Topology of Almost Complex Structures on Six-Manifolds

. We study the space of (orthogonal) almost complex structures on closed six-dimensional manifolds as the space of sections of the twistor space for a given metric. For a connected six-manifold with vanishing ﬁrst Betti number, we express the space of almost complex structures as a quotient of the space of sections of a seven-sphere bundle over the manifold by a circle action, and then use this description to compute the rational homotopy theoretic minimal model of the components that satisfy a certain Chern number condition. We further obtain a formula for the homological intersection number of two sections of the twistor space in terms of the Chern classes of the corresponding almost complex structures.


Introduction
A major open problem in differential geometry is to determine whether a closed almost complex manifold of dimension at least six always admits an integrable complex structure.By a celebrated theorem of Newlander and Nirenberg, an almost complex structure is integrable if and only if it satisfies a certain system of first order partial differential equations codified by the vanishing of the Nijenhuis tensor.An understanding of the topology of the space of all almost complex structures may be useful in the search for those which are integrable.We take this as motivation for our study of the topology of the space of almost complex structures on a sixmanifold.The present work is a continuation of [13] which focused on the six-sphere and touched upon the case of almost complex structures on six-manifolds with vanishing first Chern class.
Let M denote an oriented six-manifold and J (M ) be the space of all almost complex structures on M inducing the given orientation.Fixing a Riemannian metric on M , the inclusion J(M ) → J (M ) of the subspace of almost complex structures which are orthogonal with respect to the metric is a homotopy equivalence, allowing us to analyze J (M ) via a space of sections of a fiber bundle with compact fibers.Generally, over an oriented Riemannian 2n-manifold M one can consider the SO(2n)/U(n) bundle Z + (M ) → M whose fiber over any x ∈ M is the set of (linear) orthogonal complex structures on T x M compatible with the orientation on M ; the space Z + (M ) is known as the (positive) twistor space of M .Sections of this bundle correspond to orthogonal almost complex structures on M compatible with the orientation.
The twistor space construction can be done in any dimension, but six-manifolds enjoy the special property that the fiber of the twistor space is likewise six-dimensional.Hence two orthogonal almost complex structures on a closed Riemannian manifold M give embeddings of the manifold into its twistor space which generically intersect at a finite number of points.One of our main results (Theorem 5.10) computes this intersection number in terms of the Chern classes of the two almost complex structures.Namely, denoting by c i and c ′ i the respective Chern classes, the homological intersection number is given by In particular, the homological self-intersection of an almost complex structure is given by M c 1 c 2 − c 3 .Employing a theorem of Michelsohn and Salamon, we observe that negative self-intersection restricts the possible deformations of an orthogonal complex structure (Corollary 5.6).
In order to study the homotopy type of J(M ), in Section 3 we use the existence of spin c structures on oriented four-manifolds and almost complex six-manifolds to describe the integral cohomology of their twistor spaces.This will include both the positive and the negative twistor space Z − (M ), i.e., the total space of the bundle of linear complex structures inducing the opposite orientation on the tangent space.Presumably these results are well known, but we were not able to find them in the literature in this generality.
We take particular care with orientations throughout, due to varying historical conventions (e.g., the original [2], and [16], refer to the negative twistor space as the "twistor space", while this terminology is reserved for the positive twistor space for instance in [17] 1 and [23]), and to the authors' difficulty in verifying some formulas in [16, p. 135]; see Remark 3.9.
The twistor spaces Z + (M ) and Z − (M ) (whose homotopy types generally differ when the dimension of M is divisible by four) carry two natural almost complex structures, differing by the choice of induced orientation on the fiber (see, e.g., [23,Proposition 3.1]), known as the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures.Even though the standard complex structure and its negative on the fibers SO(4)/U(2) ∼ = CP 1 and SO(6)/U(3) ∼ = CP 3 are biholomorphic via complex conjugation, the Eells-Salamon almost complex structure is never integrable [23,Proposition 3.4], in contrast to the Atiyah-Hitchin-Singer structure (this phenomenon is not unique to dimensions four and six).
Our second main result is a description of the rational homotopy type of (a given component of) the space of almost complex structures on six-manifolds, under the additional assumption that b 1 (M ) = 0 and M c 1 c 2 − c 3 ̸ = 0 (Theorem 4.4).For this we use an expression of J(M ) as the quotient of the space of sections of an S 7 -bundle by an S 1 -action (Theorem 4.1), which one can compare with the result obtained for c 1 = 0 in [13, Proposition 5.1].Throughout we provide examples for all of the mentioned results.

Notation and conventions
The projectivization of a complex vector bundle E is denoted by P(E).We denote the Chern classes of a complex vector bundle by c i , and the Pontryagin classes of a real vector bundle by p i ; if the vector bundle is complex, by its Pontryagin classes we mean those of the underlying real bundle.The Euler class is denoted by e.The Euler characteristic of a space is denoted by χ, and σ is the signature of an oriented closed manifold.All manifolds are assumed connected unless otherwise stated.
