Quaternionic Structures in Mathematics and Physics SISSA - Trieste, 5-9 September 1994

CANONICAL COHOMOLOGY OF COMPACT G_{2}-NILMANIFOLDS ^{∗}

Marisa Fernández and Luis Ugarte

ABSTRACT

In this paper we consider the canonical complex of a G_{2}-manifold [R1, R2]. An example of a compact cocalibrated G_{2}-nilmanifold whose de Rham cohomology groups are not isomorphic to the canonical cohomology groups is constructed.

∗ Partially supported by U.P.V. Project 127.310–EA 191/94, DGICYT Project

PB94-0633-C02-02, and by U.P.V. Project 127.310–EC 248/96.

1. Introduction

G_{2}-manifolds are seven-dimensional Riemannian manifolds with a 2-fold vector cross product P, which can be considered as a natural generalization of an almost Hermitian structure. Corresponding to the Kähler form one has a fundamental 3-form φ.

Within the class of all G_{2}-manifolds one can find the G_{2} analogues of the different types of almost Hermitian manifolds. For example, if φ is coclosed then M is a cocalibrated G_{2}-manifold [HL1, HL2], a G_{2} analogue of a cosymplectic (or semi-Kähler) manifold; if d ∗ φ = θ ∧ ∗φ, where ∗ is the Hodge star operator of M and θ is the differential 1-form on M given by θ = \frac{1}{3} ∗ (∗d ∗ φ ∧ ∗φ), then M is an integrable G_{2}-manifold [FU], a G_{2} analogue of a complex manifold as we explain in Section 2. In particular, if φ is closed and coclosed then M has a subgroup of G_{2} as holonomy group [G2], and M is a G_{2} analogue of a Kähler manifold.

For a manifold with a G-structure, the theory of infinitesimal deformations of generalized instantons has been formulated by R. Reyes in [R1, R2] in terms of a canonical complex. Recently the authors of this paper have studied in [FU] the canonical cohomology groups for integrable G_{2}-manifolds and, in particular, for cocalibrated G_{2}-manifolds.

For a compact Kähler manifold L it is well-known the Hodge theorem: there is an isomorphism

H^{q}(L) ≅ ⨁_{r+s=q} H^{r,s}(L),  (1)

where H^{q}(L) denotes the q-th de Rham cohomology group of L, and H^{r,s}(L) are the Dolbeault groups. In [FU] the corresponding isomorphism to (1) is proved for compact G_{2}-manifolds whose holonomy group is a subgroup of G_{2} (see Section 2, Theorem 4). Our main purpose here is to show examples of compact integrable G_{2}-manifolds for which such isomorphism fails (see Section 3, Theorem 8 and Theorem 10). According to the classification of G_{2}-manifolds given in [FG], a manifold with Hol ⊆ G_{2} can be defined as a nearly parallel G_{2}-manifold in the class W_{3} (i.e. the class of G_{2}-manifolds for which the fundamental 3-form satisfies d ∗ φ = 0 and dφ ∧ φ = 0). In Theorem 8 we prove that the Hodge type theorem does not hold for arbitrary W_{3}-manifolds, whereas Theorem 10 shows that such result can also fail for compact nearly parallel G_{2}-manifolds.

We are grateful to Simon Salamon for several useful discussions.

2. G_{2}-manifolds and canonical cohomology

Let M be a C^{∞} Riemannian manifold of dimension 7 with metric ⟨ , ⟩. Denote by 𝔛(M) the Lie algebra of C^{∞} vector fields on M and by 𝔉(M) the algebra of C^{∞} functions on M.

A 2-fold vector cross product on M is a tensor field P: 𝔛(M) × 𝔛(M) ⟶ 𝔛(M) satisfying the axioms:

i) ⟨P (X, Y), X⟩ = ⟨P (X, Y), Y ⟩ = 0,

ii) ∥P (X, Y)∥^{2} = ∥X∥^{2}∥Y ∥^{2} − ⟨X, Y ⟩^{2},

for X, Y ∈ 𝔛(M).

CANONICAL COHOMOLOGY OF COMPACT G_{2}-NILMANIFOLDS 129

A seven dimensional Riemannian manifold M with a 2-fold vector cross product P is called G_{2}-manifold. There is a representation of G_{2} on each tangent space of M defined by means of the vector cross product P [BG, G2, G4, S].

