# Fox re-embedding and Bing submanifolds

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Kei Nakamura, Temple University
Fine Hall 314

Let $M$ be an orientable closed connected 3-manifold, and $Y$ be a connected compact 3-manifold. We show that the following two conditions are equivalent: (i) $Y$ can be embedded in $M$ so that the closure of the complement of the image of $Y$ is a union of handlebodies; and (ii) $Y$ can be embedded in $M$ so that every embedded closed loop in $M$ can be isotoped to lie within the image of $Y$. Our result can be regarded as a common generalization of Fox's reimbedding theorem (1948) and Bing's characterization of 3-sphere (1958), as well as more recent results of Hass and Thompson (1989) and Kobayashi and Nishi (1994).