We denote by Γ(E) the space of sections of a bundle E → M (where E is not necessarily a vector bundle).If s : M → E is a section, Γ(E) s denotes the connected component of s in Γ(E).For E → M a vector bundle with a metric, we write S(E) → M for the associated sphere bundle.We denote by J(M ) the connected component of a specific point in Γ(Z + (M )), which will be clear from context.For a vector space V with an inner product, J(V ) will denote the space of all orthogonal (linear) complex structures on V .If V is oriented, J + (V ) (respectively J − (V )) denotes the connected component of J(V ) consisting of elements which induce the given orientation on V (respectively the opposite orientation).By Map(X, Y ) we denote the space of unbased maps X → Y with the compact-open topology.
Cohomology is singular cohomology with integral coefficients unless another choice of coefficients is indicated.The evaluation of a cohomology class α on a homology class A is denoted by ⟨α, A⟩ or by A α.In the context of rational homotopy theoretic models, Λ(x i ) denotes the free graded-commutative algebra on generators x i .
We will make frequent use of the projective bundle formula, which states that for a complex vector bundle E → M of complex rank k, the cohomology of the projectivized bundle P(E) p − → M is the free H * (M )-algebra on one generator x subject only to the relation where x is the first Chern class of the line bundle O E * (1) (see [8,Definition 15.13]).This line bundle is dual to the tautological (Hopf) line subbundle of p * (E) whose fiber over For a real vector space V with an endomorphism J squaring to −Id, we denote by Λ * C (V ) the (complex) exterior algebra of the complex vector space (V, J).For an almost complex manifold (M, J), Λ * C (T M ) is the (complex) exterior bundle associated to (T M, J).

Preliminaries
In this section we review some of the models for the space of orthogonal complex structures on a Euclidean space following [17, Section IV.9], to which we refer for further details.Let (V, (•, •)) be a 2n-dimensional real inner product space.We will write for the space of orthogonal complex structures on V .This space naturally has the structure of a complex algebraic variety, given by the isomorphism ) with the Grassmannian of n-dimensional complex subspaces of V ⊗ C, which are isotropic with respect to the complex bilinear form on V ⊗ C obtained from the inner product on V by linear extension.The map ϕ is defined by There is a tautologous complex vector bundle on J(V ) defined by where the fiber over J ∈ J(V ) is given the complex structure determined by J.The isomorphism ϕ is covered by a bundle map to the dual (or conjugate) tautological bundle over the Grassmannian.
Lemma 2.1.Consider the universal bundle Then (i) ι classifies the tautologous complex vector bundle on J R 2n .
(ii) ι * (c 1 ) generates a subgroup of index 2 in the infinite cyclic second cohomology group of each of the two components of J R 2n .
(iii) ι * (c 1 ) is positive.In particular, ι * (c 1 ) evaluates on the orientation class of each of the two components of J R 2n to a positive integer.
Proof .(i) The universal bundle can be realized by taking BU(n) = EO(2n)/U(n).The universal bundle over BU(n) is then given by EO(2n) × U(n) R 2n → EO(2n)/U(n), and pulls back along the inclusion of the fiber ι to O(2n) × U(n) R 2n → O(2n)/U(n), which is the homogeneous expression of the tautologous bundle.
(ii) The inclusion-induced map BU(n) → BO(2n) factors through the double cover BSO(2n) → BO(2n), and the two components of O(2n)/U(n) are, respectively, the fiber SO(2n)/U(n) of BU(n) → BSO(2n) and the image of this fiber under a lift to BU(n) of the nontrivial deck transformation of BSO(2n).Therefore it suffices to prove the statement for the component SO(2n)/U(n).This follows from the Serre spectral sequence of the bundle BU(n) → BSO(2n) (see for instance [20,Theorem III.6.11]).
(iii) ι * (c 1 ) is the pullback of c 1 of the dual tautological bundle under the embedding The n th exterior power of the dual tautological bundle on Gr C 2n gives the Plücker embedding of the Grassmannian in projective space.Hence ι * (c 1 ) is the first Chern class of a line bundle on J R 2n whose sections embed J R 2n in projective space and is therefore positive.■ The above identification of orthogonal complex structures with isotropic planes in V ⊗ C leads to another description of the space J(V ) which will play an important role in this paper.
Let Cl(V ) denote the Clifford algebra determined by the quadratic form q(v) = −∥v∥ 2 , and let Cl(V ) = Cl(V ) ⊗ R C denote its complexification.Let S C denote an irreducible (2 n -dimensional) Cl(V )-module.
