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Tori in symplectic 4-manifolds

We study the question of how many embedded symplectic or Lagrangian tori can represent the same homology class in a simply connected symplectic 4-manifold.


Introduction
The basic question addressed in this paper is: Let X be a simply connected symplectic 4-manifold and let x ∈ H 2 (X, Z). How unique is an embedded symplectic or Lagrangian representative of x?
It is only in the last few years that an answer to this question has begun to emerge. The answer is 'not very' for symplectic tori of self-intersection 0 and remains elusive for higher genus surfaces. As we show below: Theorem If X is a simply connected symplectic 4-manifold containing an embedded symplectic torus T of self-intersection 0, then for each fixed integer m ≥ 2, there are infinitely many embedded symplectic tori, each representing the homology class m[T ], no two of which are equivalent under a smooth isotopy of X .
The first such examples were produced by the present authors in [6], and the technique therein was enhanced to produce further examples in [19,3]. In section 4 we give a proof of the above theorem. In section 3 we give a proof of the theorem for even m ≥ 6 which is straightfoward, and which depends on some nice theorems of Montesinos and Morton [12] and of Kanenobu [9] rather than on explicit constructions. Some intriguing questions remain. Siebert and Tian have conjectured that for symplectic 4-manifolds with b + = 1 and c 2 1 > 0 any embedded symplectic surface must be symplectically isotopic to a holomorphic curve. (Of course, no such manifold contains an embedded symplectic torus of square 0.) They have shown that in CP 2 this is true for each curve of degree ≤ 17, and they have also some results to this effect in S 2 × S 2 . However the general problem is still wide open, as is the case in general for surfaces of higher genus or for other self-intersections. However, in the case where π 1 (X) = 0 Ivan Smith has constructed examples of nonisotopic but homologous surfaces of square 0 distinguished by π 1 of their complements.
Much less is known in the case of Lagrangian tori. Until this year, it was unknown if there existed Lagrangian tori which were homologous but inequivalent (under either isotopy or orientation-preserving diffeomorphism). The first examples are due to Stefano Vidussi: Theorem [20] Let K denote the trefoil knot. Then in the symplectic manifold E(2) K there is a primitive homology class α so that for each positive integer m, there are infinitely many embedded Lagrangian tori representing mα, no two of which are equivalent under orientation-preserving diffeomorphisms.
Utilizing an invariant coming from Seiberg-Witten theory and the geometry of fibered knots, the current authors improved this theorem as follows: Theorem [7] (a) Let X be any symplectic manifold with b + 2 (X) > 1 which contains an embedded symplectic torus with a vanishing cycle. Then for each fibered knot K in S 3 , the result of knot surgery X K contains infinitely many nullhomologous Lagrangian tori, pairwise inequivalent under orientation-preserving diffeomorphisms.
(b) Let X i , i = 1, 2, be symplectic 4-manifolds containing embedded symplectic tori F i and assume that F 1 contains a vanishing cycle. Let X be the fiber sum, X = X 1 # F 1 =F 2 X 2 . Then for each fibered knot K in S 3 , the manifold X K contains an infinite family of homologically primitive and homologous Lagrangian tori which are pairwise inequivalent.
In sections 5-8 we show how this theorem works in a specific example constructed via double branched covers. The discussion here differs somewhat from the more general arguments of [7], however we feel that it is helpful to understand specific examples from different points of view.

Seiberg-Witten invariants
The Seiberg-Witten invariant of a smooth closed oriented 4-manifold X with b + 2 (X) > 1 is an integer-valued function which is defined on the set of Spin c structures over X (cf [21]). In case H 1 (X; Z) has no 2-torsion, there is a natural identification of the Spin c structures of X with the characteristic elements of H 2 (X; Z) (ie those elements k whose Poincaré dualsk reduce mod 2 to w 2 (X)).
The sign of SW X depends on an orientation of H 0 (X; R) ⊗ det H 2 + (X; R) ⊗ det H 1 (X; R); however, when X has a symplectic structure, there is a preferred sign for SW X (see [16]).
If SW X (β) = 0, then β is called a basic class of X . It is a fundamental fact that the set of basic classes is finite. Furthermore, if β is a basic class, then so is −β with SW X (−β) = (−1) (e+sign)(X)/4 SW X (β) where e(X) is the Euler number and sign(X) is the signature of X .
