.

Proof

Let $(M,\omega )$ be a Gompf–Cavalcanti manifold as given by Proposition 4. We can assume that $[\omega ]$ is an integral cohomology class. Let

$$\begin{aligned} S^1\rightarrow E\rightarrow M \end{aligned}$$

(1)

be the associated Boothby–Wang fibration. By Theorem 5, E has a K-contact structure. Now we need to prove that E cannot carry Sasakian structures.

There is an exact sequence

$$\begin{aligned} H_2(M)\rightarrow H_1\left( S^1\right) ={{\mathbb {Z}}}\rightarrow H_1(E) \rightarrow 0 \end{aligned}$$

from the Serre spectral sequence. The map $H_2(M)\rightarrow {{\mathbb {Z}}}$ is cupping with $[\omega ]\in H^2(M)$. Taking $[\omega ]$ integral cohomology class and primitive, we have that $H_2(M)\rightarrow {{\mathbb {Z}}}$ is surjective and hence $H_1(E)=0$. The long homotopy exact sequence gives $\pi _1(S^1)={{\mathbb {Z}}} \rightarrow \pi _1(E) \rightarrow \pi _1(M)=0$, hence $\pi _1(E)$ is abelian. Therefore E is simply connected.

The Gysin exact sequence associated to (1) is

$$\begin{aligned} H^1(M)=0 \mathop {\longrightarrow }\limits ^{\wedge \omega } H^3(M) \longrightarrow H^3(E) \longrightarrow H^2(M) \mathop {\longrightarrow }\limits ^{\wedge \omega } H^4(M). \end{aligned}$$

Thus

$$\begin{aligned} b^3(E)=b^3(M)+ \dim \left( \ker L_\omega :H^2(M) \rightarrow H^4(M)\right) . \end{aligned}$$

As M is a 6-manifold, we have that $b^3(M)$ is even (by Poincaré duality, the intersection pairing on $H^3(M)$ is an antisymmetric non-degenerate bilinear form, hence the dimension of $H^3(M)$ is even). By construction, $\dim (\ker L_\omega :H^2(M) \rightarrow H^4(M))$ is odd, so $b^3(E)$ is odd. As the third Betti number of a 7-dimensional Sasakian manifold has to be even [13], we have that E cannot admit a Sasakian structure. $\square$

4 Regularity and quasi-regularity

A Sasakian or a K-contact structure on a compact manifold M is called quasi-regular if there is a positive integer $\delta$ satisfying the condition that each point of M has a foliated coordinate chart (U, t) with respect to $\xi$ (the coordinate t is in the direction of $\xi$) such that each leaf for $\xi$ passes through U at most $\delta$ times. If $\delta \,=\, 1$, then the Sasakian or K-contact structure is called regular (see [3, p. 188]).

If N is a Kähler manifold whose Kähler form $\omega$ defines an integral cohomology class, then the total space of the circle bundle $S^1 \hookrightarrow M \mathop {\longrightarrow }\limits ^{\pi } N$ with Euler class $[\omega ]\in H^2(M,{\mathbb {Z}})$ is a regular Sasakian manifold with contact form $\eta$ such that $d \eta \,=\, \pi ^*(\omega )$. The converse also holds: if M is a regular Sasakian structure then the space of leaves N is a Kähler manifold, and we have a circle bundle $S^1\rightarrow M \rightarrow N$ as above. If M has a quasi-regular Sasakian structure, then the space of leaves N is a Kähler orbifold with cyclic quotient singularities, and there is an orbifold circle bundle $S^1 \rightarrow M\rightarrow N$ such that the contact form $\eta$ satisfies $d\eta =\pi ^*(\omega )$, where $\omega$ is the orbifold Kähler form.

