10 WILLIAM JACO is a level in a tower over f : K H Q , then £(f -) £(f.) and inequality occurs if and only if the level is not trivial. B In the following I shall construct a tower of two-sheeted levels in order to prove the Loop Theorem (and Dehn's Lemma). The tower for the Loop Theorem is constructed as follows: Set 2 K = D and subdivide both K and M so that f is simplicial. Set fn = f, Mn = M and Kn = fn(K). Set V*n equal to a relative regular neighborhood of Kn in MQ such that s n = V 0 ^ d (MQ) is a re ^ative regular neighborhood of fn(BK) in dM0. For ifi : Sn c - dMQ let N~ = (*n^* ^ * Having constructed a tower of two-sheeted levels of height k over f : K Mn, if there exists a two-sheeted covering of V,, then we can extend this tower to a tower of two-sheeted levels k' of height k + 1 by letting M. , be a two-sheeted covering of V, with simplicial covering projection p, -. The simplicial map f, : K V, lifts to a simplicial map f^.n : K ^V+v Set K, , = fk.-i(K). Set V, , equal to a relative regular neighborhood of K, - in M,- such that S , = V, , H dM, - is a relative regular neighborhood of f, , (dK) in d(M, .,). For \+\ : S k+i C ^ ^k+p let N k+1= (p k+l ° Vl^V' Notice that if [fkI dI)2] * \ then [f k+il^ *\+v 2 1.23. LEMMA: Suppose that f : D M satisfies the hypothesis of the Loop theorem. Let 1.15 be_ a tower of two-sheeted levels over 2 f : D M OJEheight n and defined as above. If the tower 1.15 cannot be extended to a tower of two-sheeted levels over f height 2 n + 1, then there exists an embedding. g : D V such that_ n n _
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