For each element, i.e., spinor, σ ∈ S C we have the map V ⊗ R C → S C given by right Clifford multiplication by σ.A spinor is said to be pure if the associated kernel is half-dimensional.Let PS C ⊂ S C denote the subset of pure spinors.As the subspace ker(•σ) ⊂ V ⊗C is isotropic there is a natural map from the projectivization of the set of pure spinors to the isotropic Grassmannian defined by [σ] → ker(•σ).This is a map of algebraic varieties and is in fact an SO(2n)-equivariant isomorphism [17, Proposition IV.9.7].The space J(V ) has two connected components corresponding to the two possible orientations induced by the complex structure.A choice of orientation for V leads to a decomposition with J + (V ) the component corresponding to the given orientation.In turn this decomposes the Grassmannian of isotropic subspaces into the components of positive and negative isotropic subspaces.
On the level of spinors this decomposition takes the following form.The complex volume element in Cl(V ) is the element ω C = i n e 1 • • • e 2n , where {e i } is any oriented orthonormal basis.As ω 2 C = 1, the volume element decomposes the module S C into a direct sum of +1 and −1 eigenspaces The elements of the summands are called the positive and negative spinors, respectively.Note that, since ω C anti-commutes with the action of V ⊂ Cl(V ), both S + C and S − C are spin c representations, called the positive and negative spinor representations, respectively.
Every pure spinor is necessarily either positive or negative and the isomorphisms ϕ, ψ give rise to isomorphisms of algebraic varieties In dimensions 2n = 4, 6 all nonzero spinors are pure [17, Remark IV.9.12] and the isomorphisms identify the spaces of orthogonal complex structure with the disjoint union of the projectivization of positive and negative spinors.
A point J ∈ J(V ) gives rise to a useful description (cf.[17, Chapter I, equation (5.27)]) of the irreducible module S C .We will write ⟨•, •⟩ for the Hermitian inner product on the complex vector space (V, J) whose real part is (•, •).Let Λ * C (V ) denote the exterior algebra of the complex vector space (V, J).Contraction of a vector v ∈ V with ω ∈ Λ k C (V ) is determined by the expression as one easily checks using an orthonormal basis for V having a real multiple of a nonzero v as its first element.The complex linear extension of this action gives Λ * C (V ) the structure of a Cl(V )-module of complex dimension 2 n .This is the dimension of the irreducible complex Cl(V )-module, so Λ * C (V ) is a convenient description of this module.The following is an easy computation which we include for convenience.Lemma 2.2.Let (V, J) be an oriented 2n-dimensional Euclidean real vector space with an orthogonal complex structure J compatible with the orientation and let S C = Λ * C (V ) be the irreducible Cl(V )-module described above.Then Proof .Let e 1 , . . ., e n be an orthonormal basis for (V, J).Then we can write the complex volume element as

Consider the basis e
which completes the proof.■ Remark 2.3.For (V, J) an oriented 2n-dimensional Euclidean real vector space with an orthogonal complex structure J compatible with the orientation, the line in

Twistor spaces of four-and six-manifolds
In this section we describe the integral cohomology of the twistor space of (Riemannian) almost complex six-manifolds, along with the integral cohomology of the twistor space of arbitrary oriented (Riemannian) four-manifolds.Finally we compute the Chern classes of the Atiyah-Hitchin-Singer almost complex structure on the latter.

Spinors on almost complex four-and six-manifolds
Let M be an oriented Riemannian manifold and let Z(M ) denote the space of orthogonal almost complex structures on M , called the twistor space.The space Z(M ) has two components corresponding to the structures which induce the given orientation on M or the opposite orientation.Recall, we denote these by Z ± (M ), and refer to them as the positive and negative twistor spaces.
The Atiyah-Hitchin-Singer almost complex structure on Z(M ) is defined as follows: the Levi-Civita connection induces a connection on the bundle Z(M ) → M splitting T Z(M ) into a direct sum of horizontal and vertical subspaces The almost complex structure on T v Z(M ) is determined by the algebraic variety structure of the fibers J(T x M ) of Z(M ) → M explained in Section 2. At a point J x in the fiber over x ∈ M the complex structure on The discussion in the previous section then provides a one-to-one correspondence between projectivized pure spinors on M and orthogonal complex structures inducing a given orientation: . Now suppose M is given an orthogonal almost complex structure J compatible with the orientation.Then J gives rise to a canonical spin c structure on M via the group homomomorphism U(n) κ − → Spin c (2n), which is the unique lift as a group homomorphism in the following diagram ).Using the formulas for the action of e i Je i in the proof of Lemma 2.2, we see that the spinor action of κ(g) on Λ * C R 2n , where R 2n is equipped with the complex structure given by diag 0 −1 1 0 , . . ., 0 −1 1 0 , agrees with the standard action of g on this space.We obtain the following result: Proposition 3.1.Let M be an oriented Riemannian manifold with an orthogonal almost complex structure J compatible with the orientation.Then we have the following isomorphisms of smooth fiber bundles over M : Remark 3.2.The Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures on (either) twistor space do not agree with the natural almost complex structure on the projectivization.Indeed, the projection map from the projectivization is complex linear whilst this is not the case for the twistor projection.
