Heegaard surfaces for certain graphs in compressionbodies

TL;DR

This paper extends classification methods for bridge surfaces in compressionbodies containing graphs, providing tools to analyze their structure and implications for essential surfaces in 3-manifolds.

Contribution

It generalizes existing techniques to classify bridge surfaces for graphs in compressionbodies, advancing understanding of their topological properties.

Findings

Classified bridge surfaces for graphs in compressionbodies.

Identified conditions leading to degenerate or essential surfaces.

Extended methods of Hayashi and Shimokawa to new settings.

Abstract

Let M be a compressionbody containing a graph T (with at least one edge) such that \boundary_+ M is parallel to the union of T and \boundary_- M. We extend methods of Hayashi and Shimokawa to classify bridge surfaces for T. The results of this paper are used in later work to show that if a bridge surface for a graph in a 3-manifold is c-weakly reducible then either a degenerate situation occurs or the exterior of the graph contains an essential meridional surface.