Similar properties hold in the K-contact case, substituting Kähler by symplectic (actually almost Kähler). If M has a regular K-contact structure, then it is the total space of a circle bundle $S^1 \hookrightarrow M \mathop {\longrightarrow }\limits ^{\pi } N$, where $(N,\omega )$ is a symplectic manifold, with Euler class $[\omega ]\in H^2(M,{\mathbb {Z}})$ and $d \eta \,=\, \pi ^*(\omega )$. If M has a quasi-regular K-contact structure, then it is the total space of an orbifold circle bundle $S^1 \rightarrow M\rightarrow N$ over a symplectic orbifold N with cyclic quotient singularities and Euler class $[\omega ]\in H^2(M,{\mathbb {Z}})$, where $\omega$ is the orbifold symplectic form.

A result of [22] says that if M admits a Sasakian structure, then it admits also a quasi-regular Sasakian structure. This also extends to the case of K-contact structures (Rukimbira [23], see also Theorem 7.1.10 in [3]).

Proposition 9

If a compact manifold M admits a K-contact structure, it admits a quasi-regular contact structure.

Proof

Assume that there is a K-contact structure on M. By Proposition 6, there exists a torus action $T\times M\rightarrow M$ preserving the contact distribution and a vector $X\in {\mathfrak {t}}$ such that $\langle \psi ,X\rangle >0$. Choose a vector $Y\in {\mathfrak {t}}$ with the property that it is tangent to an embedding $T'=S^1\hookrightarrow T$. Clearly, the corresponding fundamental vector field $Y_M$ has the property that the leaves of the corresponding foliations are compact. The set of such Y is dense in ${\mathfrak {t}}$. Therefore, for vectors Y which are sufficiently close to X, the condition $\langle \psi , Y\rangle >0$ is still satisfied.

So it remains to see that there is K-contact structure whose Reeb vector field is $Y_M$, since this will be quasi-regular because the leaves of the characteristic foliation are all compact. We follow the notations of the proof of Theorem 5. The action of the circle $T'$ on M preserves ${\mathcal {D}}$, hence the lifted action of $T'$ on $T^*M$ preserves ${\mathcal {D}}^0$. Since $T'$ is connected, the lifted action preserves the connected component ${\mathcal {D}}^0_+$ as well. It follows that for any 1-form $\beta$ on M with $\ker \beta ={\mathcal {D}}$, the average $\bar{\beta }$ of $\beta$ over $T'$ still satisfies $\ker \,\bar{\beta }={\mathcal {D}}$. So $\bar{\beta }\in {\mathcal {D}}^0$. Now use the formula [derived in [18], formulae (3.4) and (3.5)],

$$\begin{aligned} i_{Y_M}\bar{\beta }=\left\langle \psi \circ \bar{\beta },Y\right\rangle >0. \end{aligned}$$

Now let

$$\begin{aligned} \eta =\left( \langle \psi \circ \bar{\beta },Y\rangle \right) ^{-1}\bar{\beta }, \end{aligned}$$

which satisfies $i_{Y_M}\eta =1$. Hence $\eta$ defines the contact structure and $Y_M$ is its Killing vector field. Then $TM={\mathcal {D}}\oplus \langle Y_M\rangle$, and the splitting is $T'$-invariant. We use the splitting to define the desired Riemannian metric g. Declare ${\mathcal {D}}$ and $\langle Y_M\rangle$ to be orthogonal and define $g(Y_M,Y_M)=1$, thus $Y_M$ becomes a unit normal to ${\mathcal {D}}$. On ${\mathcal {D}}$ we choose a $T'$-invariant complex structure compatible with $d\eta |_{{\mathcal {D}}}$ and define $g|_{{\mathcal {D}}}(\cdot ,\cdot )=d\eta |_{{\mathcal {D}}}(\cdot ,\Phi \cdot )$. Then g is $T'$-invariant and hence $L_{Y_M}g=0$. Thus we have obtained a K-contact structure on M. $\square$

5 Minimal models and formality

Now we want to analyse the rational homotopy type of K-contact and Sasakian simply connected 7-manifolds, in particular the property of formality. Simply connected compact manifolds of dimension $\le$6 are always formal [12], so dimension 7 is the first instance in which formality is an issue.

