4 WILLIAM JACO o 9 9 Then there exists an embedding g : D M such that g ] dD = f | 3D . Sometimes it is helpful to be aware that: 1.7. REMARK: A contractible simple closed curve is orientation preserving. Indeed, if two simple closed curves J and K in a 3-manifold M are homologous in M, then J is orientation-preserving iff K is orientation-preserving. This follows from another useful observa- tion. Namely if a s.c.c. in the boundary of a 3-manifold M bounds a surface in M, then it is orientation preserving ^n the boundary of M. 1.8. REMARK: Dehn's Lemma is usually stated by saying that there exists a neighborhood A of 3D such that f j A is an embedding and f (f(A)) = A (i.e. that the singularities of f miss a neighbor- 2 hood of ol ). Such a version clearly follows from the Loop Theorem (1.1). However, it was pointed out to me by John Hempel that the above version of Dehn's Lemma (1.6) also follows from the Loop Theorem (1.1). The proof goes like this: Let U be a small tubular neighborhood of 2 f (SD D ) such that f is in general position with respect to dU. Set o 2 M = M - U. Since there are no singularities of f on dl , there 2 2 exists a map f! : D M1 such that ff(dD ) c BU and 2 [f1 | 3D ] i 1 in n (BU). Hence, by the Loop Theorem (1.1) there exists an embedding g : D M1 such that g(dD ) c dU and [g | 3D ] ¥ 1 in n (oU). I want to show that g | dD is a longitude 2 of U. It follows that U, along with a regular neighborhood of g(D), 2 1 3 is a punctured Lens space (allowing S X S and S both as a Lens space). Since f J 3D is the core of U (and as such generates the fundamental group of the Lens space) and is trivial in M, the only
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