The projective bundle formula now gives the following formulas for the cohomology rings of the components of the twistor space in terms of the Chern classes of (T M, J).
Corollary 3.3.Let M be an oriented Riemannian manifold with an orthogonal almost complex structure J compatible with the orientation.We have the following isomorphisms as algebras over H * (M ): Remark 3.4.
The formulas above in dimension four agree with [11,Theorem 11.2], where it is established that over a Kähler surface M , the positive twistor space is the projectivization of T 2,0 M ⊕C (note that T 2,0 M ∼ = Λ 2 C (T M )), and the negative twistor space is the projectivization of the holomorphic tangent bundle.
In [12,Section 6], Evans describes the cohomology ring of the positive twistor space of a (not necessarily almost complex) six-manifold, with complex coefficients.The description is in terms of the first Chern class of T h (Z + (M )).Using Proposition 3.8, one can check that Evans' description agrees with the one in Corollary 3.3 in the case of almost complex manifolds.We remark that Evans' computation can quickly be reproduced using naturality in the pullback diagram and expressing H * (BU(3); Q) as an H * (BSO(6); Q)-algebra using the known behavior of the right-hand side vertical map on cohomology.
Remark 3.5.In order for the formulas of Corollary 3.3(1) to hold, it is crucial to have an almost complex structure inducing the given orientation on the base manifold.For instance, the positive twistor space of CP 2 (i.e., the complex projective plane with the opposite orientation) with the Fubini-Study metric is the full flag variety U(3)/ (U(1) × U(1) × U( 1)) [23, p. 217, Example 1], whose cohomology is given by where x, y, z are in degree two [4,Proposition 31.1].This ring is not isomorphic to for any integer k, so the cohomology of Z + CP 2 can not be expressed as in the first item of Corollary 3.3 (1).This corresponds to the fact that CP 2 does not admit an almost complex structure compatible with its orientation (which can of course be seen by other means as well, e.g., Hirzebruch's congruence χ + σ ≡ 0 mod 4 for closed almost complex four-manifolds).

Spin c (4)-bundles and twistor spaces of four-dimensional manifolds
Twistor spaces of oriented Riemannian four-manifolds give interesting examples of almost complex six-manifolds which we will consider in the following sections.In this subsection we will compute their integral cohomology as well as the Chern classes of their Atiyah-Hitchin-Singer almost complex structures.Our computation requires knowledge of the integral cohomology of BSpin c (4).The cohomology of BSpin c (n) is described in detail in [9].We need only the simple case when n = 4, so we include an elementary treatment of this case.
Our aim is to compute the cohomology ring of the "universal twistor space" over BSpin c (4), namely the bundle of orthogonal complex structures on the fibers of the 4-plane bundle over BSpin c (4) classified by the canonical map BSpin c (4) → BSO(4).Since every oriented fourmanifold admits a spin c structure, naturality will yield the cohomology ring of the twistor space.
We will use unit quaternions to describe the four-dimensional spin groups and orthogonal complex structures.We consider the quaternions H as a vector space over C via right multiplication by C ⊂ H, i.e., we consider the identification of C 2 with H given by The standard complex structure on H = R 4 is then right multiplication by i, and (1, i, j, −k) is an oriented basis.
The unit quaternion q = w 1 + jw 2 acts on H by left multiplication as which corresponds via the identification with C 2 to the matrix This will be our identification of Sp(1) with SU(2).
The canonical map U(2) In terms of the coordinates above, this is given by κ q 1 , e iθ = q 1 , e iθ , e −iθ .
We will write p 1 and e for the universal Pontryagin class and Euler class in H * (BSO(4)), respectively.

Bπ
where Z(BG) denotes the bundle of orthogonal complex structures on the universal oriented 4-plane bundle over BG for G = SO(4) and its pullback for G = Spin c (4).We will see that each of the two components of Z BSpin c (4) is the projectivization of a complex plane bundle over BSpin c (4).The projective bundle formula together with naturality will give us the following description of the cohomology ring of Z(M ) as an H * (M )-algebra.Proposition 3.7.Let M be an oriented Riemannian four-manifold and α ∈ H 2 (M ) be an integral lift of w 2 (M ) (classifying the complex line bundle associated to a spin c -structure on M ).Then there is an isomorphism of H * (M )-algebras Proof .Under our identification of R 4 with H, the orthogonal complex structures on R 4 compatible with the orientation (respectively, opposite orientation) are given by right (respectively, left) multiplication by a unit imaginary quaternion.The element [q 1 , q 2 ] ∈ SO(4) acts on J + R 4 by [q 1 , q 2 ]u = q 2 uq 2 , and on J − R 4 by [q 1 , q 2 ]u = q 1 uq 1 .
