6 WILLIAM JACO infinite cyclic or splits as a nontrivial free product. A partial converse to this (the so-called Kneser Conjecture that if M is a closed, orientable 3-manifold and T T (M) is either infinite cyclic or splits as a nontrivial free product, then there exists an essential 2-sphere embedded in M) can be proved without the aid of the Sphere Theorem (see [He-], [StJ and [Wh ]). I will give the statements of the two main generalizations of 1.1 and 1.6. 1.12. GENERALIZED DEHN'S LEMMA [S-W]: Let M be,a 3-manifold and let D be a compact planar surface with boundary components J.., ..., J,. Suppose that f : D M JLS_a _ map such that f | 3D is,an embedding, f (f(BD)) = dD and f(J.) is orientation preserving in M for each i (1 i k). Then there exists a compact planar surface D1 with boundary components Jl , ..., J', and an embedding ff : D' M such that for each j (1 j kf) there exists a unique i. (li.k) so that ff(J!) = f(J. ). 1.13. GENERALIZED LOOP-THEOREM [W-] : Let, M be.a. 3-manifold and let D be a compact, planar surface with boundary components J-, ..., J . Let N1, ..., N, be normal subgroups of TL(M). Suppose that f : D - M is a map such that f (dD) c:dM, [f | J.] (EN., f(J.) is. orientation preserving in M for each i, and f(J.) 0 f(J.) = 0 for i ^ j. Then ^iven regular neighborhoods U. of f (J.) .in 3M there is a compact, planar surface D* with boundary components Jp ..., J's and an embedding g : D1 - M such that for each j C1 j S k ') there exists £ unique i. (1 i. k) that g(J!.) c u. J J J ij
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