We start by reviewing concepts about minimal models and formality from [11, 12, 15]. A differential graded algebra (or DGA) over the real numbers ${\mathbb {R}}$, is a pair (A, d) consisting of a graded commutative algebra $A=\oplus _{k\ge 0} A^k$ over ${\mathbb {R}}$, and a differential d satisfying the Leibnitz rule $d(a\cdot b) = (da)\cdot b +(-1)^{|a|} a\cdot (db)$, where |a| is the degree of a. Given a differential graded commutative algebra $({A},\,d)$, we denote its cohomology by $H^*({A})$. The cohomology of a differential graded algebra $H^*({A})$ is naturally a DGA with the product inherited from that on A and with the differential being identically zero. The DGA $({A},\,d)$ is connected if $H^0({A})\,=\,{\mathbb {R}}$, and A is 1-connected if, in addition, $H^1({A})\,=\,0$. Henceforth we shall assume that all our DGAs are connected. In our context, the main example of DGA is the de Rham complex $(\Omega ^*(M),\,d)$ of a connected differentiable manifold M, where d is the exterior differential.

Morphisms between DGAs are required to preserve the degree and to commute with the differential. A morphism $f:(A,d)\rightarrow (B,d)$ is a quasi-isomorphism if the map induced in cohomology $f^*:H^*(A,d)\rightarrow H^*(B,d)$ is an isomorphism. Quasi-isomorphism produces an equivalence relation in the category of DGAs.

A DGA $(\mathcal {M},\,d)$ is minimal if

• (1) $\mathcal {M}$ is free as an algebra, that is, $\mathcal {M}$ is the free algebra $\bigwedge V$ over a graded vector space $V\,=\,\bigoplus _i V^i$, and

• (2) there is a collection of generators $\{x_\tau \}_{\tau \in I}$ indexed by some well ordered set I, such that $|x_\mu |\,\le \, |x_\tau |$ if $\mu \,< \,\tau$ and each $d x_\tau$ is expressed in terms of preceding $x_\mu$, $\mu \,<\,\tau$.

We say that $(\bigwedge V,\,d)$ is a minimal model of the differential graded commutative algebra $({A},\,d)$ if $(\bigwedge V,\,d)$ is minimal and there exists a quasi-isomorphism $\rho :{(\bigwedge V,\,d)}\longrightarrow {({A},\,d)}$. A connected DGA $({A},\,d)$ has a minimal model unique up to isomorphism. For 1-connected DGAs, this is proved in [9]. In this case, the minimal model satisfies that $V^1=0$ and the condition (2) above is equivalent to $dx_\tau$ not having a linear part.

A minimal model of a connected differentiable manifold M is a minimal model $(\bigwedge V,\,d)$ for the de Rham complex $(\Omega ^*(M),\,d)$ of differential forms on M. If M is a simply connected manifold, then the dual of the real homotopy vector space $\pi _i(M)\otimes {\mathbb {R}}$ is isomorphic to $V^i$ for any i (see [9]).

A model of a DGA (A, d) is any DGA (B, d) with the same minimal model (that is, they are equivalent with respect to the equivalence relation determined by the quasi-isomorphisms).

A minimal algebra $(\bigwedge V,\,d)$ is called formal if there exists a morphism of differential algebras $\psi :{(\bigwedge V,\,d)}\,\longrightarrow \, (H^*(\bigwedge V),0)$ inducing the identity map on cohomology. Also a differentiable manifold M is called formal if its minimal model is formal. The formality of a minimal algebra is characterized as follows.

Proposition 10

[9] A minimal algebra $(\bigwedge V,\,d)$ is formal if and only if the space V can be decomposed into a direct sum $V\,=\, C\oplus N$ with $d(C) \,=\, 0$ and d injective on N, such that every closed element in the ideal I(N) in $\bigwedge V$ generated by N is exact.

This characterization of formality can be weakened using the concept of s-formality introduced in [12].

