Extension of incompressible surfaces on the boundary of 3-manifolds

Details Extension of incompressible surfaces on the boundary of 3-manifolds Extension of incompressible surfaces on the boundary of 3-manifolds

1997-06-30

arxiv.org/abs/math/9706222v1

Mathematics

Geometric Topology

Scientific paper

An incompressible surface $F$ on the boundary of a compact orientable 3-manifold $M$ is arc-extendible if there is an arc $\gamma$ on $\partial M - $ Int $F$ such that $F \cup N(\gamma)$ is incompressible, where $N(\gamma)$ is a regular neighborhood of $\gamma$ in $\partial M$. Suppose for simplicity that $M$ is irreducible, and $F$ has no disk components. If $M$ is a product $F\times I$, or if $\partial M - F$ is a set of annuli, then clearly $F$ is not arc-extendible. The main theorem of this paper shows that these are the only obstructions for $F$ to be arc-extendible.

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