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THREE-MANIFOLD TOPOLOGY 15 (say e x Y £ N). Let U be a small product neighborhood of T(A) such that U hasa parametrization as I X I X I with T(A) = I x I X {l/2}, U f l D = {1}X I X I, I x I X {0}P i ©Q i 0 and I X I X [l] 0 ^ ^ (8, Set ©^ = (bQ - U f l D) and set A1 = I X I X [o}. Then D1 = D' U A! has theproperty that D1 D T(D') has fewer components than D D T(D). • In thenext tworemarks, I am continuing to use thenotation of Lemma 1.25 and itsproof. 1.26. REMARK: In thecase of the sphere theorem if one tries to descend a tower of two-sheeted levels with a solution, then the situation is as 2 follows: there exists an embedding g, : S — V, with [g,] £ N, and 2 one needs to exhibit an embedding gv. i : S — ^k-1 ^ ^ fSv i ] ^ N w i • 2 If S = g,(S), then either S 0 T(S) = 0 or each component of S 0 T(S) is a simple closed curve. If C C is such a component of intersection and a = T(a), then theinduction argument may fail that is, equivariant surgery, as described in Lemma 1.25, may fail to reduce thenumber of components of intersection between the 2-sphere and its image under the covering translation T. However, in this case, it follows that V . is a nonorientable 3-manifold (infact, p, , (Ot) is an orientation reversing curve in V, n ) andone canfind a nontrivial coveringmap § k-i : s 2 - * 8 k-i (s2 ) c V i w i t h [8 k-i] * N k-i *••*- W 5 ^ i s a two-sided projective plane in V, - • 1.27. REMARK: In thecase of theGeneralized Loop Theorem (Generalized Dehn's Lemma) if one tries to descend a tower of two-sheeted levels with a solution, then there exists a planar surface F, with r, boundary