To describe the geometry of a G_{2}-manifold M it is very useful to consider the differential 3-form φ on M given by

φ(X, Y, Z) = ⟨P (X, Y), Z⟩ ,

for X, Y, Z ∈ 𝔛(M). This form φ, originally defined by Bonan [Bon], is called the fundamental 3-form of M.

Let Λ^{q}(M) denote the space of differential q-forms on M, and let ⟨ , ⟩ the inner product on Λ^{q}(M) given by

⟨α, β⟩ Ω_{M} = α ∧ ∗β,  (2)

for α, β ∈ Λ^{q}(M), where Ω_{M} denotes the volume form on M. In [FG] the first author of this paper and Gray proved that Λ^{q}(M) splits orthogonally into components Λ_{l}^{q}(M) of dimension l, which are irreducible under the action of G_{2}. The representation of G_{2} on Λ^{1}(M) is the irreducible 7-dimensional representation, and the representations of G_{2} on Λ^{q}(M) and Λ^{7−q}(M) are the same because the Hodge star operator ∗: Λ^{q}(M) ⟶ Λ^{7−q}(M) is an isometry. Therefore, it suffices to describe the representations of G_{2} on Λ^{2}(M) and Λ^{3}(M). They are (see [Br, FG, S])

Λ_{7}^{2}(M) = {∗(α ∧ ∗φ) | α ∈ Λ^{1}(M)},
Λ_{14}^{2}(M) = {β ∈ Λ^{2}(M) | β ∧ ∗φ = 0},
Λ_{1}^{3}(M) = {fφ | f ∈ 𝔉(M)},  (3)
Λ_{7}^{3}(M) = {∗(α ∧ φ) | α ∈ Λ^{1}(M)},
Λ_{27}^{3}(M) = {γ ∈ Λ^{3}(M) | γ ∧ φ = γ ∧ ∗φ = 0}.

Now, from (2) and (3), it is easy to get

Λ_{7}^{3}(M) ⊕ Λ_{27}^{3}(M) = {γ ∈ Λ^{3}(M) | γ ∧ ∗φ = 0}.  (4)

Recall [Br, C, FG, S] that the G_{2}-manifold M is said to be parallel if ∇φ = 0 (or equivalently, dφ = d ∗ φ = 0); nearly parallel if d ∗ φ = 0 and dφ = k ∗ φ, where k ∈ ℝ; W_{3}-manifold if d ∗ φ = 0 and dφ ∧ φ = 0; cocalibrated (semi-parallel or cosymplectic) if d ∗ φ = 0; integrable if d ∗ φ = θ ∧ ∗φ, where θ is the differential 1-form on M given by θ = \frac{1}{3} ∗ (∗d ∗ φ ∧ ∗φ).

Now, let us suppose that M is an integrable G_{2}-manifold. It is easy to see that dβ ∧ ∗φ = 0, for β ∈ Λ_{14}^{2}(M). Then, from (4) it follows the existence of the differential complex

\begin{aligned}0 ⟶ & Λ_{14}^{2}(M) ⟶^{D^{∧}} Λ_{7}^{3}(M) ⊕ Λ_{27}^{3}(M) ⟶^{D^{∧}} \\ & Λ^{4}(M) ⟶^{D^{∧}=d} Λ^{5}(M) ⟶^{D^{∧}=d} Λ^{6}(M) ⟶^{D^{∧}=d} Λ^{7}(M) → 0, \end{aligned}  (5)

where D^{∧} is the restriction to Λ_{14}^{2}(M) and Λ_{7}^{3}(M) ⊕ Λ_{27}^{3}(M) of the exterior derivative d of M, that is, (5) is a differential subcomplex of the de Rham complex. We shall denote by

H^{∧}^{∗}(M) the cohomology of this complex; moreover, if α is a differential q-form on M such

that D^{∧}α = 0, we shall denote by {α} the cohomology class defined by α in H^{∧}^{q}(M).

Next, we recall briefly some results about the canonical cohomology [R1, R2, S]. We also refer to [FU], where a more detailed and complete development may be found.

