Four-dimensional symplectic manifold from fibered three-manifold

In this subsection, we apply Theorem 4.2 to calculate the primitive cohomologies for another class of examples: the symplectic 4-manifold which is the product of a fibered 3-manifold with a circle. Due to McMullen and Taubes [13], such a construction provides the first example of a manifold with inequivalent symplectic forms.

The input is a closed surface Σ with an orientation preserving self-diffeomorphism τ . The map τ is called the monodromy. By Moser’s trick, we may assume that there is a τ -invariant symplectic form ω_Σ on Σ. To be more precise, the monodromy τ might be replaced by another isotopic one. Denote by Yτ the mapping torus

Yτ = Y ×τS¹ = Σ × [0, 1]

(τ (x), 0) ∼ (x, 1) . (4.12)

There is a natural map from Y_τ to S¹ induced by the projection Σ × [0, 1] → [0, 1]. Let φ be the coordinate for the base of the fibration Yτ → S¹. Then, the 4-manifold X = S¹× Y_τ admits a symplectic form defined by

ω = dt ∧ dφ + ωΣ

where t is the coordinate for the S¹-factor of X.

The only interesting filtered/primitive cohomologies of a compact symplectic 4-manifold are P H_dd² Λ(X) and P H_d+d² Λ(X). Their dimensions are given by Corollary 4.5. As we will see momentarily, the Lefschetz map L on X is determined by the map of wedging with dφ on Y_τ. Let us start with the following useful linear algebra lemma.

Lemma 4.12. Let (V²ⁿ, Ω) be a symplectic vector space, and let A : V → V be a linear symplectomorphism. Then, the Ω-orthogonal complement of ker(A − 1) is im(A − 1), where 1 is the identity map on V . As a consequence, ker(A − 1) ∩ im(A − 1) is exactly the kernel of Ω|_ker(A−1).

Proof. Suppose that u ∈ ker(A − 1), which means that Au = u. For any v ∈ V , we compute Ω(Av − v, u) = Ω(Av, u) − Ω(v, u)

= Ω(Av, Au) − Ω(v, u) = 0 .

It follows that ker(A−1) and im(A−1) are Ω-orthogonal to each other. By dimension counting, they must be the ω-orthogonal complement of each other.

Proposition 4.13. Let Yτ be the 3-manifold defined by (4.12), and let dφ be the pull-back of the canonical 1-form from Y_τ → S¹. Then,

1. H_d¹(Yτ) ∼= span{dφ} ⊕ ker (τ^∗− 1) : H_d¹(Σ) → H_d¹(Σ)
;

2. H_d²(Yτ) ∼= span{ωΣ} ⊕ coker (τ^∗− 1) : H_d¹(Σ) → H_d¹(Σ)
;

3. with the above identifications, the kernel of wedging with dφ from H_d¹(Y_τ) to H_d²(Y_τ) is span{dφ} ⊕ ker(τ^∗− 1) ∩ im(τ^∗− 1)
.

Proof. These assertions basically follow from the Wang’s exact sequence:

· · · ^// H_d⁰(Σ) ^// H_d¹(Y_τ) ^// H_d¹(Σ) ^τ

∗−1// H_d¹(Σ) ^// H_d²(Y_τ) ^// H_d²(Σ) ^//· · · (4.13) which can be proved by the Mayer–Vietoris sequence.

Let us briefly explain how to construct elements of H_d¹(Y_τ) and H_d²(Y_τ) from the map τ^∗− 1. The general theory can be found in [2]. For any [γ] ∈ ker(τ^∗− 1) ⊂ H_d¹(Σ), there exists a function g₀ such that τ^∗γ − γ = dg₀ as a differential form. It gives an element in H_d¹(Y_τ) by ˜γ = (1 − χ)γ + χτ^∗γ + χ⁰g₀dφ where χ is a cut-off function in φ ∈ (−0.1, 1, 1) which is equal to 0 on (−0.1, 0.1) and is equal to 1 on (0.9, 1.1). Originally, the 1-form ˜γ is defined on (−0.1, 1.1) × Σ. It is invariant under the identification (x, φ) → (τ (x), φ − 1), and thus defines a 1-form on Yτ.

For any [η] ∈ H_d¹(Σ), it gives the following element in H_d²(Yτ): dφ ∧ (1 − χ)η + χτ^∗η
. The element is trivial in H_d²(Y_τ) if and only if [η] ∈ im(τ^∗− 1). More explicit computation will be done later in Section 6.

