Embedded surfaces with infinite cyclic knot group

Abstract

We study locally flat, compact, oriented surfaces in $4$-manifolds whose exteriors have infinite cyclic fundamental group. We give algebraic topological criteria for two such surfaces, with the same genus $g$, to be related by an ambient homeomorphism, and further criteria that imply they are ambiently isotopic. Along the way, we prove that certain pairs of topological $4$-manifolds with infinite cyclic fundamental group, homeomorphic boundaries, and equivalent equivariant intersection forms, are homeomorphic.

v2 fixes an error in the proof of Theorem 1.3. The issue in the proof Theorem 5.10 (now Theorem 5.11) has been corrected. v3, v4 are reorganisations; new figures and applications are added. v5: Added report number. v6: Fixed the definition of a trivial 1-handle stabilisation. To appear in Geometry & Topology. v7: Fixes an error: Theorems 1.7, 1.8 on n-roll 1-twist rim surgery only hold for n=0