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To e TT 1 (M,X ( ) ), Let and 6 be a = f implies <5 is freely homotopic to f(<\$). XQ to y,. 2 y^ = . 1 shows that M=Ft x I. Lemma at and let Then at least HOMEOMORPHISMS OF 3-MANIFOLDS WITH COMPRESSIBLE BOUNDARY 49 (a) M is an I-bundle over a closed 2-manifold and deg (b) f is isotopic (rel x ) to a homeomorphism g such that g| F =l F for every x o (f)=-l incompressible boundary component F of M. Proof: Let F be an incompressible boundary component. 1. basepoint y e F.

W ± (u). ,l) - V. For each u e ft, there = T T ^ C V C U ) ^ (u)) coming Precisely, V - un ^ i=m+I is The closure of V(u)-Vi(u) is denoted by an i(u):V'(u) from is the a canonical coordinates let D1 —o D2 be the invariant Al 1=111+1 identity map j(u):Vf • V f (u). • V(u) be the inclusion maps. is i(u)# j(u)#i# . If g:(V(u),xQ(u)) the pieces of V(u). arc of - lun , and let Vf 2 Y . ^-D1). i Let = For each u there i:Vf •V and Then the canonical isomorphism • (V(uf ),x0 (u f )) is a homeoraorphism, then the automorphism induced by g on TT (V,x ) is iyrj(uf),, i(u')„ g* i(u )# j(u) # i# .

The generating set for this presentation includes the automorphisms induced by the standard homeomorphisms of V defined in Chapter II. (V,x ) = G = G * G 0 * ... * G * G , * . . * G , where G. is the infinite cyclic J 1 2 m m+1 n' i group generated by group of a closed a. G. for i < m, and connected aspherical n, i < i < ni for some k, then V. and = IT (V. 3. 3. )„ i > m+1. : F nk-1 for fundamental with x I • v.. Aut(G. ). | g. , m+1 < j < n} generates G. To define automorphisms of G, we will describe their effects on each of these generators.