The de Rham cohomologies of the 4-manifold X = S¹× Y_τ can be found by the K¨unneth formula:

H_d¹(X) ∼= span{dt, dφ} ⊕ ker (τ^∗− 1) : H_d¹(Σ) → H_d¹(Σ)
, H_d²(X) ∼= dt ∧ H_d¹(Y_τ)
⊕ H_d²(Y_τ) ,

H_d³(X) ∼= span{dφ ∧ ωΣ, dt ∧ ω_Σ} ⊕ coker (τ^∗− 1) : H_d¹(Σ) → H_d¹(Σ)
.

For a compact symplectic 4-manifold, the only interesting Lefschetz map is the one from H_d¹(X) to H_d³(X). In the current case, the map is determined by the third item of Proposition 4.13.

With the help of Lemma 4.12, Theorem 4.2 leads to the following proposition.

Proposition 4.14. Suppose that Σ is a closed surface, τ is a monodromy, and ωΣ is a τ -invariant area form. Let X be S¹× Y_τ with the symplectic form ω = dt ∧ dφ + ω_Σ. It has the following properties.

1. Consider τ^∗−1 acting on H_d¹(Σ). The dimension of ker(τ^∗− 1)/ ker(τ^∗− 1) ∩ im(τ^∗− 1)
is even, and denote it by 2p. Let q − p be the dimension of ker(τ^∗− 1) ∩ im(τ^∗− 1).

2. dim H_d¹(X) = dim H_d³(X) = q + p + 2 and dim H_d²(X) = 2q + 2p + 2.

3. dim P H_dd² Λ(X) = dim P H_d+d² Λ(X) = 3q + p + 1 and dim P H_∂¹

+(X) = dim P H_∂¹₋(X) = q + p + 2.

We remark that the dimensions of the de Rham cohomologies only depend on the dimension of the τ^∗-invariant subspace of H_d¹(Σ). The dimensions of the primitive cohomologies involve the degeneracy of the intersection pairing on the τ^∗-invariant subspace of H_d¹(Σ). We will return to this example in Section 6 to demonstrate aspects of the product structures which we shall describe next.

5 A

-algebra structure on filtered forms

The exact triangle (1.4) relates the filtered cohomologies closely with the de Rham cohomologies through Lefschetz maps. It is thus tempting to think that some of the algebraic properties of the de Rham cohomology should also be present for filtered cohomologies. For instance, an important property of the de Rham cohomology is its ring structure with the product operation taken to be the exterior product on forms. Underlying this ring structure is the standard differential graded algebra on the space of differential form, (Ω^∗, ∧, d), with the two operation being the exterior product and the exterior derivative. So could the filtered cohomology groups also be rings? As we shall see in this section, the answer turns out to be yes. However, there is not a differential graded algebra for filtered forms. What we have instead is a generalization, that of an A∞-algebra on the space of p-filtered forms.

Let us first recall the definition of an A∞-structure (see, for example [16, 11]). An A∞ -algebra is a Z-graded vector space A = ⊕j∈ZA^j, with graded maps,

m_k: A^⊗k → A , k = 1, 2, 3, . . .

of degree 2 − k that satisfy the strong homotopy associative relation:

X

r, t ≥ 0 , s>0

(−1)^{r+s t}mr+t+1 1^⊗r⊗ m_s⊗ 1^⊗t
= 0 , (5.1)

for each k = r + s + t . Here, when acting on elements, the standard Koszul sign convention applies:

(ϕ1⊗ ϕ₂)(v1⊗ v₂) = (−1)^|ϕ2||v1|ϕ1(v1) ⊗ ϕ2(v2) , (5.2) where ϕi are graded maps, vi are homogeneous elements, and the absolute value denotes their degree.

Explicitly, relation (5.1) implies the following for the first three m_k maps:

• m₁: A → A satisfies m₁m₁ = 0 . Since m₁ increases the degree of the grading by one and squares to zero, it is a differential with (A, m1) a differential complex.

• m₂: A^⊗2→ A satisfies

m₁m₂ = m₂(m₁⊗ 1 + 1 ⊗ m₁) . (5.3) Here, m₂ preserves the grading, so it is considered a multiplication operator in A. With m1 as the differential, condition (5.3) is just the requirement that the Leibniz product rule holds.

• m₃: A^⊗3→ A satisfies

m2(1 ⊗ m2− m₂⊗ 1) = m₁m3+ m3(m1⊗ 1 ⊗ 1 + 1 ⊗ m₁⊗ +1 ⊗ 1 ⊗ m₁) (5.4) The left-hand-side measures the associativity of the multiplication m2. Equation (5.4) effectively stipulates that m₂ is associative up to homotopy.

