On properly embedding planes in arbitrary $3$-manifolds

Abstract

We prove an analog of the loop theorem for an arbitrary noncompact 3-manifold. In particular, we show that the existence of a proper map of a plane into a 3-manifold implies the existence of a nearby nontrivial embedding of a plane into the 3-manifold. Introduction. In (2) we showed that if M is an eventually end-irreducible 3-mani- fold and f: R2 -- M is a proper essential map, then there is a proper essential embedding g: R2 -_ M. In this paper we remove the restriction to manifolds which are eventually end-irreducible. This permits us to choose the image of g to lie in a preassigned neighborhood of the image of f. This could not be done with the weaker theorem, for even when M is eventually end-irreducible, a regular neighborhood of f(R2) need not be. We also extend the theorem in another direction; roughly, we show that if a normal subgroup of the fundamental group of the complement of a compact set in M is given and large loops around the origin in R2 are mapped by f to homotopy classes not in the subgroup, then g can be chosen to have the same property. The proof of the main theorem, (2.2), parallels that of (2). At one point, (2) makes major use of the eventually end-irreducible hypothesis. In this paper we prove Lemma (2.1) to substitute where eventually end-irreducible was used before. We believe the technique used in Lemma (2.1) is new. We expect it to be generally applicable to 3-manifolds which are not eventually end-irreducible. 1. Notational conventions and a preliminary lemma. We work in the category of simplicial complexes and piecewise linear maps. A map f: X -* Y is proper if f -1(C) is compact for every compact C C Y. If X C Y we use Fr( X) to mean the frontier of X in Y and Cl(X) to mean the closure of X in Y. If M is a manifold, we use aM for the boundary of M. We follow Waldhausen's convention (6) on regular neighbor- hoods; specifically, if X c Y, choose a triangulation of Y in which all previously mentioned subspaces are subcomplexes, and let U(X) be the simplicial neighbor- hood of X in the second barycentric subdivision. If M is a 3-manifold, an exhausting sequence for M will be a sequence { M, } of compact 3-manifolds in M with Mn C Mn+1 - Fr(Mn+1), Mn n aM a 2-manifold,