In this paper we prove an analog of the loop theorem for a certain class of noncompact 3-manifolds. In particular, we show that the existence of a "nontrivial" proper map of a plane into a 3-manifold implies the existence of a nontrivial proper embedding of a plane into a 3-manifold. Introduction. In this paper we prove an analog of the loop theorem for a certain class of noncompact 3-manifolds. More precisely we show that the existence of a "nontrivial" proper map of a plane into a noncompact

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... ventually end-irreducible 3-manifold implies the existence of a "nontrivial" proper embedding of a plane into that 3-manifold. We remark that an eventually end-irreducible 3-manifold is essentially a 3-manifold which has an infinite hierarchy. A discussion of proper homotopy and related topics is given in [1]. Notation. All spaces are simplicial complexes and all maps are piecewise linear. We use the notation of Brown and Tucker [1] without change. A 3-manifold is eventually end-irreducible at the end [a] if there is an exhausting sequence {Mn} of compact 3-dimensional submanifolds of M and a compact subset C c int{Mx) with the following property, (a) If A is the component of M -Mn determined by the end [a] and if F is a component of ¥r{A), then the inclusion map trx{F) -» ttx{M -C) induces a monomorphism.