Let M be a G_{2}-manifold with fundamental 3-form φ, and denote by π_{7}: Λ^{2}(M) ⟶ Λ_{7}^{2}(M) and π_{1}: Λ^{3}(M) ⟶ Λ_{1}^{3}(M) the orthogonal projections of Λ^{2}(M) and Λ^{3}(M) onto

Λ_{7}^{2}(M) and Λ_{1}^{3}(M), respectively. The maps D^{∨}_{0}: Λ^{0}(M) ⟶ Λ^{1}(M), D^{∨}_{1}: Λ^{1}(M) ⟶

Λ_{7}^{2}(M) and D^{∨}_{2}: Λ_{7}^{2}(M) ⟶ Λ_{1}^{3}(M) are defined by

D^{∨}_{0} = d,  (6)
D^{∨}_{1}(α) = π_{7}(dα),  (7)

for α ∈ Λ^{1}(M), and

D^{∨}_{2}(β) = π_{1}(dβ),  (8)

for β ∈ Λ_{7}^{2}(M).

It will be convenient to make no distinction to denote the maps D^{∨}_{0}, D^{∨}_{1} and D^{∨}_{2}. In

fact, unless clarity is required, we shall write D^{∨} for these maps.

The following proposition is proved in [R1], [R2], [S] and [FU].

Proposition 1. Let M be an integrable G_{2}-manifold. Then

0 ⟶ Λ^{0}(M) ⟶^{D^{∨}=d} Λ^{1}(M) ⟶^{D^{∨}} Λ_{7}^{2}(M) ⟶^{D^{∨}} Λ_{1}^{3}(M) ⟶ 0  (9)

is a differential complex.

Now, we stablish the notion of canonical cohomology. Definition. Let M be an integrable G_{2}-manifold with fundamental 3-form φ. The complex (9) is called canonical complex of M. (It can be considered as the “Dolbeault complex” for G_{2}-manifolds.)

We denote by H^{∨}^{∗}(M) the cohomology of this complex, and we call it canonical coho-

mology of M. For a differential q-form α such that D^{∨}α = 0 we shall denote by [α] the

cohomology class in H^{∨}^{q}(M) defined by α.

From now on, M is an integrable G_{2}-manifold, and H^{∗}(M) denotes the de Rham cohomology of M. For α ∈ Λ^{q}(M) such that dα = 0 we shall denote by [α]_{dR} the cohomology class in H^{q}(M) defined by α.

Now, as a consequence of Proposition 1 we have that H^{∨} ^{0}(M) = H^{0}(M) and the

homomorphism μ: H^{1}(M) ⟶ H^{∨}^{1}(M) given by

μ([α]_{dR}) = [α],  (10)

for [α]_{dR} ∈ H^{1}(M), is injective.

Since the differential complexes defined by (5) and (9) are elliptic (see [R1, R2]),

the cohomology groups H^{∨} ^{q}(M) and H^{∧} ^{q}(M) have dimension < ∞ for every compact integrable G_{2}-manifold M. The following results [FU] show special properties of the canonical cohomology groups of particular classes of integrable G_{2}-manifolds.

Theorem 2. Let M be a compact cocalibrated G_{2}-manifold. Then

i) H^{∨}^{1}(M) ≅ H^{∨}^{2}(M),

ii) H^{∨}^{3}(M) ≅ H^{0}(M).

Theorem 3. Let M be a compact nearly parallel G_{2}-manifold.

i) The homomorphism μ: H^{1}(M) ⟶ H^{∨}^{1}(M) given by (10) is an isomorphism. There-

fore, H^{∨}^{1}(M) ≅ H^{1}(M).

ii) Furthermore, if M is nearly parallel non-parallel G_{2}-manifold, we have

H^{∨}^{1}(M) ≅ H^{1}(M) = {0}  and  H^{2}(M) ≅ H^{∧}^{2}(M).

We finish this section with a Hodge type theorem for compact G_{2}-manifolds whose holonomy group is a subgroup of G_{2}. Its proof is given in [FU].

Theorem 4. For a compact parallel G_{2}-manifold M we have

\begin{aligned}H^{1}(M) & ≅ H^{∨}^{1}(M), \\ H^{2}(M) & ≅ H^{∨}^{2}(M) ⊕ H^{∧}^{2}(M), \\ H^{3}(M) & ≅ H^{∨}^{3}(M) ⊕ H^{∧}^{3}(M), \\ H^{4}(M) & ≅ H^{∧}^{4}(M). \end{aligned}  (11)

Corollary 5. For any compact parallel G_{2}-manifold M the Betti numbers b_{q}(M) satisfy

b_{2}(M) ≥ b_{1}(M),
b_{3}(M) ≥ b_{1}(M) + b_{0}(M).