A key vanishing theorem for the Seiberg-Witten invariants is: Theorem [21] Let X be a smooth closed 4-manifold which admits a decomposition X = A ∪ B into 4-manifolds with ∂A = ∂B = Y . Suppose that b + 2 (A) > 0, b + 2 (B) > 0, and that Y admits a metric of positive scalar curvature, then SW X = 0.
Another important and extremely useful fact about Seiberg-Witten invariants is the adjunction inequality: If X is a smooth closed 4-manifold with b + 2 > 1 and Σ is an embedded surface of positive genus g in X representing a nontrivial element of H 2 (X; R) then for any basic class β of X We next recall the link surgery construction of [5]. This construction starts with an oriented n-component link L = {K 1 , . . . , K n } in S 3 and n pairs (X i , T i ) of smoothly embedded self-intersection 0 tori in simply connected 4-manifolds.
(In the original article [5], an extra condition ('c-embedded') was placed on these tori; however, recent work of Cliff Taubes [17] has shown this condition to be unnecessary.) Let α L : π 1 (S 3 \ L) → Z denote the homomorphism characterized by the property that it sends the meridian m i of each component K i to 1, and let ℓ i denote the longitude of K i . The curves γ i = ℓ i − α L (ℓ i )m i on ∂N (K i ) form the boundary of a Seifert surface for the link, and in case L is a fibered link, the γ i are given by the boundary components of a fiber.
In S 1 × (S 3 \ N (L)) let T m i = S 1 × m i , and define the link-surgery manifold X(X 1 , . . . X n ; L) by where S 1 × ∂N (K i ) is identified with ∂N (T i ) so that for each i It is not clear whether or not this determines X(X 1 , . . . X n ; L) up to diffeomorphism, however any such manifold will have the same Seiberg-Witten invariant: Theorem [5] If each π 1 (X\T i ) = 1, then X(X 1 , . . . X n ; L) is simply-connected and its Seiberg-Witten invariant is SW X(X 1 ,...Xn;L) = ∆ sym L (t 2 1 , . . . , t 2 n ) · n j=1 SW X j · (t j − t −1 j ) where t j = t [T j ] and ∆ sym L (t 1 , . . . , t n ) is the symmetric multivariable Alexander polynomial of the link L.
In case each (X i , T i ) ∼ = (X, T ), a fixed pair, we write X(X 1 , . . . X n ; L) = X L (We implicitly remember T , but it is removed from the notation.) As an example, consider the case where each X i = E(1), the rational elliptic surface (E(1) ∼ = CP 2 #9CP 2 ) and each T i = F is a smooth elliptic fiber. Since In case the link L is actually a knot K , we call the procedure 'knot surgery' and the resulting manifold X K . The formula for the Seiberg-Witten invariant looks slightly different in this case due to the difference in the relationship of the Seiberg-Witten invariant of a 3-manifold and its Alexander polynomial when b 1 > 1 and b 1 = 1.

Tori and simple covers
Our first construction utilizes an extremely interesting theorem of José Montesinos and Hugh Morton which characterizes fibered links in the 3-sphere. To begin, let X be a simply connected symplectic 4-manifold containing an embedded symplectic torus T of self-intersection 0, and identify a tubular neighborhood of T with S 1 × (S 1 × D 2 ). A closed braid may be viewed as contained in S 1 × D 2 ⊂ S 3 = (S 1 × D 2 ) ∪ (D 2 × S 1 ) and then its axis is {0} × S 1 . The theorem of Montesinos and Morton is: Theorem (Montesinos and Morton [12]) Every fibered link in S 3 with k components can be obtained as the preimage of the braid axis for a d-sheeted simple branched cover of S 3 branched along a suitable closed braid, where d = max{k, 3}.
(Recall that a simple branched cover of degree d is one whose branch points have exactly d − 1 points in their preimages.) A second important ingredient in this construction is a theorem of Kanenobu concerning the Hosokawa polynomial of fibered links. The Alexander polynomial of a link L of k components is a polynomial ∆ L (t 1 , . . . , t k ) in k variables (corresponding to the meridians of the components of the link). The polynomial ∆ L (t, . . . , t) obtained by setting all the variables equal is always divisible by (t − 1) k−2 , and the Hosokawa polynomial of L is defined to be ∇ L (t) = ∆ L (t, . . . , t)/(t − 1) k−2 .