Definition 11

A minimal algebra $\left( \bigwedge V,\,d\right)$ is s-formal ($s> 0$) if for each $i\le s$ the space $V^i$ of generators of degree i decomposes as a direct sum $V^i=C^i\oplus N^i$, where the spaces $C^i$ and $N^i$ satisfy the three following conditions:

• (1) $d(C^i) = 0$,

• (2) the differential map $d:N^i\longrightarrow \bigwedge V$ is injective, and

• (3) any closed element in the ideal $I_s=I\left( \bigoplus \nolimits _{i\le s} N^i\right)$, generated by the space $\bigoplus \nolimits _{i\le s} N^i$ in the free algebra $\bigwedge \left( \bigoplus \nolimits _{i\le s} V^i\right)$, is exact in $\bigwedge V$.

A differentiable manifold M is s-formal if its minimal model is s-formal. Clearly, if M is formal then M is s-formal, for any $s>0$. The main result of [12] shows that sometimes the weaker condition of s-formality implies formality.

Theorem 12

[12] Let M be a connected and orientable compact manifold of dimension 2n or $(2n-1)$. Then M is formal if and only if it is $(n-1)$-formal.

By Corollary 3.3 in [12] a simply connected compact manifold is always 2-formal. Therefore, Theorem 12 implies that any simply connected compact manifold of dimension not more than six is formal. For simply connected 7-dimensional compact manifolds, we have that M is formal if and only if M is 3-formal.

Theorem 12 also holds for compact connected orientable orbifolds, since the proof of [12] only uses that the cohomology $H^*(M)$ is a Poincaré duality algebra.

6 Homotopy properties of simply connected Sasakian 7-manifolds

Proposition 13

Let M be a simply connected compact K-contact 7-dimensional manifold. Then a model for M is $(H\otimes \bigwedge (x),d)$, where H is the cohomology algebra of a simply connected symplectic 6-dimensional orbifold and $dx=\omega \in H^2$ is the class of the symplectic form.

If M is Sasakian, then H is the cohomology algebra of a simply connected 6-dimensional Kähler orbifold.

Proof

Suppose M admits a Sasakian structure. Then M admits a quasi-regular Sasakian structure [22]. Therefore, there is an orbifold circle bundle $S^1 \rightarrow M \rightarrow B$, where B is a compact Kähler orbifold of dimension 6, with Euler class given by the Kähler form $\omega \in H^2(B)$. We note that B is simply connected because M is so (see [3, Theorem 4.3.18]). In particular, $S^1 \rightarrow M \rightarrow B$ is a rational fibration, hence if ${\mathcal {M}}$ is a model for B, then ${\mathcal {M}}\otimes \bigwedge (x)$, with $|x|\,=\,1$, $dx\,=\,\omega$, is a model for M.

Now B is a simply connected compact orbifold of dimension 6. So it is 2-formal. Theorem 12 also holds for orbifolds, hence B is formal. Therefore ${\mathcal {M}}\sim (H,0)$, where $H=H^*(B)$ is the cohomology algebra of B. So a model for M is of the form $(H\otimes \bigwedge (x),d)$, $dx=\omega \in H^2$.

The case where M admits a K-contact structure is similar. By Proposition 9, it admits a quasi-regular K-contact structure. Therefore, M is an orbifold $S^1$-bundle over a symplectic orbifold $S^1 \rightarrow M \rightarrow B$, with Euler class given by the orbifold symplectic form $\omega \in H^2(B)$. As above, a model for M is $(H\otimes \bigwedge (x),d)$, $dx=\omega \in H^2$, where $H=H^*(B)$. $\square$

We prove now Corollary 2.

Corollary 14

Let M be a simply connected compact K-contact 7-dimensional manifold. Suppose that the cup product map $H^2(M)\times H^2(M)\longrightarrow H^4(M)$ is non-zero. Then M does not admit a Sasakian structure.

Proof

Let us compute the cohomology of M from its model $({\mathcal {M}},d)=(H\otimes \bigwedge (x),d)$, $dx=\omega$, where $H=H^*(B)$ is the cohomology algebra of a 6-dimensional simply connected symplectic manifold. Note that $\omega \in H^2$ is a non-zero element with $\omega ^3\in H^6$ generating the top cohomology.