3. Compact integrable G_{2}-nilmanifolds

The main problem to construct counterexamples to Theorem 4 is the difficulty to compute

the cohomology groups H^{∨}^{q}(M), H^{q}(M) and H^{∧}^{q}(M) of an integrable G_{2}-manifold M.

For a compact nilmanifold M = Γ\K it is well-known Nomizu’s theorem [N], which asserts that the cohomology H^{∗}(𝔎^{∗}) of the complex Λ^{∗}(𝔎^{∗}) of left invariant differential forms on the nilpotent Lie group K (𝔎 being the Lie algebra of K) is isomorphic to the de Rham cohomology H^{∗}(M).

For the authors is still unknown if there is a similar result for the canonical cohomology of an integrable G_{2}-nilmanifold. In [FU] they have proved some relations among the

dimensions of the groups H^{q}(M), H^{∨}^{q}(M)⊕H^{∧}^{q}(M) and H^{∨}^{q}(𝔎^{∗})⊕H^{∧}^{q}(𝔎^{∗}), for any compact integrable G_{2}-nilmanifold M = Γ\K. We explain below such relations, and we shall use them to show that (11) fails for compact integrable non-parallel G_{2}-nilmanifolds.

Let M = Γ\K be a compact nilmanifold of dimension 7, that is, K is a connected and simply-connected nilpotent Lie group of dimension 7, and Γ is a discrete subgroup of K such that the quotient Γ\K is a compact space.

Now, let us suppose that K is a G_{2}-manifold with a left invariant metric ⟨ , ⟩, and a left invariant 2-fold vector cross product P. Denote by φ the left invariant fundamental 3-form on K. Then ⟨ , ⟩, P and φ descend to a metric, a 2-fold vector cross product and a 3-form on M, respectively, for which we shall use the same notation, that is, ⟨ , ⟩, P and φ, respectively. Therefore, M is a G_{2}-nilmanifold with vector cross product P and fundamental 3-form φ. Moreover, M is integrable, cocalibrated or, in particular, W_{3}-manifold if and only if K is integrable, cocalibrated or W_{3}-manifold, respectively.

Denote by 𝔎 the Lie algebra of K. Let

⋯ ⟶ Λ^{q−1}(𝔎^{∗}) ⟶^{d} Λ^{q}(𝔎^{∗}) ⟶^{d} Λ^{q+1}(𝔎^{∗}) ⟶ ⋯

be the differential complex, where Λ^{q}(𝔎^{∗}) denotes the left invariant differential q-forms on K. The Chevalley-Eilenberg cohomology is defined by

H^{q}(𝔎^{∗}) = \frac{Ker\{d: Λ^{q}(𝔎^{∗}) ⟶ Λ^{q+1}(𝔎^{∗})\}}{Im\{d: Λ^{q−1}(𝔎^{∗}) ⟶ Λ^{q}(𝔎^{∗})\}}.

On the other hand, we introduce the differential complexes

0 ⟶ ℝ ⟶^{D^{∨}=0} Λ^{1}(𝔎^{∗}) ⟶^{D^{∨}} Λ_{7}^{2}(𝔎^{∗}) ⟶^{D^{∨}} Λ_{1}^{3}(𝔎^{∗}) ⟶ 0,  (12)

and

\begin{aligned}0 ⟶ & Λ_{14}^{2}(𝔎^{∗}) ⟶^{D^{∧}} Λ_{7}^{3}(𝔎^{∗}) ⊕ Λ_{27}^{3}(𝔎^{∗}) ⟶^{D^{∧}} \\ & Λ^{4}(𝔎^{∗}) ⟶^{D^{∧}=d} Λ^{5}(𝔎^{∗}) ⟶^{D^{∧}=d} Λ^{6}(𝔎^{∗}) ⟶^{D^{∧}=d} Λ^{7}(𝔎^{∗}) → 0, \end{aligned}  (13)

where the spaces Λ_{l}^{q}(𝔎^{∗}) are defined by similar relations to (3). (For example, Λ_{7}^{2}(𝔎^{∗}) = {∗(α ∧ ∗φ) | α ∈ Λ^{1}(𝔎^{∗})}.) The complexes (12) and (13) are differential subcomplexes of (9) and (5), respectively.

