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Fox reimbedding and Bing submanifolds


Author: Kei Nakamura
Journal: Trans. Amer. Math. Soc. 367 (2015), 8325-8346
MSC (2010): Primary 57N10, 57M27; Secondary 57N12, 57M50
DOI: https://doi.org/10.1090/tran/6044
Published electronically: September 1, 2015
MathSciNet review: 3403057
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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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Additional Information

Kei Nakamura
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Address at time of publication: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08901
Email: nakamura@temple.edu

DOI: https://doi.org/10.1090/tran/6044
Received by editor(s): February 18, 2012
Received by editor(s) in revised form: August 24, 2012, and December 2, 2012
Published electronically: September 1, 2015
Article copyright: © Copyright 2015 American Mathematical Society