THREE-MANIFOLD TOPOLOGY 11 g (oD2)cs c l and [g I dD2] t N . 6 n n n LO Proof: By 1.18 and the hypothesis of this lemma, it follows that V has no two-sheeted covering spaces. Hence, by 1.19 each component of dV is a 2-sphere. It follows that the 2-manifold S is spherical and therefore T T (s ) is generated by bS . Since n i n n o [f | dD ] £ N , the conjugacy class determined by some component of hS does not belong to N . This class has a representative which n n is a simple closed curve in S and each simple closed curve in S r n n bounds a disk embedded in oV (and therefore bounds a disk embedded n 2 in V ). Let g : D — V be an embedding which realizes such n' °n n ° a disk. B 1.24. REMARK: If we consider the "top" of a tower as the level at which the tower cannot be extended in a desired fashion (in the case of the Loop Theorem a level at which V has no two-sheeted coverings), then we can interpret Lemma 1.23 as simply stating that there exists a solution to the Loop Theorem at the top of a tower of two-sheeted levels. In the case of the Generalized Loop Theorem (Generalized Dehn's Lemma) again the method of proof is to exhibit a solution at the top of a tower of two-sheeted levels. However, in this case the top occurs at a level in which it may not be true that each component of dV_ is xx a 2-sphere therefore there is more work in exhibiting a solution. The top of the tower is precisely where this approach falters in the case of the Sphere Theorem. Even though it is necessary that a solution to the Sphere Theorem exists at the top of a tower of two-sheeted levels, I do not know of a direct method for finding it. Notice that it is true,

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