We denote by H^{∧}^{∗}(𝔎^{∗}) the cohomology of the complex (13), and by H^{∨}^{∗}(𝔎^{∗}) the coho-

mology of the complex (12). Let h^{∧}_{q}( ) = dim H^{∧}^{q}( ) and h^{∨}_{q}( ) = dim H^{∨}^{q}( ); the following theorem is proved in [FU].

Theorem 6. Let M = Γ\K be a compact integrable G_{2}-nilmanifold with fundamental 3-form φ, which arises from a left invariant fundamental 3-form φ on K. Then

b_{1}(M) ≤ h^{∨}_{1}(𝔎^{∗}) ≤ h^{∨}_{1}(M).  (14)

Moreover, if M is cocalibrated, we have

b_{2}(M) ≤ h^{∨}_{2}(𝔎^{∗}) + h^{∧}_{2}(𝔎^{∗}) ≤ h^{∨}_{2}(M) + h^{∧}_{2}(M).  (15)

In order to construct a compact integrable G_{2}-nilmanifold for which (11) is not satisfied, we proceed as follows.

Consider the Lie group K = H(2, 1) × ℝ^{2}, where H(2, 1) is the generalized Heisenberg group [Ha] of dimension 5 consisting of the matrices of the form

a = ( \begin{matrix}1 & a_{1} & a_{2} & c \\ 0 & 1 & 0 & b_{1} \\ 0 & 0 & 1 & b_{2} \\ 0 & 0 & 0 & 1 \end{matrix} ) ,

where a_{i}, b_{i}, c ∈ ℝ, 1 ≤ i ≤ 2. A global system of coordinates {x_{1}, x_{2}, y_{1}, y_{2}, z} on H(2, 1) is defined by

x_{i}(a) = a_{i},  y_{i}(a) = b_{i},  z(a) = c,

for 1 ≤ i ≤ 2. Denote by {t_{1}, t_{2}} the natural coordinates on ℝ^{2}. Then it is easy to verify that {dx_{1}, dx_{2}, dy_{1}, dy_{2}, dz −x_{1}dy_{1} −x_{2}dy_{2}, dt_{1}, dt_{2}} is a basis for the left invariant 1-forms on K = H(2, 1) × ℝ^{2}.

Now, to get a compact quotient, we take M = Γ\K, where Γ is the uniform subgroup of K given by Γ = Γ(2, 1) × ℤ^{2}, Γ(2, 1) being the subgroup of H(2, 1) which consists of those matrices a whose entries a_{1}, a_{2}, b_{1}, b_{2}, c are integers.

The 1-forms dx_{i}, dy_{i}, dz − x_{1}dy_{1} − x_{2}dy_{2}, dt_{i} all descend to M; denote the 1-forms induced on M by α_{i}, β_{i}, γ, η_{i} (1 ≤ i ≤ 2), respectively. Therefore {α_{i}, β_{i}, γ, η_{i}; 1 ≤ i ≤ 2} is a basis for the 1-forms on M such that

\{ \begin{matrix}dα_{i} = dβ_{i} = dη_{i} = 0, 1 ≤ i ≤ 2, \\ dγ = −α_{1} ∧ β_{1} − α_{2} ∧ β_{2}. \end{matrix}  (16)

Theorem 7. There exists a (non-parallel) vector cross product on M such that the fundamental 3-form φ is coclosed and dφ∧φ = 0. Therefore, M is a cocalibrated G_{2}-nilmanifold and a W_{3}-manifold.

Proof. Let φ be the 3-form on M defined by

\begin{aligned}φ = & & β_{1} ∧ α_{2} ∧ η_{1} + α_{2} ∧ α_{1} ∧ η_{2} + α_{1} ∧ η_{1} ∧ β_{2} + η_{1} ∧ η_{2} ∧ γ \\ & + & η_{2} ∧ β_{2} ∧ β_{1} + β_{2} ∧ γ ∧ α_{2} + γ ∧ β_{1} ∧ α_{1}. \end{aligned}  (17)

On the right hand side of (17) all of the terms are coclosed; and so d ∗ φ = 0. Define a metric on M by

⟨ , ⟩ = α_{1}^{2} + α_{2}^{2} + β_{1}^{2} + β_{2}^{2} + γ^{2} + η_{1}^{2} + η_{2}^{2}.