Consider the Lefschetz map $L_\omega :H^*\rightarrow H^{*+2}$, and let $K^*=\ker L_\omega$, $Q^*={{\mathrm{coker}}}L_\omega$. We have a (non-canonical) isomorphism $H^i(M) \cong Q^i \oplus K^{i-1} x$. Note that $Q^3=K^3=H^3$ and $H^6={\mathbb {R}}$. Also $Q^2=H^2/\langle \omega \rangle$, and $K^4=\ker (L_\omega : H^4\rightarrow {\mathbb {R}})$ are vector spaces of codimension one. We have the following:

$$\begin{aligned} H^0(M)&={\mathbb {R}}, \\ H^1(M)&=0, \\ H^2(M)&= Q^2, \\ H^3(M)&=H^3 \oplus K^2 x, \\ H^4(M)&=Q^4 \oplus H^3 x, \\ H^5(M)&=K^4 x, \\ H^6(M)&=0, \\ H^7(M)&=\langle \omega ^3 x\rangle . \end{aligned}$$

The map $H^2(M)\times H^2(M)\rightarrow H^4(M)$ factors through $Q^2 \times Q^2 \rightarrow Q^4$. Hence if it is non-zero then $Q^4\ne 0$. In particular, the Lefschetz map $L_\omega :H^2\rightarrow H^{4}$ is not an isomorphism, so B is not hard Lefschetz.

If M admits a Sasakian structure, then there is a quasi-regular fibration $S^1 \rightarrow M \rightarrow B$ with B satisfying the hard Lefschetz property (it is a Kähler orbifold, so [26] is applicable). This contradicts the above. $\square$

Now we shall study the case of Sasakian 7-manifolds in more detail. Let M be a simply connected compact Sasakian 7-dimensional manifold. Then

$$\begin{aligned} {\mathcal {M}}=\left( H\otimes \bigwedge (x),d\right) \end{aligned}$$

is a model for M, by Proposition 13, where $H=H^*(B)$ is the cohomology algebra of a simply connected compact 6-dimensional Kähler orbifold. This algebra H has a very rich structure:

• (1) there is a canonical isomorphism $H^6 \cong {\mathbb {R}}$, which is given by integration $\int _M:H^6\rightarrow {\mathbb {R}}$;

• (2) H is a Poincaré duality algebra, hence $H^3 \otimes H^3 \rightarrow {\mathbb {R}}$ is an antisymmetric bilinear pairing;

• (3) there is a scalar product on each $H^j$. This is given by the Hodge star operator $* :H^j \rightarrow H^{6-j}$ combined with wedge and integration;

• (4) H has a Hodge structure, that is, $H\otimes {\mathbb {C}}$ has a bigrading such that $H^k\otimes {\mathbb {C}}=\oplus _{p+q=k} H^{p,q}$, where $H^{p,q}=\overline{H^{q,p}}$, and the wedge product respects the bigrading;

• (5) there is a distinguished element $\omega \in H^2$ which is in $H^{1,1}$. This defines the space of primitive forms $P=\langle \omega \rangle ^\perp \subset H^2$. Hence $H^2=\langle \omega \rangle \oplus P$. Moreover $P=P^{1,1}\oplus P^{2,0}$, where $P^{1,1}=P\cap H^{1,1}$ and $P^{2,0}=P \cap (H^{2,0}\oplus H^{0,2})$;

