THREE-MANIFOLD TOPOLOGY 3 compact surface in dM. Set G = Im(T T (S) c - n (M)). Then G * F * G. * ... * G, where F is a free group and G. « T T (S.) for some closed surface S. (1 i k). Proof: One simply applies 1.1 to prove that K = ker(TT (S) C— TL(M)) is normally generated as a subgroup of T T (s) by a finite, pairwise disjoint collection of simple closed curves in S. I 1.4. EXERCISE: Show 1.3 is true in the case where S is not necessarily compact. Of course, the conclusion may need to be modified to admit a possibly infinite number of factors G.. If M is a 3-manifold, a subgroup H of " T (M) is peripheral if there exists a surface S c 0 M such that H is conjugate in T T (M) into a subgroup of Im(TT (S) c - T T (M)). 1.5. EXERCISE: Any finitely generated peripheral subgroup H of a 3-manifold group has the form H ^ F * H- *... * H where F is a I n free group and H. w T T (S.) for some closed surface S. (1 i n The next result is Dehn's Lemma, first formulated by M. Dehn [D-,] in 1910. However, his proof contained a serious gap which was pointed out by H. Kneser [Kn1] in 1927. A satisfactory solution to Dehn!s Lemma was given by C. D. Papakyriakopoulos in 1956 along with his versions of the Loop Theorem and the Sphere Theorem [P-,P«]. 1.6. DEHN'S LEMMA: Suppose that M is, a 3-manifold and that 2 9 f : D M is a , map such that f | 3D i^ an embedding and -1 2 2 2 f (f(dD )) = 3D (i.e. the singularities of f do not meet 3D).
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