16 WILLIAM JACO k k components b ..., b and an embedding g : F, — V, such that k k i k k g (b.)c S. c: £M, and for some j [g, | b.] f. N.. One needs to exhibit a planar surface F, , with r, . boundary components b. . , ..., b k-1 k-1c ^"^ and an embedding g ^ : F ^ - V ^ with g^O ^ ) c s n ^-i and for some m, [g. - I b ] ? N °k- Set F = g^O^)* then either F H T(F) = 0 or each component of intersection is a spanning arc or a simple closed curve. Now, I have been able to make the method of equivariant surgery presented here successful in this case by using that F is planar and if J, K are components of dF, then either J f l T(K) = 0 or J = K. This latter fact follows from the hypothesis that the singular images of distinct components of the boundary of the planar surface do not meet. Hence, if a component of F f T(F) is a s.c.c. a, then both a and T (a) separate both F and T(F) and equivariant surgery can be done. If a component of F H T(F) is a spanning arc (X , then either a has both of its end points in the same component J of dF (and, by the above observation (hypothesis), T(a) has both of its end points in T(J)c dT(F)), so both a and T (a) separate both F and T(F) and equivariant surgery can be done or both a and T (a) have one end point in J c dF and the other end point in K c BF (J i K) (again, by the above observation (hypothesis)), so together c c and T(cc) separate both F and T(F) and equivariant surgery can be done. 1.28. QUESTION: Is the Generalized Loop Theorem (1.13) valid if we eliminate the hypothesis that f(J.) f f(J.) = 0 for i i j? 1.29. EXERCISE: Suppose that M is a simply connected 3-manifold and

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