• (6) the Lefschetz map $L_\omega :H^2\rightarrow H^4$ is an isomorphism. Therefore $H^4=\langle \omega ^2\rangle \oplus \omega P^{1,1}\oplus \omega P^{2,0}$. By Theorem 3.16 of Chapter V of [27], for $\alpha _1\in P^{1,1}$ we have $* \alpha _1=-\alpha _1\wedge \omega$, for $\alpha _2\in P^{2,0}$ we have $* \alpha _2=\alpha _2\wedge \omega$, and $* \omega = \frac{1}{2} \omega ^2$. This implies that $L_\omega : \langle \omega \rangle \oplus P^{1,1}\oplus P^{2,0} \rightarrow \langle \omega ^2\rangle \oplus \omega P^{1,1}\oplus \omega P^{2,0}$ is of the form $L_\omega (\alpha )=L_\omega (\alpha _0+\alpha _1+\alpha _2)=\frac{1}{2} *\alpha _0-*\alpha _1+* \alpha _2$, where $\alpha =\alpha _0+\alpha _1+\alpha _2$ is the decomposition according to $H^2=\langle \omega \rangle \oplus P^{1,1}\oplus P^{2,0}$.

The Lefschetz map $L_\omega :H^2\rightarrow H^4$ is an isomorphism so there is an inverse $L_\omega ^{-1}:H^4\rightarrow H^2$. Using it, we can define a map ${\mathcal {F}}: P \times P\times P\times P \rightarrow {\mathbb {R}}$ by

$$\begin{aligned} {\mathcal {F}}(\alpha ,\beta ,\gamma ,\delta )=\int _M L_\omega ^{-1}(\alpha \wedge \beta ) \wedge \gamma \wedge \delta . \end{aligned}$$

This clearly factors through ${{\mathrm{Sym}}}^2P \times {{\mathrm{Sym}}}^2P$. Using (5) above, we have the alternative description

$$\begin{aligned} {\mathcal {F}}(\alpha ,\beta ,\gamma ,\delta )= 2\langle \left( \alpha \wedge \beta \right) _0, \left( \gamma \wedge \delta \right) _0 \rangle - \langle \left( \alpha \wedge \beta \right) _1, \left( \gamma \wedge \delta \right) _1 \rangle +\langle \left( \alpha \wedge \beta \right) _2, \left( \gamma \wedge \delta \right) _2 \rangle , \end{aligned}$$

from where it follows that ${\mathcal {F}}$ factors as a map  ${{\mathrm{Sym}}}^2({{\mathrm{Sym}}}^2P)\rightarrow {\mathbb {R}}$.

Let ${\mathcal {K}}_M$ be the kernel of the map ${{\mathrm{Sym}}}^2({{\mathrm{Sym}}}^2 P)\rightarrow {{\mathrm{Sym}}}^4 P$. Then we define a map

$$\begin{aligned} {\mathcal {F}}_M={\mathcal {F}}|_{{\mathcal {K}}_M}: {\mathcal {K}}_M\rightarrow {\mathbb {R}}. \end{aligned}$$

We have the following result.

Theorem 15

Let M be a simply connected compact Sasakian 7-dimensional manifold. Then M is formal if and only if ${\mathcal {F}}_M=0$.

Proof

Using Theorem 12, we only have to check whether M is 3-formal. For this we have to construct the minimal model $\rho :(\bigwedge V,d) \rightarrow {\mathcal {M}}= (H\otimes \bigwedge (x),d)$ up to degree 3. This is easy:

$$\begin{aligned} V^1&=0, \\ V^2&= P, \\ V^3&= H^3 \oplus N^3, \qquad \text {where } N^3={{\mathrm{Sym}}}^2P, \end{aligned}$$

where the differential is given by $d=0$ on P and $H^3$, and $d:N^3 \rightarrow \bigwedge V^2$ is the isomorphism ${{\mathrm{Sym}}}^2P \rightarrow \bigwedge ^2 P$. The map $\rho$ is given as follows. $\rho : V^2 =P\rightarrow {\mathcal {M}}^2=H^2$ is defined as the obvious (inclusion) map, $\rho : H^3 \rightarrow {\mathcal {M}}^3=H^3 \oplus H^2 x$ is the inclusion on the first summand, and $\rho :N^3={{\mathrm{Sym}}}^2P \rightarrow {\mathcal {M}}^3=H^3 \oplus H^2 x$ is defined as $\rho (\alpha \cdot \beta )=L_{\omega }^{-1}(\alpha \wedge \beta ) \, x$. Note that

