12 WILLIAM JACO and easy to prove, that for any compact 3-manifold V with T T (V) finite and Tt (V) not trivial, then dV is a nonempty collection of 2-spheres and T T (V) is generated as a T T (V)-module by the components of BV. Next, I consider the method of descending a tower with a solution. For simplicity's sake, as well as continuity of my presenta- tion, I shall give detail only in the case of proving the Loop Theorem (Dehnfs Lemma). It is true, however, that particularly subtle points appear in descending a tower with a solution to the Sphere Theorem or with a solution to the Generalized Loop Theorem (Generalized Dehnfs Lemma). I will say more about this in the remarks immediately following the proof of Lemma 1.25. 1.25. LEMMA: Using the notation established above, suppose that 1.15 is a tower of two-sheeted levels for the Loop Theorem. _If 1.15 has a solution at height k (k 1), then 1.15 has £ solution at height 2 k - 1 i.e. if there exists an embedding g, : D — M, such that gk(S© ) c s, c:dM, and [g, | 3BD ] f N,, then there exists an 2 2 embedding g, , : 1 © —- M, , such that g 1 (dD ) = s, , c: °*1 i and [s k-l I dl2] ? h i - proof: Let T be the nontrivial covering translation. The 2 idea is that a solution g, : D — M, can be found such that 2 D D T(D) = 0 where D = g,(D )• Then it follows that g. _ = p,o g 2 is the desired embedding of B into M, -. 2 To begin, I take the given embedding g : D —^ M, and set 2 D = g (D ). I may assume that adjustments have been made so that D

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