Theorem (Kanenobu [9]) Let f (t) be any symmetric polynomial of even degree with integral coefficients satisfying f (0) = ±1, then for any k ≥ 2 there is a fibered link L of k components in S 3 with ∇ L (t) = f (t).
We now use these two theorems to build symplectic tori homologous to multiples of T . We have described a tubular neighborhood N of T as N = S 1 × (S 1 × D 2 ). Fix a three-component fibered link L in S 3 and let B L be the braid corresponding to L by the Montesinos-Morton Theorem. As above, we view Figure 1 for an example.) Then T L = S 1 × B L ⊂ N is a symplectic torus [6], and if B L has m strands, then T L is homologous to mT . Let π : (S 3 , L) → (S 3 , A) be the threefold branched cover with branch set B L given by the Montesinos-Morton Theorem. Because L = π −1 (A) is a threecomponent link, the covering restricted to A is trivial. This means that the restriction of π over ∂(S 1 × D 2 ) is a trivial covering, and the induced branched cover over N = S 1 ×(S 1 ×D 2 ) extends trivially over X . We thus get a threefold simple branched cover p = p L : X → X with branch set T L . We have This means that X is obtained via link surgery on the link L using (X, T ). The Seiberg-Witten invariant of X (viewed as an element of ZH 2 ( X)) may be calculated via the techniques of [17,14,5]: The induced map p * : ZH 2 ( X) → ZH 2 (X) satisfies p * (SW X i ) = SW X . Also, since t i is the element of ZH 2 ( X) corresponding to the homology class of S 1 ×µ i where µ i are the meridians of the components of L, p * (t i ) is the element of Now suppose that we are given another three-component link L ′ which is a threefold simple cover of S 3 with branch set B L ′ and symplectic torus T L ′ = S 1 × B L ′ . The covering projections p L , p L ′ are determined by homomorphisms ϕ L (or ϕ L ′ ) from π 1 (X \ T L ) (or π 1 (X \ T L ′ )) to the symmetric group S 3 such that each meridian of T L (or T L ′ ) is sent to a transposition.
Any isotopy of X taking T L to T L ′ and which carries the covering data for p L to that of p L ′ gives rise to where f (T L ) = T L ′ and f * is the identity on homology.
Since f * (SW X L ) = SW X L ′ it follows from (2) and (3) that . Using Kanenobu's theorem, one sees that there are infinite families of fibered links {L i } whose ∇ sym L i (t) are distinct and have arbitrary fixed even degree (> 0). The genus g L of the fibered link L is half the degree of its Hosokawa polynomial. (See, for instance, [2].) Furthermore, the fiber of L is the thrice-punctured surface which is a simple threefold branched cover of D 2 (a normal fiber to S 1 × {0}) with m branch points. Thus the number of strands m of B L is determined by m = 2g L + 4. This means that for any even m ≥ 6 we get an infinite family {T i } of symplectic tori homologous to mT with distinct threefold simple branched covers. Note that each braided torus T L admits at most finitely many simple threefold branched covers of X with T L as branch set, since there are finitely many distinct homomorphisms π 1 (X \ T L ) → S 3 . Thus we have: Theorem 3.1 Let X be a simply connected symplectic 4-manifold containing an embedded symplectic torus T of self-intersection 0. Then for each even m ≥ 6 there are infinitely many pairwise nonsmoothly isotopic embedded symplectic tori homologous to mT .

Fiber sums
We begin this section with the same hypotheses as the last: We are given a simply connected symplectic 4-manifold X containing an embedded symplectic torus T of self-intersection 0. The construction of new symplectic tori is similar to that of the last section (and of [6]). For each m ≥ 2 consider closed braids B with m strands. Then the braided torus is symplectic and homologous to mT .
Suppose that B and B ′ are m-strand closed braids and that T B ′ is smoothly isotopic to T B in X . Then there is a diffeomorphism f : X → X satisfying: and f * = id on H * (X). (Here µ B and µ B ′ are meridians to the braids; so they also may be viewed as meridians to the tori T B and T B ′ .) Our goal is to use relative Seiberg-Witten invariants SW (X,T B ) to distinguish the tori T B up to isotopy. Let E(1) denote the rational elliptic surface. Because of the gluing theorems of [17,14] and the fact that the relative Seiberg-Witten invariant of E(1) minus a a smooth elliptic fiber is SW E(1)\F = 1 (see eg [10]), the relative Seiberg-Witten invariant of (X, T B ) may be expressed as the absolute Seiberg-Witten invariant of the fiber sum of X and E(1) along T B and F : Now write N (T B ) for a tubular neighborhood of T B in X and also writeN (T ) = S 1 × (S 1 × D 2 ), the original tubular neighborhood of T . We have Let L B be the link in S 3 consisting of the closed braid B together with its axis A. If µ A denotes a meridian to A, then T is homologous to S 1 × µ A . Let t = t T denote the corresponding element in ZH 2 (X).