Let {E_{0}, …, E_{6}} be the basis dual to {β_{1}, α_{2}, α_{1}, η_{1}, η_{2}, β_{2}, γ}. Then, a 2-fold vector cross product P on M is given by P (E_{i}, E_{j}) = −P (E_{j}, E_{i}), and P (E_{i}, E_{i+1}) = E_{i+3}, P (E_{i+3}, E_{i}) = E_{i+1}, P (E_{i+1}, E_{i+3}) = E_{i} (i ∈ ℤ_{7}). It is not hard to show that P satisfies the axioms for a 2-fold vector cross product, and moreover the form φ, given by (17), is the fundamental 3-form. To show that P is not parallel we prove that φ is not closed. Indeed, a calculation shows that dφ = (−α_{1} ∧ β_{1} − α_{2} ∧ β_{2}) ∧ η_{1} ∧ η_{2} ≠ 0.

Moreover, from (17) it follows that

dφ ∧ φ = (−α_{1} ∧ β_{1} − α_{2} ∧ β_{2}) ∧ (β_{2} ∧ γ ∧ α_{2} + γ ∧ β_{1} ∧ α_{1}) ∧ η_{1} ∧ η_{2}
= (−α_{1} ∧ β_{1} ∧ α_{2} ∧ β_{2} ∧ γ + α_{1} ∧ β_{1} ∧ α_{2} ∧ β_{2} ∧ γ) ∧ η_{1} ∧ η_{2}
= 0,

which implies that M is a W_{3}-manifold.

Theorem 8. For the compact W_{3}-manifold M we have

H^{1}(M) ≅ H̸^{∨}^{1}(M),  (18)
H^{2}(M) ≅ H̸^{∨}^{2}(M) ⊕ H^{∧}^{2}(M).  (19)

Proof. It follows from (16) and Nomizu’s theorem [N] that the de Rham cohomology groups H^{1}(M) and H^{2}(M) are

H^{1}(M) = {[α_{1}]_{dR}, [α_{2}]_{dR}, [β_{1}]_{dR}, [β_{2}]_{dR}, [η_{1}]_{dR}, [η_{2}]_{dR}},
H^{2}(M) = {[α_{1} ∧ α_{2}]_{dR}, [α_{1} ∧ β_{1}]_{dR}, [α_{1} ∧ β_{2}]_{dR}, [α_{1} ∧ η_{1}]_{dR},
[α_{1} ∧ η_{2}]_{dR}, [α_{2} ∧ β_{1}]_{dR}, [α_{2} ∧ η_{1}]_{dR}, [α_{2} ∧ η_{2}]_{dR},
[β_{1} ∧ β_{2}]_{dR}, [β_{1} ∧ η_{1}]_{dR}, [β_{1} ∧ η_{2}]_{dR}, [β_{2} ∧ η_{1}]_{dR},
[β_{2} ∧ η_{2}]_{dR}, [η_{1} ∧ η_{2}]_{dR}}..

Thus we have

b_{1}(M) = 6,  b_{2}(M) = 14.  (20)

Next, because the left invariant differential forms on K all descend to differential forms on M, it will be convenient to make no distinction to denote the left invariant 1-forms dx_{i}, dy_{i}, dz − x_{1}dy_{1} − x_{2}dy_{2}, dt_{i} (1 ≤ i ≤ 2) on K and the 1-forms α_{i}, β_{i}, γ, η_{i} (1 ≤ i ≤ 2) on M, and as well as for the wedge products.

To compute H^{∨}^{1}(𝔎^{∗}) we see that α_{i}, β_{i} and η_{i} (1 ≤ i ≤ 2) also define nonzero elements

in H^{∨} ^{1}(𝔎^{∗}). Moreover, we note that γ does not define a cohomology class in H^{1}(M), because γ is not closed. However, by using that ∗φ is given by

\begin{aligned}∗φ = & −α_{1} ∧ η_{2} ∧ β_{2} ∧ γ + β_{1} ∧ η_{1} ∧ β_{2} ∧ γ − β_{1} ∧ α_{2} ∧ η_{2} ∧ γ \\ & +β_{1} ∧ α_{2} ∧ α_{1} ∧ β_{2} + α_{2} ∧ α_{1} ∧ η_{1} ∧ γ − β_{1} ∧ α_{1} ∧ η_{1} ∧ η_{2} \\ & −α_{2} ∧ η_{1} ∧ η_{2} ∧ β_{2}, \end{aligned}  (21)

one can check that dγ ∧ ∗φ = 0, and hence dγ ∈ Λ_{14}^{2}(𝔎^{∗}), that is, D^{∨}γ = 0 or, equivalently