Let us note that a differential graded algebra is just a special case of an A∞-algebra with the multiplication m2 being associative, and hence, m_k = 0 for all k ≥ 3. Moreover, even though the multiplication m₂ is in general not associative on A, it is always associative on the associated homology H^∗A = H^∗(A, m1). This follows directly from (5.4), since acting on elements of H^∗(A, m₁) which are m₁-closed, the right-hand-side is zero modulo the m₁-exact term, m1m3.

F_p⁰ F_p¹ . . . F_p^n+p−1 F_p^n+p F_p^n+p+1 F_p^n+p+2 . . . Fp^2n+2p F_p^2n+2p+1 F^pΩ⁰ F^pΩ¹ . . . F^pΩ^n+p−1 F^pΩ^n+p F^pΩ^n+p F^pΩ^n+p−1 . . . F^pΩ¹ F^pΩ⁰

Table 3: The Fp^j subspaces of a p-filtered graded algebra F_p following the notation of (5.5).

We now construct an A∞-algebra on p-filtered forms. We will denote it by F_p. The first step is to specify the Fp^j subspaces. We shall use the assignment suggested by the p-filtered elliptic complex (3.2) and its associated filtered cohomology

F^pH = {F^pH₊⁰, F^pH₊¹, . . . , F^pH₊^n+p, F^pH₋^n+p, F^pH₋^n+p−1, . . . , F^pH₋⁰}

which consists of 2(n + p) + 1 distinct objects. Assigning each to be the homology of a subspace, the nontrivial Fp^j subspaces should have degree in the range 0 ≤ j ≤ 2(n + p) + 1 . Specifically, we shall label the subspaces in the following way. (See also Table 3.)

A_j ∈ F^pΩ^j = F_p^j for 0 ≤ j ≤ n + p , A¯j ∈ F^pΩ^j = F_p^2n+2p+1−j for 0 ≤ j ≤ n + p .

(5.5)

For clarity, since a p-filtered j-form may be in either Fp^j or Fp^2n+2p+1−j subspace, we have dis-tinguished the two spaces by using a bar notation to denote a j-form ¯A_j ∈ F^pΩ^j = Fp^2n+2p+1−j

instead of being an element in Fp^j = F^pΩ^j. We will follow this convention for the rest of this paper as well.

Further, mimicking closely the filtered elliptic complex, we choose the differential of the A∞-algebra d_j : Fp^j → F_p^j+1, i.e. the m₁ map, to be as follows.

dj =











d₊ if 0 ≤ j < n + p − 1 ,

−∂₊∂− if j = n + p ,

−d− if n + p + 1 ≤ j ≤ 2n + 2p + 1

(5.6)

This differential clearly satisfies d_j+1d_j = 0 on the space {F^pΩ^∗, F^pΩ^∗} . It only differs from the differential operators of the elliptic complex by a negative sign in front of the “minus”

operators ∂− and d−. We will see below in Section 5.2 that the negative signs are needed for satisfying the Leibniz rule condition.

5.1 Product on filtered forms

The symplectic elliptic complex (3.2) motivated the definition of the grading and the m₁ map.

To obtain the m2 multiplication map, we turn to the long exact sequence of cohomology (4.1) and its underlying chain of short exact sequences (2.24). These exact sequences are suggestive of how to define a product on F^pΩ^∗ for they contain maps between F^pΩ^∗ and Ω^∗ such as Π^p, ∗r, L^−(p+1)d , and Π^p∗_rd L^−(p+1). So to define a product on filtered forms, we can first map F^pΩ^∗ to Ω^∗, then apply the wedge product on Ω^∗⊗ Ω^∗, and finally map the resulting form back to F^pΩ^∗ with the desired grading. (See Figure 3 for more details.) In this way, we are led to defining the following product operation on F_p= {F^pΩ^∗, F^pΩ^∗}:

Let us note that the product of p-filtered forms F^pΩ^∗ in (5.7) simplifies depending on the value of j + k. For if j + k > n + p, then Π^p(Aj∧ A_k) = 0. On the other hand, the terms in the Notice also that the expressions on the right hand side of (5.8) and (5.9) are automatically p-filtered. This can be seen simply by applying the p-filtered condition (2.22) and using (2.8) and (∗_r)² = 1 . Furthermore, the product A_j × ¯A_k of (5.8) is identically zero unless j ≤ k . (Similarly, for (5.9), a non-trivial product only occurs for k ≤ j .) This property is natural since the product F_p^j× F_p^k = 0 if j + k > 2n + 2p + 1, as subspaces with grading greater than

Column A B C D

Figure 3: Consider as an example the above commutative diagram of Lemma 2.3 in dimension 2n = 8 for the degree two Lefschetz map which involves the p = 1 filtered forms {F¹Ω^∗, F¹Ω^∗}.

