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Of) t = glS 1 × O must be (a m u l t i p l e of) k I = glO × S I. We argue a n a l o g o u s l y Let G to a map of a tube into Precisely ~Q~e end-point o_~f k 4 , 1 Q corollary. since as in Case i. (M,~), F, M lies in a curve of f-iF. (M,m),= and i__ss aspherical. d. 9. Let f: is admissibly (G~) deformed + in (M~m) be (M~) so that the number of curves of f-iF is as small as possible. Then fiG: (G,~) ~ surface obtained Proof. 9. Thus, that fIGl is inessential in if splitting a__t f-IF. of essential (GI,~I) (M,~)~ then (G,~) is the curves in is a c o m p o n e n t (GI~I) (G~), of and (G,~) such must be either a 46 square, or an annulus.

In (M,~). Then, by definition, g: (D,~) ~ (M,~) such that and that the r e s t r i c t i o n essential deformation in (D,~). g singular curve in of g i__n ( M ~ ) ~ (D,~) + if and (M,m) such i i 3, g- 1 F is equal to one side in F. Therefore let F be inessential is an i-faced disc, to one side~ k, of F. sinqular disc g: that and useful g curves is chosen so that it has the above proper- in addition, the number of components of g -I F is as small as possible. Assume g-iF ~ k. Then there exists at least one curve, kl, of g-iF w h i c h is d i s j o i n t (DI,~I) ~ from (D,d)= such that glkl cannot be essential instead of (DI,~I) ~ and applying in k.

2. E v e r y curve of f-iF is e s s e n t i a l in 3. If the r e s t r i c t i o n o__f f an admissible o___f f G. t__o every c o m p o n e n t o__f G sinqular square, (M~m) w h i c h is e s s e n t i a l in Remark. 4 can be g e n e r a l i z e d to essential maps. Proof. 1-3. :8 Proposition. Let (M,m) be an irreducible 3-manifold. b e an a d m i s s i b l e surface in o_f F Let F (M~m), w i t h F n 5M = 5F~ b u t no c o m p o n e n t ~ 2-sphere or an i-faced disc, the m a n i f o l d o b t a i n e d bv splittinq 1 i i i 3.