THREE-MANIFOLD TOPOLOGY 17 TL(M) f 0. Using the Loop theorem prove that there exists an essential 2-sphere embedded in M. 1.30. EXERCISE: Prove that Dehn!s Lemma, the Loop Theorem, and their generalized versions (1.12 and 1.13) follow from the Sphere Theorem. The next exercises give direct and elementary applications of the theorems discussed in this Chapter. 3 1.31. EXERCISE: Using Dehn's Lemma prove that the knot k e s is 3 trivial if and only if T T (S - k) is infinite cyclic. 1.32. EXERCISE: Assume that M is a compact 3-manifold such that every 2-sphere in M bounds a 3-cell in M (M is irreducible). Use the Loop Theorem to prove that M is a cube-with-handles if and only if T T (M) is a free group. 1.33. EXERCISE: Assume that M is a compact, irreducible 3-manifold with connected boundary. Prove that M is a cube-with-handles if and only if T T (dM) — T T (M) is onto. 3 1.34. EXERCISE: Let k be a knot in S . Use the Sphere Theorem to 3 prove that S - k is aspherical. 1.35. EXERCISE: Let M be an irreducible 3-manifold. Let F be a closed component of dM and let F1 ^ F be a component of dM. Suppose that every loop in F is nomotopic to a loop in Ff. Use the Generalized Loop Theorem to prove that if both ker(TT (F) ^ ^ T T (M)) = {l} and kerOT^F1) ^ ^ ^(M)) = {l}, then M is homeomorphic to F X I via a homeomorphism taking F onto F X {0}.

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