The filtered product of Definition 5.1 can be heuristically understood as first mapping the filtered forms in Columns A and D into Columns B and C. Once in the middle two columns, the wedge product can be applied and then the resulting form can be projected back to the outer columns. For the case of F^pΩ^j × F^pΩ^k where j + k > n + p, the product crosses the middle row of the diagram which notably is without filtered forms and therefore has no grading assignment. Hence, in order to obtain the desired product grading of j +k > n+p, the definition of the product must involve a derivative map which shifts forms down by a row. The three terms in (5.12) correspond to the three different ways one can apply the derivative map to a product of two filtered forms.

2n + 2p + 1 are defined to be the empty set. This also explains why the product in (5.10) is trivial. Lastly, the factor of (−1)^j in (5.8) ensures that the product is graded commutative.

That is,

F_p^j× F_p^k= (−1)^jkF_p^k× F_p^j .

Now we can check that our definition of the filtered product × is consistent with the exact triangle of (4.2). At the level of cohomology, the long exact sequence (4.1) locally has the following form (setting r = p + 1 in (4.1))

and we have denoted the filtered cohomology by

F^pH^j =

Heuristically, we can view the product of two filtered cohomologies as tensoring two long exact sequences locally in the following way:

correct degrees for it to be a wedge product. Instead, for our purpose, it has the interpretation as the standard Massey triple product with the middle element of the triple product fixed to be ω^p+1. To see this, since our concern is the filtered product, we are mainly interested in the subset of elements of H_d^j−2p−1 and H_d^k−2p−1 that are in the image of g from F^pH^j and F^pH^k, respectively. By the exactness of the sequences, these elements are in the kernel of the Lefschetz map:

[ξ_j−2p−1] ∈ ker[L^p+1: H_d^j−2p−1→ H_d^j+1] and [ξ_k−2p−1] ∈ ker[L^p+1: H_d^k−2p−1 → H_d^k+1] . Therefore, there must exist an η_j ∈ Ω^j and an η_k ∈ Ω^k such that

dη_j = ω^p+1∧ ξ_j−2p−1 and dη_k= ω^p+1∧ ξ_k−2p−1 .

Given this, it is then natural to consider the Massey triple product hξj−2p−1, ω^p+1, ξk−2p−1i.

With the symplectic structure element ω^p+1 kept fixed, this Massey triple product then defines what we shall simply call the Massey product:

hξ_j−2p−1, ξk−2p−1i_p = ξj−2p−1∧ η_k+ (−1)^jηj∧ ξ_k−2p−1∈ H_d^∗(M )

I(ξ_j−2p−1, ξ_k−2p−1) , (5.17) where I(ξ_j−2p−1, ξ_k−2p−1) is the ideal generated by ξ_j−2p−1 and ξ_k−2p−1. We note that this Massey product is only well-defined on the quotient since different choices of ηj and ηk may differ by a d-closed form.

We can now ask whether the filtered product × is compatible with the wedge and Massey product that surrounds it in (5.16). For the filtered product we defined in Definition (5.1), the diagram in (5.16) in fact commutes. The precise statement of this is given in the following theorem whose proof is provided in Appendix A.

Theorem 5.2. Let (M, ω) be a symplectic manifold. The product operator × on F^pH^∗(M ) is compatible with the topological products. That is, it satisfies the following properties:

1. (Wedge product) For any two [ξ_j] ∈ H_d^j(M ) and [ξ_k] ∈ H_d^k(M ), f (ξ_j ∧ ξ_k) = f (ξ_j) × f (ξk). The equality is considered on the filtered cohomology class corresponding to f : H_d^j+k(M ) → F^pH^j;

2. (Massey product) For any two [A] ∈ F^pH^∗(M ) and [A⁰] ∈ F^pH^∗(M ), hg(A), g(A⁰)i_p = g(A × A⁰). To be more precise, the equality is considered on H_d^∗(M )/I(g(A), g(A⁰));

where the maps f and g are defined by (5.14) and the Massey product is defined by (5.17).