That is, SO(4) acts on orthogonal complex structures via the two projections to SO(3).The commutative diagrams Spin c (4) U(2) q 1 , q 2 , e iθ q 2 , e −iθ SO( 4) SO(3), q 1 , q 2 q 2 .π χ + π right and Spin c (4) U(2) q 1 , q 2 , e iθ q 1 , e −iθ SO(4) SO(3), q 1 , q 2 q 1 .π χ − π left allow us to express each component of the bundle of orthogonal complex structures associated to an oriented 4-plane bundle with a spin c structure as the projectivization of a complex plane bundle.The formula (3.3) will follow from the projective bundle formula once we compute the Chern classes of these plane bundles.By naturality it suffices to compute the images of c 1 , c 2 under Bχ * ± .As the composition Spin c (4) In view of Lemma 3.6, this will complete the proof.The maps are given respectively by (q 1 , q 2 ) → [q 2 , 1] and (q 1 , q 2 ) → [q 1 , 1] (i.e., they correspond respectively to the inclusions of the right and left copies of SU(2) in U( 2)).Therefore, with respect to the standard basis for H 4 (BSp(1) × BSp(1)) used in the proof of Lemma 3.6, we have for some λ, µ ∈ Z.
We can determine the coefficients λ and µ by mapping to are given by q, e iθ → e iθ , e iθ and q, e iθ → q, e iθ , respectively.The second map is the identity, while the first map is the representation U(2) Recall that T v Z ± (M ) and T h Z ± (M ) denote the vertical and horizontal subbundles of T Z ± (M ) which are complex vector bundles via the Atiyah-Hitchin-Singer almost complex structure.
Proposition 3.8.Let M be an oriented Riemannian four-manifold and α ∈ H 2 (M ) a choice of lift of w 2 (M ).Then, in terms of the expression for H * (Z ± (M )) in Proposition 3.7, the total Chern class of the standard almost complex structure on Z ± (M ) is (3.4) Proof .It suffices to compute the total Chern classes of the vertical and horizontal subbundles of T Z ± (M ).We saw in the proof of Proposition 3.7 that Z ± (M ) is the projectivization of a rank two complex vector bundle E with total Chern class As As for the horizontal bundle, consider the pullback diagrams M BSpin c (4) BSO(4).).We saw in the proof of Proposition 3.7 that In terms of this identification we have τ * (c 2 ) = Bπ * e = −S 1 + S 2 and τ * (c 1 ) = γx + λα for some γ, λ ∈ Z.
Let k ∈ O(4) be an element with determinant −1 and let φ : SO(4) → SO(4) denote conjugation by k.As a homogeneous space of SO(4) we have 2)) and we have a commutative diagram M BSpin c (4) BSO( 4) BSO( 4) with the middle and left-hand squares both pullback squares and the right hand square an isomorphism of fiber bundles induced by the automorphism φ −1 .Note that Bφ −1 is covered by an isomorphism between the (complex) universal bundles ESO(4 Considering the action of Bφ on the maximal torus we see that Bφ : BSO(4) → BSO( 4) has the following effect on cohomology: Arguing as before, we conclude that τ * (c 2 ) = −Bπ * (e) = S 1 −S 2 and τ * (c 1 ) = 2x+α, leading to the formula (3.4) in the case of Z − (M ).■ Remark 3.9.
The formulas in Proposition 3. In [16, equation (1.4)], there is a description of the real cohomology ring of the negative twistor space of a four-manifold X, as the free H * (X; R)-module generated by h = 1 2 c 1 T v Z − (X) subject only to the relation h 2 = 1 2 e(X) − 1 4 p 1 (X).However, this description is at odds with the identification of the negative twistor space of CP 2 as the full flag variety U(3)/ (U(1) × U(1) × U(1)) [16, p. 133], [23, p. 217, Example 1].Namely, H * CP 2 ; R ∼ = R[x]/ x 3 , where x 2 , CP 2 = 1, so e CP 2 = p 1 CP 2 = 3x 2 .Therefore the real cohomology of the negative twistor space is, according to [16, equation (1.4) One can check directly, however, that this (graded) ring is not isomorphic to the cohomology of the flag variety, i.e., to R In terms of Propositions 3.7 and 3.8, we have h = x+ α 2 and therefore h 2 = 1 4 p 1 (x)− 1 2 e(X).Remark 3.10.Let M be an almost complex Riemannian six-manifold.The fiberwise diffeomorphism a : ) induces an anti-holomorphic map between the fibers and hence The spaces Z ± (M ) are projectivizations of rank 4 complex vector bundles with first Chern class 2c 1 (M ) (see Corollary 3.3), and so as in Proposition 3.8 we see that c 1 (T v Z ± (M )) = 2c 1 (M )+4x.Therefore As there is no torsion in H 2 of the universal example Z(BU(3)), it follows that a * (x) = −x − c 1 .One can check that a * yields an isomorphism between the H * (M )-algebras of Corollary 3.3.
