Communications in Analysis and Geometry

Volume 21 (2013)

Number 2

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

Pages: 295 – 330



Scott A. Taylor (Department of Mathematics, Colby College, Waterville, Maine, U.S.A.)

Maggy Tomova (Mathematics Department, University of Iowa, Iowa City, Ia., U.S.A.)


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 show that if $H$ is what we call $(T,\Gamma)$-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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