2011/08/21 by Irmer, Ingrid
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1108.4206
Various curve complexes with vertices representing multicurves on a surface S have been defined, for example [3], [4] and [8]. The homology curve complex HC(S,α) defined in [7] is one such complex, with vertices corresponding to multicurves in a nontrivial integral homology class α. Given two multicurves m1 and m2 corresponding to vertices in HC(S,α), it was shown in [8] that a path in HC(S,α) connecting these vertices represents a surface in S× ℝ, and a simple algorithm for constructing minimal genus surfaces of this type was obtained. In this paper, a Morse theoretic argument will be used to prove that all embedded orientable incompressible surfaces in S× ℝ with boundary curves homotopic to m2-m1 are homotopic to a surface constructed in this way. This is used to relate distance between two vertices in HC(S,α) to the Seifert genus of the corresponding link in S× ℝ.