$$\begin{aligned}&d( \rho (\alpha \cdot \beta ))=L_{\omega }^{-1}(\alpha \wedge \beta ) \, dx= L_{\omega }^{-1}(\alpha \wedge \beta )\omega \\&\qquad = \alpha \wedge \beta =\rho (\alpha )\wedge \rho (\beta )=\rho (\alpha \wedge \beta )= \rho \left( d(\alpha \cdot \beta )\right) , \end{aligned}$$

so $\rho$ is a DGA map. Clearly it is a 3-equivalence (it induces an isomorphism on cohomology up to degree 3 and an inclusion on degree 4).

The space of closed elements is $C^3=H^3$. Now let us check when the elements $z\in I(N^3)$ with $dz=0$ satisfy $[\rho (z)]=0\in H^*(M)$. The only cases to check is when z has degree 5 or 7. If z has degree 5, then $[\rho (z)]\ne 0$ if and only if there exists some $\beta \in P,\,[\rho (\beta )]\in H^2(M)$, such that $[\rho (z)]\wedge [\rho (\beta )]\ne 0$, by Poincaré duality. Hence $[\rho (z\beta )]\ne 0$. This means that we can restrict to elements z of degree 7, that is $z \in N^3\cdot \bigwedge ^2 P$.

Let $z \in N^3\cdot \bigwedge ^2 P \cong {{\mathrm{Sym}}}^2P \times {{\mathrm{Sym}}}^2 P$. Then the map $d: N^3\cdot \bigwedge ^2P \rightarrow \bigwedge ^4P$ coincides the full symmetrization map ${{\mathrm{Sym}}}^2P \times {{\mathrm{Sym}}}^2 P \rightarrow {{\mathrm{Sym}}}^4P$. So

$$\begin{aligned} Z= \ker d|_{I(N^3)^7}={\mathcal {K}}_M \oplus {{\mathrm{Ant}}}^2\left( {{\mathrm{Sym}}}^2P\right) , \end{aligned}$$

where ${{\mathrm{Ant}}}^2(W)$ denotes the antisymmetric 2-power of a vector space W.

Now we have to study the map

$$\begin{aligned} \rho : Z \rightarrow H^7(M) =H^6 x, \end{aligned}$$

and see if this is non-zero. This is given (on the basis elements) by

$$\begin{aligned} \rho \left( (\alpha \cdot \beta )\cdot (\gamma \cdot \delta )\right) = \left( L_{\omega }^{-1}(\alpha \wedge \beta ) \wedge \gamma \wedge \delta \right) x, \end{aligned}$$

so ${\mathcal {F}}_M=\rho |_{{\mathcal {K}}_M}$. Note that $\rho$ automatically vanishes on ${{\mathrm{Ant}}}^2({{\mathrm{Sym}}}^2P)$, hence M is formal if and only if $\rho$ vanishes on ${\mathcal {K}}_M$ if and only if ${\mathcal {F}}_M=0$.

According to Theorem 12, to check non-formality we have to test the relevant property (2) on any splitting $V^3=C^3+ N'{}^3$. If we take another splitting $V^3=C^3+ N'{}^3$, then the projection $\pi :V^3\rightarrow N^3$ gives an isomorphism $\pi :N'{}^3 \rightarrow N^3$, and so an isomorphism $N'{}^3\cdot {{\mathrm{Sym}}}^2P \cong N^3 \cdot {{\mathrm{Sym}}}^2P$. Clearly, $d\circ \pi =d$ on $N'{}^3$, so the spaces of cycles correspond ${\mathcal {K}}' \cong {\mathcal {K}}$. On the other hand $H^3 \cdot H^2\cdot H^2 =0$, so the maps $\rho :{\mathcal {K}}\rightarrow H^6 x$ and $\rho :{\mathcal {K}}'\rightarrow H^6 x$ also correspond. This means that the corresponding ${\mathcal {F}}$ and ${\mathcal {F}}'$ coincide under the isomorphism ${\mathcal {K}}\cong {\mathcal {K}}'$. This means that the choice of splitting is not relevant. $\square$

This result means that the formality or non-formality of M only depends on the cohomology algebra H. Theorem 15 can be applied to the examples in Section 5.3 of [2]. For instance for $B={\mathbb {C}}P^1\times {\mathbb {C}}P^1\times {\mathbb {C}}P^1$, we have a simply connected Sasakian 7-manifold which is non-formal (Theorem 12 of [2]). For $B={\mathbb {C}}P^3$, we have obviously $P=0$ and hence M is formal.

