Extending immersions of curves to properly immersed surfaces

Abstract

Proper generic immersions of compact one-dimensional manifolds in surfaces are studied. Suppose an immersion γ of a collection of circles is given with an even number of double points in a closed surface G. Then γ extends to various proper immersions of surfaces in three-manifolds that are bounded by G. Some of these extensions do not have triple points. The minimum of the genera of the triple point free surfaces is an invariant of the curve. An algorithm to compute this invariant is given. Necessary and suffecient conditions determine if a given collection δ of immersed arcs in a surface F maps to the double points set of a proper immersion. In case the conditions are satisfied, an immersion of F into a three-manifold that depends on δ is constructed explicitly. In the process, the possible triple points of immersed surfaces in three-manifolds are categorized. The techniques are applied to find examples of curves in surfaces that do not bound immersed disks in any three-manifold.