Smooth surfaces with non-simply-connected complements (Q1007191)

This paper studies two constructions involving surfaces embedded in simply connected \(4\)-manifolds whose complements are not simply connected. The first is an iteration of an operation, known as \(m\)-twist rim surgery, which was originated by the first author [Geom. Topol. 10, 27--56 (2006; Zbl 1104.57018)]. Given a surface \(\Sigma\) embedded in a \(4\)-manifold \(X\), a knot \(K\) in \(S^3\), and an integer \(m\), \(m\)-twist rim surgery produces a surface \(\Sigma_K(m)\) in the same \(4\)-manifold. This twisted version of rim surgery generalizes an untwisted rim surgery technique pioneered by \textit{R. Fintushel} and \textit{R. J. Stern} [Math. Res. Lett. 4, No.6, 907--914 (1997; Zbl 0894.57014)] who used it, beginning with a symplectic \(\Sigma\) in a simply connected symplectic \(4\)-manifold, to find families of smoothly embedded surfaces which are not smoothly isotopic. In this paper, the authors consider the effect of \(m\)-twist rim surgery on the knot groups \(G=\pi_1(X-\Sigma)\), showing that in some cases (e.g. \(m=1\)) it is unchanged, but in other cases new groups arise. In the case where \(G\) is a ``good'' group -- one for which the \(5\)-dimensional \(s\)-cobordism theorem holds -- they show it is possible to iterate twisted rim surgeries to find families of topologically equivalent surfaces. However, by choosing a symplectically embedded \(\Sigma\) in a symplectic \(X\), Seiberg-Witten invariants can be used to show that these topologically equivalent families are smoothly nonequivalent. An interesting specific example is given, beginning with a specific complex \(\Sigma\) in \(S^2 \times S^2\), and producing for each odd \(p\) a family of topologically equivalent but smoothly nonequivalent surfaces with knot group the dihedral group \(D_{2p}\). The second construction is to show for any group \(G\) satisfying a necessary condition, there is a simply connected symplectic \(4\)-manifold \(X\) and a symplectically embedded surface \(S\) with knot group \(\pi_1(X-S)=G\). Combining this construction with the first shows that for any good \(G\) satisfying the necessary condition (required for \(S\) to be symplectic), there is a simply connected symplectic \(4\)-manifold and an infinite family of smoothly embedded surfaces with knot group \(G\), with the members of the family topologically equivalent but smoothly nonequivalent.