γ defines a nonzero cohomology class in H^{∨}^{1}(𝔎^{∗}). Thus

H^{∨}^{1}(𝔎^{∗}) = {[α_{1}], [α_{2}], [β_{1}], [β_{2}], [γ], [η_{1}], [η_{2}]},

which implies

h^{∨}_{1}(𝔎^{∗}) = 7.  (22)

Now, from (20), (22) and (14) it follows that

b_{1}(M) = 6 < h^{∨}_{1}(𝔎^{∗}) = 7 ≤ h^{∨}_{1}(M),

which proves (18).

In order to prove (19) we compute a basis of the space Λ_{14}^{2}(𝔎^{∗}) = {β ∈ Λ^{2}(𝔎^{∗}) | β ∧ ∗φ = 0}. From (21) we get

Λ_{14}^{2}(𝔎^{∗}) = {α_{1} ∧ β_{2} − α_{2} ∧ β_{1}, α_{1} ∧ β_{2} − γ ∧ η_{2}, α_{1} ∧ β_{1} + α_{2} ∧ β_{2},
α_{1} ∧ β_{1} + η_{1} ∧ η_{2}, α_{1} ∧ η_{1} + α_{2} ∧ γ, α_{1} ∧ η_{1} − β_{1} ∧ η_{2},
α_{2} ∧ η_{1} + β_{2} ∧ η_{2}, α_{1} ∧ γ + β_{2} ∧ η_{2}, α_{1} ∧ α_{2} − β_{1} ∧ β_{2},
α_{1} ∧ α_{2} + γ ∧ η_{1}, α_{2} ∧ η_{2} − β_{1} ∧ γ, α_{2} ∧ η_{2} − β_{2} ∧ η_{1},
α_{1} ∧ η_{2} + β_{1} ∧ η_{1}, α_{1} ∧ η_{2} − β_{2} ∧ γ}.

Now, we compute H^{∧}^{2}(𝔎^{∗}) = {β ∈ Λ_{14}^{2}(𝔎^{∗}) | dβ = 0}. It is given by

H^{∧}^{2}(𝔎^{∗}) = {{α_{1} ∧ β_{2} − α_{2} ∧ β_{1}}, {α_{1} ∧ β_{1} + α_{2} ∧ β_{2}}, {α_{1} ∧ β_{1} + η_{1} ∧ η_{2}},
{α_{1} ∧ η_{1} − β_{1} ∧ η_{2}}, {α_{2} ∧ η_{1} + β_{2} ∧ η_{2}}, {α_{1} ∧ α_{2} − β_{1} ∧ β_{2}},
{α_{2} ∧ η_{2} − β_{2} ∧ η_{1}}, {α_{1} ∧ η_{2} + β_{1} ∧ η_{1}}}.

Thus, we obtain

h^{∧}_{2}(𝔎^{∗}) = 8.  (23)

On the other hand, from Theorem 2 and Theorem 7 we have h^{∨}_{2}(𝔎^{∗}) = h^{∨}_{1}(𝔎^{∗}). Then, from (22) we conclude

h^{∨}_{2}(𝔎^{∗}) = 7.  (24)

Now, from (20), (23), (24) and (15) it follows that

b_{2}(M) = 14 < h^{∨}_{2}(𝔎^{∗}) + h^{∧}_{2}(𝔎^{∗}) = 15 ≤ h^{∨}_{2}(M) + h^{∧}_{2}(M),

which completes the proof.

Finally, we show an example of a compact nearly parallel G_{2}-manifold N for which

H^{3}(N) ≅ H̸^{∨}^{3}(N) ⊕ H^{∧}^{3}(N), and H^{4}(N) ≅ H̸^{∧}^{4}(N). It is not a G_{2}-nilmanifold.

