@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.