Fox reimbedding and Bing submanifolds

Nakamura, Kei

doi:10.1090/tran/6044

Fox reimbedding and Bing submanifolds
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by Kei Nakamura PDF

Trans. Amer. Math. Soc. 367 (2015), 8325-8346 Request permission

Abstract:

Let $M$ be an orientable closed connected $3$-manifold. We introduce the notion of an amalgamated Heegaard genus of $M$ with respect to a closed separating $2$-manifold $F$, and use it to show that the following two statements are equivalent: (i) a compact connected 3-manifold $Y$ can be embedded in $M$ so that the exterior of the image of $Y$ is a union of handlebodies; and (ii) a compact connected $3$-manifold $Y$ can be embedded in $M$ so that every knot in $M$ can be isotoped to lie within the image of $Y$.

Our result can be regarded as a common generalization of the reimbedding theorem by Fox (1948) and the characterization of $3$-sphere by Bing (1958), as well as more recent results of Hass and Thompson (1989) and Kobayashi and Nishi (1994).

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