We may now rewrite (4) as The manifold X# T B =F E(1) is obtained from the same components as link surgery using the link L B and the manifolds (E(1), F ) and (X, T ); however the gluings are not necessarily those specified in section 2. Since E(1) has big diffeomorphism group with respect to F (see eg [8]), each diffeomorphism ∂N (F ) → ∂N (F ) extends to a self-diffeomorphism of E(1) \ N (F ); so the diffeomorphism used to glue in E(1) \ N (F ) is inconsequential. However, it is useful to demand that the fiber F of E(1) should be identified with According to [17,14], SW X · (t − t −1 ) is the relative Seiberg-Witten invariant of (X, T ), and by [5], as described in section 2, the relative invariant of the manifold . Applying [5] and [17] we obtain: . When applying this formula, we need to remember that t T = t and t F = t m .
Theorem 4.1 Let X be a simply connected symplectic 4-manifold with b + 2 > 1 containing an embedded symplectic torus T of self-intersection 0. For a fixed integer m ≥ 2, let B and B ′ be closed m-strand braids in S 3 . Then T B and T B ′ are embedded symplectic tori in X which are homologous to mT . If there is an isotopy of X taking Let A = T ⊥ B = T ⊥ ⊂ H 2 (X). Note that the adjunction inequality (1) implies that no basic class of X has nontrivial intersection with [T ]. Thus SW X ∈ ZA. We have H 2 (X \T B ) = A⊕R B . Finally, suppose that [T ] is n times a primitive homology class, and let S denote the class in H 2 (X# T B =F E(1)) which has a representative built from mn punctured sections in E(1) \ F and a surface in X \ T B which has boundary mn copies of the meridian µ B to T B .
A Mayer-Vietoris argument shows that the homology of X# T B =F E(1) splits as where the E 8 comes from H 2 (E(1) \ F ). There is a similar splitting of the homology of H 2 (X# T B ′ =F E(1)).
If there is an isotopy of T B to T B ′ , there is a diffeomorphism is a basic class, and so α · τ ′ = 0, and α · [T ] = 0. Furthermore each class in R B ′ is orthogonal to T .
We have as a corollary: Theorem 4.2 Let X be a simply-connected symplectic 4-manifold satisfying b + 2 (X) > 1 and containing an embedded symplectic torus T of self-intersection 0. For each m ≥ 2 there are infinitely many pairwise nonisotopic embedded symplectic tori in X which are homologous to mT .
Proof This follows from the above theorem provided for each m ≥ 2 there are infinitely many closed m-strand braids B whose 2-component links L B = A∪ B have distinct 2-variable Alexander polynomials. Such examples are given, for example, in the work of Etgu and Park [3].

Lagrangian tori
In this section we use branched covers as a means for constructing examples of Lagrangian tori in symplectic 4-manifolds whose homology classes are equal but which are not equivalent under symplectic diffeomorphisms. There are already two papers [20,7] dealing with this phenomenon, and the invariants of [7] can be used to distinguish the examples given in this section. However we believe that the constructions below are interesting in their own right and are certainly different from those cited.
To begin, let K be the trefoil knot, and M K the 3-manifold obtained from 0-framed surgery on S 3 along K . Since K is a genus-1 fibered knot, M K fibers over the circle with fiber a torus, where m 0 is a meridian to K . This manifold has a symplectic structure induced from that on E(1) and the structure on S 1 × M K in which the fiber and section are symplectic submanifolds. (See [5].)    Let m 1 and m 2 be meridians of K as in Figure 2, and let X denote the double branched cover of E(1) K with branch set S 1 × (m 1 ∪ m 2 ). Since S 1 × m i is a section to the fibration S 1 × M K → S 1 × S 1 , the branch set of this cover is symplectic, hence X inherits a symplectic structure. We have It follows that M K also fibers over the circle, and its fiber is the double branched cover of the fiber of M K , branched over two points. Thus the fiber of M K → S 1 has genus 2. We can say more:    Since 1) ′′ , and the complement of a fiber in E(1) is simply-connected, we have where µ ′ and µ ′′ are the meridians to m ′ 0 and m ′′ 0 . The group in the middle, ), is normally generated by the classes of S 1 × pt, any meridian to K#K , and by µ ′ and µ ′′ . These loops all lie on S 1 × m ′ 0 × µ ′ or S 1 × m ′′ 0 × µ ′′ ; so we see that X is simply-connected.