In general [G2], for an orientable hypersurface N of ℝ^{8}, with unit normal vector field U on N, there is a 2-fold vector cross product P on N given by

P (X, Y) = \overline{P}_{±}(U, X, Y),  (25)

for X, Y ∈ 𝔛(N), where \overline{P}_{±} denote the two non-isomorphic 3-fold vector cross products on ℝ^{8} [Z].

Proposition 9. [FG] Let N be an orientable hypersurface of ℝ^{8} with 2-fold vector cross product P given by (25). Then

i) N is cocalibrated G_{2}-manifold,

ii) N is parallel G_{2}-manifold if N is totally geodesic,

iii) N is nearly parallel G_{2}-manifold if N is totally umbilic (i.e. N is a part of a sphere).

Theorem 10. Let S^{7} be the 7-sphere with 2-fold vector cross product given by (25).

Then S^{7} is a nearly parallel (non-parallel) G_{2}-manifold with H^{3}(S^{7}) ≅ H̸^{∨}^{3}(S^{7}) ⊕ H^{∧}^{3}(S^{7})

and H^{4}(S^{7}) ≅ H̸^{∧}^{4}(S^{7}).

Proof. From Proposition 9 it follows that S^{7} is a nearly parallel (non-parallel) G_{2}-manifold.

If (B^{∗}(S^{7}), D^{∧}) and (A^{∗}(S^{7}), D^{∨}) denote the differential complexes (5) and (9), respectively, then the short exact sequence of complexes

0 ⟶ (B^{∗}(S^{7}), D^{∧}) ⟶^{i} (Λ^{∗}(S^{7}), d) ⟶^{p} (A^{∗}(S^{7}), D^{∨}) ⟶ 0,

where i and p are the natural inclusion and the orthogonal projection, respectively, induces the long exact sequence of cohomology groups

\begin{matrix}0 & ⟶ & H^{1}(S^{7}) & ⟶^{H^{1}(p)} & H^{∨}^{1}(S^{7}) & ⟶^{H^{1}(d)} \\ H^{∧}^{2}(S^{7}) & ⟶^{H^{2}(i)} & H^{2}(S^{7}) & ⟶^{H^{2}(p)} & H^{∨}^{2}(S^{7}) & ⟶^{H^{2}(d)} \\ H^{∧}^{3}(S^{7}) & ⟶^{H^{3}(i)} & H^{3}(S^{7}) & ⟶^{H^{3}(p)} & H^{∨}^{3}(S^{7}) & ⟶^{H^{3}(d)} \\ H^{∧}^{4}(S^{7}) & ⟶^{H^{4}(i)} & H^{4}(S^{7}) & ⟶ & 0 . \end{matrix}  (26)

Here H^{q}(i) and H^{q}(p) are the naturally induced maps, and H^{q}(d) is the connecting

homomorphism given by H^{q}(d)([α]) = {dα} ∈ H^{∧}^{q+1}(S^{7}), for [α] ∈ H^{∨}^{q}(S^{7}).

Since H^{q}(S^{7}) = 0 for 0 < q < 7, from (26) it follows that H^{q}(d) is an isomorphism for q = 1, 2, 3. Thus, from Theorem 2 and Theorem 3, we obtain

H^{∨}^{1}(S^{7}) ≅ H^{∨}^{2}(S^{7}) ≅ H^{∧}^{2}(S^{7}) ≅ H^{∧}^{3}(S^{7}) ≅ {0},
H^{∨}^{3}(S^{7}) ≅ H^{∧}^{4}(S^{7}) ≅ ℝ.

Therefore, we have H^{∨}^{3}(S^{7}) ⊕ H^{∧}^{3}(S^{7}) ≅ ℝ ≅ H̸^{3}(S^{7}) = {0} and H^{∧}^{4}(S^{7}) ≅ ℝ ≅ H̸^{4}(S^{7}) =

{0}.

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Authors’ addresses: Marisa Fernández: Departamento de Matemáticas Facultad de Ciencias Universidad del País Vasco Apartado 644 48080 Bilbao, SPAIN

Luis Ugarte: Departamento de Matemáticas (Geometría y Topología) Facultad de Ciencias Universidad de Zaragoza Campus Plaza San Francisco 50009 Zaragoza, SPAIN

E-mails: Marisa Fernández: mtpferol@lg.ehu.es Luis Ugarte: mtpugvil@lg.ehu.es , ugarte@posta.unizar.es