Totally peripheral 3-manifolds

Matthew G. Brin, Klaus Johannson and Peter Scott

Vol. 118 (1985), No. 1, 37–51

DOI: 10.2140/pjm.1985.118.37

Abstract

We will say that a 3-manifold M is totally peripheral, or TP, if every loop in M is freely homotopic into the boundary ∂M of M. In this paper, we show that if M is a compact, orientable, 3-manifold which is TP, then there is a component F of ∂M such that the natural map π₁(F) → π₁(M) is surjective. In the non-orientable case, this result is almost true but there is essentially one counterexample.

Mathematical Subject Classification 2000

Primary: 57N10

Secondary: 57M05

Milestones

Received: 1 July 1983

Revised: 14 December 1983

Published: 1 May 1985