The element ${\mathcal {F}}_M$ of Theorem 15 is the principal Massey product defined by Crowley and Nordström [6] for simply connected compact 7-manifolds in general. The principal Massey product is the full obstruction to formality for simply connected compact 7-manifolds.

Now we deduce Corollary 3.

Corollary 16

Let M be a simply connected compact Sasakian 7-dimensional manifold. Then M is formal if and only if all triple Massey products are zero.

Proof

Suppose that ${\mathcal {F}}_M\ne 0$. We choose an orthonormal basis for $H^2=\langle e_0,e_1,\ldots , e_m\rangle$, where $e_0=\frac{1}{\sqrt{3}} \omega ,\,P^{1,1}=\langle e_1,\ldots ,e_s\rangle ,\,P^{2,0}=\langle e_{s+1},\ldots , e_m\rangle$. The vector space ${\mathcal {K}}_M$ is generated by elements of the form

$$\begin{aligned} a_{ijkl}= \left( e_i \cdot e_j\right) \cdot (e_k\cdot e_l) - (e_k \cdot e_j)\cdot \left( e_i\cdot e_l\right) , \end{aligned}$$

for $0\le i,j,k,l \le m$ (here, as usual, the dot product means symmetric product). Now define the numbers

$$\begin{aligned} \lambda _{ijk}= \int _M e_i\wedge e_j\wedge e_k \in {\mathbb {R}}, \end{aligned}$$

for $0\le i,j,k \le m$. Note that these numbers are fully symmetric on i, j, k. Also $\lambda _{000}= \frac{2}{\sqrt{3}}$ and $\lambda _{ij0}=\frac{1}{\sqrt{3}}\varepsilon _i\delta _{ij}$, for $(i,j)\ne (0,0)$, where $\varepsilon _i=-1$ for $1\le i \le s$ and $\varepsilon _i=1$ for $s+1\le i \le m$. Then

$$\begin{aligned} L_\omega ^{-1}(e_i\wedge e_j)=2*(e_i\wedge e_j)_0 - (e_i\wedge e_j)_1+(e_i\wedge e_j)_2 =2\lambda _{ij0}e_0 +\sum _{t>0} \varepsilon _t \lambda _{ijt}e_t. \end{aligned}$$

So

$$\begin{aligned} {\mathcal {F}}_M ( (e_i \cdot e_j)\cdot (e_k\cdot e_l))=2\lambda _{ij0}\lambda _{kl0} +\sum _{t>0}\varepsilon _t\lambda _{ijt}\lambda _{klt}. \end{aligned}$$

Evaluating ${\mathcal {F}}_M$ on $a_{ijkl}$ gives a set of equations to determine the formality of M. M is non-formal when there exists some $a_{ijkl}$ with ${\mathcal {F}}_M(a_{ijkl})\ne 0$. By [6], we have that the triple Massey product $\langle e_i, e_j, e_k\rangle$ is a well-defined element of $H^5(M)$ and it satisfies

$$\begin{aligned} {\mathcal {F}}_M(a_{ijkl}) =\langle e_i, e_j, e_k\rangle \cup e_l. \end{aligned}$$

So $\langle e_i, e_j, e_k\rangle \ne 0$, as required. $\square$

This result is of relevance since it is not known if for general simply connected compact 7-dimensional manifolds there are obstructions to formality different from triple Bianchi-Massey tensor, as remarked in [6]. It is true that for higher dimensional manifolds, there are obstructions to formality even when all Massey products (triple and higher order) can be zero.