Exotic knottings and symmetries of surfaces in 4-manifolds

Abstract

We study exotic knottings of surfaces in 4-manifolds through their ambient symmetries. We first give a general recipe for producing projectively rigid surfaces, for which every smoothly extendable self-diffeomorphism acts on first homology by plus or minus the identity. For every integer g >0, a refinement of this construction yields a finite sequence of genus-g surfaces F_0, ..., F_2g contained in a 4-manifold X_g. These surfaces are topologically isotopic and topologically flexible: every orientation-preserving self-diffeomorphism of F_i can be realized by a self-homeomorphism of X_g preserving F_i. Successive knotting, however, rules out increasingly many projective homological symmetries, revealing a finer knottedness phenomenon. The first two constructions combine iterated rim surgery with the convex geometry of Newton polytopes of relative Seiberg-Witten invariants. We also use hyperbolic geometry to construct a totally geodesic surface of positive genus whose smooth and topological extendable mapping class groups are both trivial.