In view of the previous remark, from now on we will restrict our attention to the positive twistor space of almost complex Riemannian six-manifolds.
4 On the homotopy type of the space of almost complex structures on six-manifolds In this section we study the homotopy type of the components of the space of almost complex structures on connected six-manifolds M , complementing our previous treatment [13] of the case when c 1 = 0.
Theorem 4.1.Let M be an oriented Riemannian manifold of dimension four or six, J an orthogonal almost complex structure on M compatible with the orientation, and J(M ) its component in the space of orthogonal almost complex structures.Let S + C (M ) be the positive spinor bundle on M determined by J. Then the map Γ S S + C (M ) → J(M ) is surjective, and for any lift s of J, is a fiber sequence.
M ) defined by s(x) = 1 lifts the section J.As S S + C → Z + (M ) is a fiberwise fibration over M , the map induced on sections is a fibration.Since the fiber over J is nonempty, any choice of lift of J identifies the fiber with Map M, S 1 .■ Remark 4.2.The above argument remains valid in all dimensions if one replaces S S + C (M ) with S S + C (M ) ∩ PS + C (M ).Now let (M, J) be a closed almost complex Riemannian six-manifold with H 1 (M ; Z) = 0 (equivalently, b 1 = 0) satisfying c 1 c 2 − c 3 ̸ = 0.By Theorem 4.1, we have the fibration Map M, S 1 → Γ S S + C (M ) s → J(M ).Since H 1 (M ; Z) = 0, evaluation at a chosen basepoint gives a homotopy equivalence Map M, S 1 ev − → S 1 , so we have a principal fiber sequence S 1 → Γ S S + C (M ) s → J(M ).We can thus consider instead the fiber sequence Now, S S + C is an oriented fiber bundle over M with fiber S 7 , and hence it is classified by a map M → BAut + S 7 to the classifying space of orientation-preserving homotopy automorphisms of S 7 .It is known that , where the subscript of Q denotes (Sullivan) rationalization [25,Section 11].By [21, Theorem 5.3], the rationalization of Γ S S + C s is homotopy equivalent to (the connected component corresponding to s in) the space of sections of the fiberwise rationalized S 7 bundle over M .Since H 8 (M ; Q) = 0, this latter bundle is trivial, and hence Γ S S + C s Q is homotopy equivalent to a connected component of Map M, S 7 Q .Denoting the Betti numbers of M by b i , by Thom's theorem on the space of maps into an Eilenberg-Maclane space, Map M, S 7  Q is in fact connected and has the homotopy type of We can describe the fundamental group of J(M ) by using a theorem of Crabb and Sutherland: Theorem 4.3 ([7, Theorem 2.12(i)]).Let M be a closed connected 2n-manifold and ξ a complex (n + 1)-plane bundle over M .Denote by N ξ any component of the space of sections of P(ξ) whose elements lift to sections of ξ.Then π 1 (N ξ) is a central extension By Theorem 4.1, we can apply this to ξ = S + C (M ) ∼ = C ⊕ Λ 2 C T M , which satisfies N ξ = J(M ).Therefore, the fundamental group of J(M ) is given by Z modulo M c 1 c 2 −c 3 (where c i = c i (T M )), see Corollary 3.3.By [21], J(M ) is a nilpotent space as it is the space of sections of a fibration with nilpotent fiber namely CP 3 over a finite-dimensional base.We may thus rationalize the fibration (4.1) to obtain the fibration Consider the degree one rational class corresponding to the factor S 1 Q in the fiber.If c 1 c 2 −c 3 ̸ = 0, then since π 1 (J(M ) Q ) = 0, this class must hit (a nonzero multiple of) the degree two generator in K(Q, 2) in the Serre spectral sequence.A model for the total space is given by the tensor product of the base and fiber's models with a perturbed differential on the fiber generators [25,Section 4]; the previous sentence thus tells us that a model for J(M ) is of the form where (x 2 ) denotes the ideal generated by x 2 , and i, j, k range over sets of size b 4 , b 3 , b 2 , respectively.
