On the existence of infinitely many essential surfaces of bounded genus

Ulrich Oertel

Vol. 202 (2002), No. 2, 449–458

DOI: 10.2140/pjm.2002.202.449

Abstract

A theorem of William Jaco and Eric Sedgwick states that if M is an irreducible, ∂-irreducible 3-manifold with boundary a single torus, and if M contains no genus one essential (incompressible and ∂-incompressible) surfaces, then M cannot contain infinitely many distinct isotopy classes of essential surfaces of uniformly bounded genus. The main result in this paper is a generalization: If M is an irreducible ∂-irreducible 3-manifold with boundary, and M contains no genus one or genus zero essential surfaces, then M cannot contain infinitely many isotopy classes of essential surfaces of uniformly bounded genus.

Milestones

Received: 16 May 2000

Revised: 6 September 2000

Published: 1 February 2002