Knotted surfaces in 4-manifolds (Q2836447)

The main result is that for a symplectic surface \(\Sigma_0\) of genus \(g \geq 1\) smoothly embedded in a closed symplectic 4-manifold \(X\) such that the complement \(X-\Sigma_0\) is simply-connected and the self-intersection of \(\Sigma_0\) is equal to or greater than \(2-2g\), there exist infinitely many smooth surfaces \(\Sigma_n \subset X\), \(n=1,2,\ldots\), homologous to \(\Sigma_0\), that are topologically equivalent but smoothly distinct, i.e. there exist homeomorphisms of pairs \((X, \Sigma_n) \simeq (X, \Sigma_m)\) for all \(n,m \geq 0\), but no diffeomorphisms between these pairs for \(n \neq m\). This is an extended result of a similar result with a stronger assumption that the self-intersection is non-negative, shown by \textit{R. Fintushel} and \textit{R. J. Stern} [Math. Res. Lett. 4, No. 6, 907--914 (1997; Zbl 0894.57014)].NEWLINENEWLINEThe main result is shown by using several auxiliary results that may be of independent interest. Specifically, the author gives a version of the knot surgery formula for Seiberg--Witten invariants, in the cases of closed manifolds and manifolds with boundary, analogous to a result of \textit{R. Fintushel} and \textit{R. J. Stern} [Invent. Math. 134, No. 2, 363--400 (1998; Zbl 0914.57015)]. This result is shown by an argument similar to the original one based on the skein relation for the Alexander polynomial, together with the following gluing results: One is a result given in this paper, which describes the effect of a logarithmic transformation on the relative and absolute Ozsváth--Szabó 4-manifold invariants, analogous to a result in Seiberg--Witten theory due to \textit{J. W. Morgan, T. S. Mrowka} and \textit{Z. Szabó} [Math. Res. Lett. 4, No. 6, 915--929 (1997; Zbl 0892.57021)], and the other is a result on Ozsváth--Szabó 4-manifold invariants of fiber sums due to the author and \textit{S. Jabuka} [Geom. Topol. 12, No. 3, 1557--1651 (2008; Zbl 1156.57026)]. Further, the author gives a calculation of the twisted Heegaard Floer homology for circle bundles over surfaces, whose degree is sufficiently large in absolute value, by using the techniques as applied in the calculation of knot Floer homology with integer coefficients by \textit{P. Ozsváth} and \textit{Z. Szabó} [Adv. Math. 186, No. 1, 58--116 (2004; Zbl 1062.57019)].