Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

In 1948, Fox showed that a 3-submanifold of the 3-sphere can be reimbedded so that the complement is a union of handlebodies. In 1958, Bing showed that a closed 3-manifold is the 3-sphere if and only if every knot in the manifold can be isotoped to lie within an embedded 3-ball in the manifold. Generalizing these two classical theorems in 3-dimensional topology, as well as the more recent results of Hass and Thompson (1989) and Kobayashi and Nishi (1994), we prove the following: if every knot in a closed connected orientable 3-manifold can be isotoped to lie within a compact connected 3-submanifold, then this submanifold can be reimbedded so that the complement is a union of handlebodies. One of the main tools is the notion of amalgamated Heegaard genus, which arise naturally from amalgamation of Heegaard splittings.