Notice that such a model is not minimal, and in fact a minimal model is obtained by quotienting out the differential ideal generated by z 1 .Indeed, by the argument in [27, Proposition 2] we see that the map (Λ, d) → Λ/(z 1 , dz 1 )Λ, d induces an isomorphism on cohomology.Here by (Λ, d) we denote the differential graded algebra displayed above, and by d the induced differential on the quotient.We see that where the right-hand side is minimal.To summarize, we have the following: We emphasize that the above makes no nilpotency assumption on M .Alternatively, a model for (a given component of) the space of almost complex structures can be obtained with the Haefliger-Sullivan model for the space of sections of a fibration [25,Section 11], [15], applied to CP 3 → Z + (M ) → M .Our approach above lets one quickly describe the minimal model of J(M ) using the geometry of the setup.1. Observe that the space of almost complex structures inducing a given orientation on the connected sum g S 3 × S 3 is connected.By Theorem 4.4, for g ̸ = 1, the rationalization of this space is K(Q, 7) × K(Q, 4) 2g .In particular, the space of almost complex structures on S 6 has the same rational homotopy type as S 7 , i.e., K(Q, 7); cf.[13] where it is shown that a certain natural inclusion RP 7 → J S 6 induces an isomorphism on rational homotopy groups and fundamental groups.(Note that the covering S 7 → RP 7 induces an isomorphism on rational homotopy groups.) For g = 1, one can calculate directly using the Haefliger-Sullivan model that the space of almost complex structures on S 3 × S 3 has the same rational homotopy type as 2. The components of almost complex structures on CP3 are parametrized by c 1 = 2kx, c 2 = 2k 2 − 2 x 2 , where x is the generator of H 2 CP 3 such that x 3 , CP 3 = 1 (i.e., x = c 1 (O(1)) for the standard complex structure).Hence c 1 c 2 − c 3 = 4k 3 − 4k − 4, which is never zero.Hence every component of almost complex structures has rationalization K(Q, 7) × K(Q, 5) × K(Q, 3).Contrast this with the trivial CP 3 bundle over CP 3 , whose (infinitely many) components of sections exhibit two distinct rational homotopy types [22,Example 3.4]; the space of maps CP 3 → CP 3 homotopic to any fixed essential map has rationalization K(Q, 7) × K(Q, 5) × K(Q, 3).
5 Intersections of almost complex structures on six-manifolds as sections of the twistor space We will now use the results of Section 3 to calculate the homological intersection of two almost complex structures on a given six-manifold M , after identifying them with the image of M under the corresponding section of the twistor bundle.
where c i denote the Chern classes of E. 3   Proof .The Poincaré dual to s * [M ] is the image of the Thom class of the normal bundle to the submanifold s(M ) ⊂ P(E ⊕ C) under the first map below in the long exact sequence of a pair, where H 2n (P(E ⊕ C), P(E ⊕ C) \ s(M )) has been identified with H 2n (U, U \ s(M )) for some tubular neighborhood U of s(M ) via excision; see [5,Section 6.11].This is a nonzero class, as s * [M ] is nonzero since it projects to [M ].Now notice that P(E ⊕ C) \ s(M ) deformation retracts to P(E).Therefore the effect of the second map on cohomology is the natural quotient and thus the kernel is generated by Hence the Thom class maps to To check the sign, we restrict to a fiber and evaluate against the fundamental class CP n .Since the Thom class is the orientation of the normal bundle, which is given by the orientation of the fibers, and x n , CP n = 1, we see that the sign is positive.■ Proof .In terms of the identification in Proposition 3.1, s is the canonical section of P S + C = P Λ 2 C T M ⊕C (see the proof of Theorem 4.1), where T M is equipped with the complex structure determined by s.To compute the homological self-intersection of s(M ), we integrate the square of the expression obtained in Lemma 5.1 for its Poincaré dual in Z + (M ).We use Corollary 3.3, and observe that since Using Theorem 5.2, we obtain the following: Example 5.3.
1.For any closed oriented Riemannian four-manifold M , by Proposition 3.8, we have the following Chern numbers of the Atiyah-Hitchin-Singer almost complex structure on Z + (M ): Hence the self-intersection of the Atiyah-Hitchin-Singer almost complex structure on By [1, Lemma 3], closed oriented four-manifolds admitting a maximally nonintegrable almost complex structure satisfy 5χ+6σ = 0. Hence for these manifolds, the corresponding self-intersection number is zero.
2. By a theorem of Taubes [26, Theorem 1.1], given any closed oriented four-manifold M , for every large enough k the manifold N = M #kCP 2 carries a self-dual metric.Hence its negative twistor space Indeed, we obtain this again from Proposition 3.8, namely This agrees with [16, equation (1.5)].Alternatively, from [10, Section 4] we see that the holomorphic Euler characteristic of X is given by 1 , which equals the Todd genus X 1 24 c 1 c 2 (X).The Euler characteristic of X is twice that of N , as seen immediately, e.g., using the Leray-Hirsch theorem.
From here we obtain Hence for any M , for large enough k we obtain compact complex threefolds X with negative homological self-intersection in their own positive twistor space. of oriented 2-planes in R 8 , Calabi and Gluck [6] show that these "octonion" J's send S 6 to the family of 2-planes of the form (o, v), where o is a fixed unit octonion depending on J (so, the image of S 6 is the unit sphere in the hyperplane orthogonal to a given line in R 8 .)Two such nearby families intersect transversally.Therefore, the −2 homological self-intersection obtained in Example 5.4 shows the octonion J's are not integrable.Of course, this also follows by a direct computation of the Nijenhuis tensor, and the result is a special case of the classical theorem of Blanchard [3] and LeBrun [18] that there is no integrable complex structure on S 6 orthogonal with respect to the round metric.
