Surfaces in 4-Manifolds

In this paper we introduce a technique, called {\em rim surgery}, which can change a smooth embedding of an orientable surface $\Sig$ of positive genus and nonnegative self-intersection 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 $S^1\times E(K)$, where $E(K)$ is the exterior of a knot $K\subset S^3$. The smooth change can be detected easily for certain pairs $(X,\Sig)$ called {\em SW-pairs}. For example, $(X,\Sig)$ is an SW-pair if $\Sig$ 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: \smallskip \noindent{\bf Theorem.} {\em Consider any SW-pair $(X,\Sig)$. For each knot $K\subset S^3$ there is a surface $\Sig_K\subset X$ such that the pairs $(X,\Sig_K)$ and $(X,\Sig)$ are homeomorphic. However, if $K_1$ and $K_2$ are two knots for which there is a diffeomorphism of pairs $(X, \Sig_{K_1})\to (X,\Sig_{K_2})$, then their Alexander polynomials are equal: $\DD_{K_1}(t)=\DD_{K_2}(t)$.}

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Published 1 January 1997