Extension of incompressible surfaces on the boundaries of 3-manifolds

Michael Freedman, Hugh Howards and Ying-Qing Wu

Vol. 194 (2000), No. 2, 335–348

DOI: 10.2140/pjm.2000.194.335

Abstract

An incompressible bounded surface F on the boundary of a compact, connected, orientable 3-manifold M is arc-extendible if there is a properly embedded arc γ on ∂M − IntF 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.

Milestones

Received: 16 June 1998

Revised: 1 February 1999

Published: 1 June 2000