2 WILLIAM JACO For here a singular mapping of a planar surface (a surface is planar 2 if it can be embedded in R , the plane) or of a sphere into a 3- manifold is replaced by a non-singular mapping, while preserving certain prescribed conditions. Of first importance, however, is the case when the surface is one of the basic 2-elements, the 2-disk or the 2-sphere. The version of the Loop Theorem given here is after J. Stallings [St-], 1.1. LOOP THEOREM: Suppose that M Is^ a _ 3-manifold, S is a connected 2 surface in dM and N is^a ^ normal subgroup of T T (S). Let f : D — M 2 i 2 be a ma£ such that f(dD ) c S c dM and [ f | 3D ] j N. Then there 2 2 exists a n embeddin g g : D — M suc h tha t g(d D ) c S c ^ M an d [ g | 3D 2 ] J E N. To obtain a possibly better understanding of I.l consider the following, otherwise not so obvious, consequences. 1.2. COROLLARY: Suppose that M jjs a 3-manifold and that S is a connected surface in dM. Set K equal to the normal subgroup ker{TT (S)c- T T (M)}. If K i [l], then a nontrivial element of K can be represented by a simple closed curve (s.c.c.) in S. Proof: One simply applies I.l in the case N = {l}. • Contrast this to the fact that many normal subgroups of T T (S) have no such element. Indeed, this is already the case for S = S X S and any normal subgroup of TT (S) generated by a nontrivial, non-primitive element. In fact, it is a very interesting question as to when the kernel of a homomorphism that is induced by a mapping between two closed, orientable surfaces needs to contain such an element (see [Ed-] and [Tu9]). 1.3. COROLLARY: Suppose that M is a 3-manifold and that S is a

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