Consider the paths P and P ′ shown in Figure 5, each running from a point y 1 ∈ m 1 to y 2 ∈ m 2 . These paths lie in a fiber of the fibration of S 1 × M K over S 1 × S 1 ; thus the construction of the symplectic structure (essentially 'area form on base' plus 'area form on fiber') implies that the surfaces S 1 × P and S 1 × P ′ are Lagrangian.      We thus obtain Lagrangian tori T = T γ = S 1 × γ and T ′ = T ′ γ = S 1 × γ ′ in S 1 × M K#K . These tori are disjoint from the lifts of m 0 , where the gluing in the construction of X takes place, so T and T ′ are Lagrangian tori in X .     The meridian m 0 in M K lifts to a pair of meridians, m ′ 0 , m ′′ 0 in M K#K as in Figure 6. Referring to Figure 6 Because δ ′ and δ ′′ link neither m ′ 0 (or m ′′ 0 ) nor µ ′ (or µ ′′ ), the loops δ ′ and δ ′′ are nullhomologous in M K#K \ (m ′ 0 ∪ m ′′ 0 ). This means that the corresponding Lagrangian tori, Σ ′ = S 1 × δ ′ and Σ ′′ = S 1 × δ ′′ are nullhomologous in X . Since T ′ − T is homologous to Σ ′ + Σ ′′ , we see that T and T ′ are homologous in X .
The loop γ is a separating curve in the fiber of M K#K → S 1 . See Figure 7. We see that γ is homologous to µ ′ in the fiber of 0 , one of the tori along which the fiber sum In such a fiber sum, the rim tori give essential homology classes -thus we see that the Lagrangian tori T and T ′ are essential in X .   We claim that there is no diffeomorphism of X which takes T to T ′ . To see this we shall use an invariant obtained from Seiberg-Witten theory. To do this we need the notion of 'surgery on T '. As usual, this means the result of removing a tubular neighborhood N (T ) ∼ = T 2 × D 2 and regluing it.
The key quantity in this operation is the class ω ∈ H 1 (∂N (T )) which is killed by the composition of ψ : ∂N (T ) → T 2 × ∂D 2 with the inclusion T 2 × ∂D 2 → T 2 × D 2 . This class determines X(T, ψ) up to diffeomorphism; so we write X T (ω) instead of X(T, ψ).
Note that if there is a diffeomorphism f of X taking T to T ′ , then each manifold X T (ω) corresponds to a unique X T ′ (f * (ω)). Thus the collection of all manifolds is a diffeomorphism invariant of (X, T ). Our invariant I(X, T ), defined and computed below, will be the set of Seiberg-Witten invariants of these manifolds.

Product formulas for the Seiberg-Witten invariant
Before formally defining I(X, T ) we need to discuss techniques for calculating the Seiberg-Witten invariants of the manifolds X T (ω). Fix simple loops α, β , δ on ∂N (T ) whose homology classes generate H 1 (∂N (T )). If ω = pα + qβ + rδ write X T (p, q, r) instead of X T (ω). An important formula for calculating the Seiberg-Witten invariants of surgeries on tori is due to Morgan, Mrowka, and Szabo [13] (see also [11], [17]). Given a class k ∈ H 2 (X): In this formula, T denotes the torus which is the core T 2 × 0 ⊂ T 2 × D 2 in each specific manifold X(a, b, c) in the formula, and k (a,b,c) ∈ H 2 (X T (a, b, c)) is any class which agrees with the restriction of k in H 2 (X \ T × D 2 , ∂) in the diagram: Let π(a, b, c) : H 2 (X T (a, b, c)) → H 2 (X \T × D 2 , ∂) be the composition of maps in the above diagram, and π(a, b, c) * the induced map of integral group rings.
