14 WILLIAM JACO small product neighborhood of T(A) such that U has a parametrization as I X I X I with T(A) = I x I X {l/2}, U H D = d(I X I) X I, I X I X {0} n DQ ¥ 0 and I x I X {l} 0 D][ i 0. Set T l Q = (bQ - (U D D)) and set A1 = I x I X {0}. Then D1 = D' U A' has the property that D! f T(D') has fewer components than D D T(D). Case 2. D H T(D) has no s.c.c. components (see Figure 1.2). T(a) T(A) (a) (b) (c) Figure 1.2 Let a be a component of D H T(D) (a is a spanning arc of D) so that there exists an arc 3 in dD having the property that dec = dg o and a U P bounds a disk A in D where A u T(D) =0 i.e. a is "outermost11 on D. The spanning arc T(cc) c ffi H T(D) (which is disjoint from a) divides D into two components Dn and D1, each of which is a disk, and divides dD into two arcs e x and e x such that &Q c dBQ and a. c 3© . Set Y = T(P). By properly choosing orientation, we have [3D] = aQa- = (a Y) (Y~ Oi^ . Therefore, a Y £ N or Y~ a. £ N

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