C-essential surfaces in (3-manifold, graph) pairs

Scott Taylor; Maggy Tomova

doi:10.4310/cag.2013.v21.n2.a2

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arXiv · 0910.3251

C-essential surfaces in (3-manifold, graph) pairs

Scott Taylor·Maggy Tomova

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Abstract

Let $T$ be a graph in a compact, orientable 3--manifold $M$ and let $Γ$ be a subgraph. $T$ can be placed in bridge position with respect to a Heegaard surface $H$. We show that if $H$ is what we call $(T,Γ)$-c-weakly reducible in the complement of $T$ then either a "degenerate" situation occurs or $H$ can be untelescoped and consolidated into a collection of "thick surfaces" and "thin surfaces". The thin surfaces are c-essential (c-incompressible and essential) in the graph exterior and each thick surface is a strongly irreducible bridge surface in the complement of the thin surfaces. This strengthens and extends previous results of Hayashi-Shimokawa and Tomova to graphs in 3-manifolds that may have non-empty boundary.

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BibTeX RIS

Scott Taylor, Maggy Tomova. 2012-10-26. C-essential surfaces in (3-manifold, graph) pairs. https://doi.org/10.4310/cag.2013.v21.n2.a2

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