The indeterminacy due to the sum in (6)   We first specify a basis for H 1 (∂N (T )) as follows: (Figure 7). Then N (T ) may be identified with S 1 × γ × D 2 , and we take the basis α = [S 1 × pt × pt], β = [pt × γ L ], where γ L is a pushoff of γ in the fiber of the fibration M K#K \ (m ′ 0 ∪ m ′′ 0 ) → S 1 (this is called the 'Lagrangian framing' in [7]), and δ = [m T ], where m T = pt × pt × ∂D 2 , the meridian to T . It is then clear from (7) that in order to calculate I(X, T ), one needs to calculate SW Y for Y = X T (1, 0, 0), X T (0, 1, 0), and X T (0, 0, 1); however, from our choice of basis, we have X T (0, 0, 1) ∼ = X . This leaves us with two invariants to calculate below. For a different approach to these calculations see [7].
The calculation of SW X T (0,1,0) depends on some basic facts about double covers of 3-manifolds branched over closed braids. Suppose that B is a braid in a solid torus with 2m strands, ie B ⊂ S 1 × D 2 ⊂ M 3 is a link such that each disk pt × D 2 intersects B in exactly 2m points. There is then a double cover Y B → M 3 branched over B for which each meridian to B is covered nontrivially, and this cover is trivial outside the solid torus.
The pertinent question is: 'What is the effect on Y B of putting half-twists into the braid B ?' In other words, suppose that ζ is an arc in (pt × D 2 ) ⊂ S 1 × D 2 whose endpoints lie on B , but so that no other point of ζ is on B . Then we can put half-twists in B by twisting in a small neighborhood of ζ . Figure 8 shows a local picture. According to our framing convention, 0-surgery on ζ is the one that kills the homology class of a pushoff of ζ in S , ie the class of ζ × pt in ζ × D 2 . Thus the result of 0-surgery is . Such a map ϕ is given by ϕ(z, w) = (w, z). Define the involution σ on S 1 × D 2 by σ(z, w) = (ρ(z),w). Then we see that the diagram commutes. Thus, the restriction of τ to Y B \ ( ζ × D 2 ) extends to an involution τ ′ over all of Z via σ .
On the solid torus S 1 ×D 2 the fixed set of τ ′ = σ is {±i} ×{the real axis ∩ D 2 }. Thus the picture in the quotient is exactly that of Figure 9.
We now apply this proposition to the case at hand, where the 3-manifold is M K , the braid is the trivial braid with components m 1 and m 2 , and the arc ζ is the path P of Figure 5. It follows that X T (0, 1, 0) is the double branched cover of E(1) K with branch set S 1 × C where C is the loop shown in Figure 10.
Notice that C is an unknotted circle which is unlinked from K .    The double cover of a 3-ball, branched over an unknot, is S 2 × I so it follows that the double cover of M K branched over C is M K #M K . Thus This means that X T (0, 1, 0) is split by S 1 × S 2 with b + 2 positive on each side. It follows that SW X T (0,1,0) = 0.
Next we need to make a similar calculation for X T ′ (0, 1, 0). This time the arc ζ is the path P ′ of Figure 5, and X T ′ (0, 1, 0) is the double branched cover of E(1) K with branch set S 1 × C ′ where C ′ is the loop shown in Figure 11.       An isotopy of Figure 11 gives Figure 12, and the corresponding double branched cover is shown in Figure 13. (For techniques for determining this double cover, see [15].) Using Fox calculus, one calculates the torsion of the link m ′ 0 ∪ m ′′ 0 in the double cover (M K ) C ′ (the 3-manifold shown in Figure 13). This torsion is 1. According to [11] this means that the Seiberg-Witten invariant of S 1 × ((M K ) C ′ \ (m ′ 0 ∪ m ′′ 0 )) is 1. Since the Seiberg-Witten invariant of E(1) \ F is also equal to 1, the gluing theorem of Taubes [17] tells us that the Seiberg-Witten invariant of E(1)# F =S 1 ×m ′ 0 (S 1 × (M K ) C ′ )# S 1 ×m ′ 0 =F E(1) is equal to 1. However, we have just shown that this manifold is X T ′ (0, 1, 0). The key calculation of this section will show that the Seiberg-Witten invariants of the manifolds X T (1, 0, 0) and X T ′ (1, 0, 0) vanish. Our approach here is to describe the surgered manifolds in terms of a branched covering. (It would be useful to compare with [7], where a more general approach is utilized.) Proposition 8.1 Letγ denote either γ or γ ′ , and let Z be the result of the surgery on T = S 1 ×γ ⊂ X which kills S 1 × pt × pt ⊂ S 1 ×γ × ∂D 2 ⊂ S 1 × (M K#K \ (m ′ 0 ∪ m ′′ 0 )). Then Z is the double branched cover of the manifold W obtained from E(1) K by a surgery on a circle S 1 × {point onγ } (trading a neighborhood S 1 × D 3 for D 2 × S 2 ). The branch set in W of this cover consists of a pair of disjoint 2-spheres of self-intersection 0.
