Essential surfaces in (3-manifold, graph) pairs and leveling edges of Heegaard spines

Details Essential surfaces in (3-manifold, graph) pairs and leveling edges of Heegaard spines Essential surfaces in (3-manifold, graph) pairs and leveling edges of Heegaard spines

2009-10-17

arxiv.org/abs/0910.3251v2

Mathematics

Geometric Topology

31 pages, 5 figures. Minor improvements in the exposition

Scientific paper

Let $T$ be a graph in a compact, orientable 3--manifold $M$ and let $\Gamma$ be a subgraph. $T$ can be placed in bridge position with respect to a Heegaard surface $H$. We use untelescoping and consolidation operations to show that if $H$ is what we call $(T,\Gamma)$-c-weakly reducible in the complement of $T$ then either the exterior of $T$ contains an essential meridional surface or one of several ``degenerate'' situations occurs. This extends previous results of Hayashi-Shimokawa and Tomova to graphs in 3-manifolds which may have non-empty boundary. We apply this result to the study of leveling edges of trivalent Heegaard spines.

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