THREE-MANIFOLD TOPOLOGY 13 and T(D) meet transversally. Then D D T(D) is either empty (the desirable situation) or consists of a pairwise disjoint collection of simple closed curves and spanning arcs. I then use the union B U T(D) to find a disk D1 such that a map realizing P1 as an embedding of 2 D into M, is a solution to the Loop theorem, D1 meets T(D§) transversely and D1 f l T(Df) has fewer components than D f l T(D). If I continue to call this new embedding g,, then it is clear that inductively, I will obtain the desired solution. Suppose that D f l T(D) i 0. Case 1. D C\ T(D) has a simple closed curve component (Figure 1.1)• T(A) (a) (b) Figure 1.1 (c) In this case let a be a simple closed curve component of o D f l TQ) such that a bounds a disk A on D and A f T(D) = 0 i.e. A is "innermost" on D. The simple closed curve T(ct) (which may be equal to a if a is invariant) divides B into two components: an annulus Dn, which contains dD, and a disc D-. Let U be a

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