EXTENSION OF INCOMPRESSIBLE SURFACES ON THE BOUNDARY OF 3-MANIFOLDS

MICHAEL FREEDMAN HUGH HOWARDS AND YING-QING WU

Abstract. An incompressible surface F on the boundary of a compact orientable 3-manifold M is arc-extendible if there is an arc γ on ∂M− Int F such that F ∪N(γ) is incompressible, where N(γ) is a regular neighborhood of γ in ∂M. Suppose for simplicity that M is irreducible, and F has no disk components. If M is a product F × I, or if ∂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.