On Embedding Annuli in M 3

Abstract

Let M be a 3-manifold. In this note we give a condition when a proper map of an into M3 can be replaced by a proper embedding of an in M3. Let 2L1 and A2 be disjoint simple loops embedded in the boundary of an orientable 3-manifold M. We suppose that 2, and 22 are freely homotopic in M and that )1 is not nullhomotopic in M. Then it is a consequence of a well-known theorem in [3] that L1u)2 is the boundary of an embedded in M. Unfortunately we can obtain no new information from this theorem when )1 and AS bound an in the boundary of M. The following theorem allows us to construct a annulus in a few special cases when the free homotopy of 2, and ii2 cannot be deformed to bd(M). This theorem seems to be closely related to a problem of R. H. Fox discussed in [1]. The proof of Theorem I is basically an application of the proof of the sphere theorem in [2] and [4]. One might hope that one could use a tower of two sheeted coverings and obtain a more general result. The difficulty here is analogous to the one encountered when trying to prove the sphere theorem via a tower of two sheeted coverings, i.e. one obtains an embedded surface at the top of the tower, but the projection of the surface may not be nontrivial in the original sense. Throughout the remainder of this paper all spaces will be simplicial complexes and all maps will be piecewise linear. Let I be a simple loop embedded in a 3-manifold M and based at a point x. Let ([l]) be the subgroup of 7r,(M, x) generated by [1]. Let N(l) be the normal closure of ([l]) in 7T1(M, x) and C(l) the centralizer of K[I]) in 71(M, X). THEOREM 1. Let M be a compact, orientable 3-mnanifoldsuch that 7T2(M) = 0. Let F be a surface, other than a torus, which is a boundary component of M. Let I be a simple loop embedded in F and not nullhomotopic in M. Then Received by the editors January 14, 1972. AMS 1970 subject classifications. Primary 55A35; Secondary 57C35. ' The author is partially supported by NSF Grant #GP-15357. Q American Mathematical Society 1972