THREE-MANIFOLD TOPOLOGY 5 o 2 possibility is that the Lens space is S . Therefore, g | dD is a 2 2 longitude of U. The s.c.c. g(3© ) and f(dD ) cobound an annulus in U and so the map f | dD can be extended to an embedding of D into M. The following theorem is often referred to as the Projective Plane Theorem. The version here is after Epstein [Ep^] and incorporates essentially all of the other major versions [P., Wh , St^J. 1.9. SPHERE THEOREM: Suppose that M is a compact 3-manifold and 2 N i^ j a T T (M)-invariant subgroup of r r (M). Let f : S — M be a map such that [f] £ N. Then there exists £ covering map g : S — g(S ) c M such that g(S2) is,two-sided in M and [g] £ N. 1.10. REMARK: The covering map g in the conclusion of 1.9 has image 2 g(S ) either a 2-sphere or a projective plane. In the case of the 2 projective plane it guarantees that g(S ) is two-sided in M (a 2-sphere is always two-sided in a 3-manifold). If the manifold M is assumed to be orientable (or at least does not admit an embedded, two- sided projective plane), then the conclusion of the Sphere Theorem is 2 that there exists an embedding g : S — M with [g] £. N. The manifold 2 1 M = P X S (which has the property that every embedded 2-sphere in M bounds a 3-cell in M) provides an example where the covering map 2 2 g : S — M must be nontrivial and therefore g(S ) is a two-sided projective plane. 1.11. REMARK: If M is an orientable 3-manifold and T T (M) is not trivial, then it follows from the Sphere Theorem that TT-(M) is either

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