Symplectic submanifolds from surface fibrations. (Q5943436)

There exist homology classes in (non-simply connected) 4-manifolds, which admit infinitely many pairwise non-isotopic symplectic representatives. An example is provided by \(2m[F_g]\) in \(F_g\times S^2 (m>1,g>0)\) with the geometric representative having genus \(r\) such that \(2-2r=2m(2-2g)\). For \(g=1\), \(m=2\), this follows from results of H. Geiges on \(T^2\)-bundles over \(T^2_1\) which the author then generalises to higher values of \(g\) and \(m\). As a corollary one has that a blow up to any simply-connected complex projective surface contains a symplectic surface not isotopic to any complex curve. The reviewer also finds the constrution interesting for the study of symplectic forms on 4-manifolds admitting geometric structures in the sense of W. Thurston.