In order to study intersections of almost complex structures in different components of the space of sections of the twistor space we will now analyze the dependence of our formula for the cohomology (Corollary 3.3) on the choice of almost complex structure.
As the canonical map κ : U(3) → Spin c (6) is an isomorphism on π 1 , we see that ȷ is an isomorphism on π 2 and hence the 5-lemma implies that Bκ is a 7-equivalence.Therefore the set of isomorphism classes of almost complex structures on six-manifolds is in natural bijective correspondence with the set of isomorphism classes of spin c structures via the canonical map from the former to the latter.The set of isomorphism classes of spin c -structures on a manifold M is a torsor over the group H 2 (M ) of isomorphism classes of complex line bundles (corresponding to the action of the fiber on the principal fibration along the right-hand column of (5.1)).Letting [s] denote the spin c structure associated to the almost complex structure s : M → Z + (M ), and writing a • [s] for the action of the cohomology class a ∈ H 2 (M ) on [s], we have the following result: Lemma 5.8.Let M be an oriented Riemannian six-manifold and let s, s ′ : M → Z + (M ) be two almost complex structures on M compatible with the orientation of M .Let a ∈ H 2 (M ) be (the unique cohomology class) such that [s ′ ] = a • [s].Then the canonical isomorphism between the expressions for H * (Z + (M )) given in Corollary 3.3 in terms of s, s ′ is given by Proof .Let E and E ′ be the positive spinor bundles associated to the spin c structures [s] and [s ′ ], respectively, and let L denote a complex line bundle with c 1 (L) = a.
The relation a • [s] = [s ′ ] implies that there is an isomorphism The kernel in T x M ⊗ C of right multiplication by a positive spinor v ∈ E x is the same as the kernel of right multiplication by λ ⊗ v ∈ L x ⊗ E x for any λ ̸ = 0, so the canonical isomorphism given by the composition is the obvious one sending the line L ⊗ (Cv) to Cv.
We conclude that on Z + (M ) the tautological (Hopf) line bundles H and H ′ coming respectively from the identifications with P(E) and P(E ′ ) satisfy the relation ) so the difference class a ∈ H 2 (M ) between two almost complex structures s, s ′ is a specific "square root" of the difference c ′ 1 − c 1 .We can suggestively write which is unambiguous on rational cohomology (or when there is no 2-torsion in H 2 (M )).
Proof .By Corollary 3.3, the presentation of H * (Z + (M )) with reference to J is given by and with reference to J ′ it is given by Combining Corollary 3.3 with Lemma 5.1, we have that the Poincaré dual of s * [M ] is given by and the Poincaré dual of s ′ * [M ] is given by Recall from (5.2) that x ′ = x + 1 2 (c 1 − c ′ 1 ).(Our calculation will be insensitive to torsion, so we may in fact work in the rational cohomology ring where there is no ambiguity in writing 1 2 (c 1 − c ′ 1 ).)Using this to rewrite the expression for s ′ * [M ], and multiplying with s * [M ], yields Pairing with the fundamental class of Z + (M ) gives the result.■

8 for c 2
and c 3 of Z − (M ) differ by a sign from those given in[16, p. 135]; however, the fundamental class being used in loc.cit.seems to also differ by a sign from the one induced by the Atiyah-Hitchin-Singer almost complex structure on Z − (M ).Hence the values of the Chern numbers given in[16, equation (1.5)] coincide with those obtained with the Chern classes in Proposition 3.8.

Theorem 4 . 4 .
Let M be a connected closed six-manifold with b 1 = 0 equipped with an almost complex structure J such that M c 1 c 2 − c 3 ̸ = 0. Then the space of almost complex structures on M in the component of J is a nilpotent space with finite cyclic fundamental group and minimal model given by Λ z 7 , z i 3 , z j 4 , z k 5 , d = 0 ; here i, j, k range over sets of size b 4 (M ), b 3 (M ), b 2 (M ) (= b 4 (M )), respectively.In particular, this space is formal.

Lemma 5 . 1 .
Let E π − → M be a complex vector bundle of rank n over a closed oriented manifold M .Let s : M → P(E ⊕ C) denote the canonical section given pointwise by [0 : • • • : 0 : 1].Then the Poincaré dual of s * [M ] in H *

Theorem 5 . 2 .
Let s be a section of the positive twistor bundle over a closed Riemannian sixmanifold M , and give M the orientation determined by s.Then the homological self-intersection number s * [M ] • s * [M ] of s(M ) in Z + (M ) is given by M c 1 c 2 − c 3 , where the c i are the Chern classes of the almost complex structure determined by s.