WE DESCRIBE a homotopy 4-sphere X4, built with the usual zero and 4-handle and two l-handles and two 2-handles (Figure 28). Of course X4 is homeomorphic to S4 [Freedman] but considerable effort has not led to a proof that X4 is diffeomorphic to S4. X4 has the following virtues: (A) Although it is easy to construct smooth homotopy 4-spheres (e.g. the Gluck… (More)
T T T T T T T T T T T T T T T Abstract Given a hyperbolic 3–manifold M containing an embedded closed geodesic, we estimate the volume of a complete hyperbolic metric on the complement of the geodesic in terms of the geometry of M. As a corollary, we show that the smallest volume orientable hyperbolic 3–manifold has volume > .32.
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It is known that about 70% of surgeries on the figure 8 knot give manifolds which contain immersed incompressible surfaces. We improve this to about 80% by giving a very simple proof that all even surgeries give manifolds containing such a surface. Moreover, we give a quick proof that every (6k, t) surgery is virtually Haken, thereby partially dealing with… (More)