Proof As we have seen in the proof of Proposition 7.1, the deck transformation of X → E(1) K in a neighborhood S 1 ×γ × D 2 of T is given by τ (t, z, w) = (t,z, ρ(w)) where ρ is reflection through the imaginary axis. The manifold Z is: where ϑ(t, z, w) = (w, z, t). Then the diagram commutes, where υ(t, z, w) = (ρ(t),z, w). Thus υ extends the deck transformation τ over the surgered manifold Z .
The quotient (S 1 ×S 1 ×D 2 )/τ ∼ = S 1 ×(D 2 ×I) ∼ = S 1 ×D 3 , but (S 1 ×S 1 ×D 2 )/υ ∼ = S 2 × D 2 , since the action of υ restricted to S 1 × S 1 × {pt} is equivalent to the action of the deck transformation of the double covering T 2 → S 2 with four branch points. Thus the effect of the surgery on the base is to perform surgery on the circle S 1 × {pt} ⊂ S 1 × D 3 . Before performing the surgery, the branch set consists of two tori. Since the fixed point set of υ on S 1 × S 1 × D 2 is {±i} × {±1} × D 2 , the surgery trades a pair of annuli for four disks. Removing the annuli leaves us with a pair of complementary annuli in the branch set, and the addition of the four disks caps them off, giving a pair of 2-spheres.
To see that the components of the branch set of W have self-intersection 0, first consider the branch torus S 1 × m 1 of X . Write m 1 = J 1 ∪ J 2 , the union of two intervals meeting only at their endpoints. We do this so that the intersection of S 1 ×m 1 with (S 1 ×S 1 ×D 2 )/τ ∼ = S 1 ×D 3 is S 1 ×J 2 where J 2 ∩∂D 3 = ∂J 2 . Then the corresponding component of the branch set in W is (S 1 × J 1 ) ∪ (D 2 × ∂J 2 ). In X we can isotop S 1 × m 1 slightly by moving m 1 tom 1 = I 1 ∪ I 2 in M K ; so thatm 1 ∩ m 1 = ∅,m 1 ∩ D 3 = I 2 , and I 2 ∩ ∂D 3 = ∂I 2 . Then in W , (S 1 × I 1 ) ∪ (D 2 × ∂I 2 ) is disjoint from (S 1 × J 1 ) ∪ (D 2 × ∂J 2 ).
Let Γ i denote the components of the branch set in Z . The Γ i are also 2-spheres of self-intersection 0.
In X there is a 'section class' C which arises from the sections of the elliptic fibrations on the copies of E(1). To build a representative for C , start with a fixed Seifert surface B 0 of K#K whch is a fiber of the fibration of S 3 \K#K → S 1 . The boundary of B 0 is capped off by the 2-disk introduced when we do 0-surgery on K#K to form M K#K . The tori S 1 × m ′ 0 and S 1 × m ′′ 0 which are identified with fibers of E(1) ′ and E(1) ′′ each intersect {pt} × B 0 in a single point. Remove disks in {pt} × B 0 about each of these points. The boundaries then bound disks of self-intersection −1, sections of E(1) minus the neighborhood of a fiber. The union of these surfaces gives a genus-2 surface of self-intersection −2 representing C .
The loopγ ⊂ X is contained in a Seifert surface for K#K , and we may assume that it is disjoint from B 0 . Thus the surgery torus, S 1 ×γ is disjoint from C . Since C · (S 1 × m 1 ) = C · F ′ = 1, after surgery in Z, we still have C · Γ 1 = 1; so Γ 1 is an essential 2-sphere in Z (and similarly for Γ 2 ). Thus Z contains an essential 2-sphere of self-intersection 0, and that means that SW Z = 0 [4].
Theorem 8.2 T and T ′ are essential and homologous Lagrangian tori of X ; however, there is no orientation-preserving diffeomorphism f of X with f (T ) = T ′ .