@MISC{Fintushel97surfacesin, author = {Ronald Fintushel and Ronald and J. Stern}, title = {Surfaces in 4-manifolds}, year = {1997} }

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Abstract. In this paper we introduce a technique, called rim surgery, which can change a smooth embedding of an orientable surface Σ of positive genus and nonnegative selfintersection in a smooth 4-manifold X while leaving the topological embedding unchanged. This is accomplished by replacing the tubular neighborhood of a particular nullhomologous torus in X with S 1 × E(K), where E(K) is the exterior of a knot K ⊂ S 3. The smooth change can be detected easily for certain pairs (X,Σ) called SW-pairs. For example, (X,Σ) is an SW-pair if Σ is a symplectically and primitively embedded surface with positive genus and nonnegative self-intersection in a simply connected symplectic 4-manifold X. We prove the following theorem: Theorem. Consider any SW-pair (X,Σ). For each knot K ⊂ S 3 there is a surface ΣK ⊂ X such that the pairs (X,ΣK) and (X,Σ) are homeomorphic. However, if K1 and K2 are two knots for which there is a diffeomorphism of pairs (X,ΣK1) → (X,ΣK2), then their Alexander polynomials are equal: ∆K1(t) = ∆K2(t). 1.

... �= 1, then ΣK is not smoothly ambient isotopic to a symplectic submanifold of X. Proof. Since Σ and Sg are symplectic submanifolds of X and Yg, the fiber sum Xn#Σn=SgYg is also a symplectic manifold =-=[G]-=-. Thus SWXn#Σn=Sg Yg �= 0 [T1]; so (X,Σ) forms an SWpair. This proves the first statement of the theorem. Next, suppose that ΣK is smoothly ambient isotopic to a symplectic submanifold Σ ′ of X. This ...

...y ambient isotopic to a symplectic submanifold of X. Proof. Since Σ and Sg are symplectic submanifolds of X and Yg, the fiber sum Xn#Σn=SgYg is also a symplectic manifold [G]. Thus SWXn#Σn=Sg Yg �= 0 =-=[T1]-=-; so (X,Σ) forms an SWpair. This proves the first statement of the theorem. Next, suppose that ΣK is smoothly ambient isotopic to a symplectic submanifold Σ ′ of X. This isotopy carries the rim torus ...

...of Xn#Σ ′ n=SgYg can be grouped into collections Cb = {b +2mT ′ }, and if ∆K(t) �= 1 then each Cb contains more than one basic class. Note, however, that 〈ω, b +2mT ′ 〉 = 〈ω, b〉. Now Taubes has shown =-=[T2]-=- that the canonical class κ of a symplectic manifold with b + > 1isthe basic class which is characterized by the condition 〈ω, κ〉 > 〈ω, b ′ 〉 for any other basic class b ′ . But this is impossible for...

...SW-pairs Recall that the Seiberg-Witten invariant SWX of a smooth closed oriented 4-manifold X with b + > 1 is an integer valued function which is defined on the set of spin c structures over X, (cf. =-=[W]-=-). In case H1(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). In this case we view the Seiberg-Witten invariant ...