A Curve Complex and Incompressible Surfaces in $S\times \mathbb{R}$

TL;DR

This paper uses Morse theory to show that all incompressible surfaces in $S\times \mathbb{R}$ with boundary curves homotopic to given multicurves are homotopic to surfaces constructed from paths in a homology curve complex, linking complex distance to Seifert genus.

Contribution

It proves that all incompressible surfaces with specified boundary are homotopic to those constructed from the homology curve complex, connecting geometric and topological invariants.

Findings

All embedded incompressible surfaces are homotopic to constructed surfaces.

Distance in the homology curve complex relates to Seifert genus.

Provides a Morse theoretic proof for surface classification.

Abstract

Various curve complexes with vertices representing multicurves on a surface $S$ have been defined, for example [3], [4] and [8]. The homology curve complex $\mathcal{HC}(S,\alpha)$ defined in [7] is one such complex, with vertices corresponding to multicurves in a nontrivial integral homology class $\alpha$. Given two multicurves $m_1$ and $m_2$ corresponding to vertices in $\mathcal{HC}(S,\alpha)$, it was shown in [8] that a path in $\mathcal{HC}(S,\alpha)$ connecting these vertices represents a surface in $S\times \mathbb{R}$, 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 $S\times \mathbb{R}$ with boundary curves homotopic to $m_{2}-m_1$ are homotopic to a surface constructed in this way. This is used to relate distance between two vertices in $\mathcal{HC}(S,\alpha)$ to the Seifert genus of the corresponding link in $S\times \mathbb{R}$.