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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Essential embeddings of annuli and Möbius bands in $ 3$-manifolds


Authors: James W. Cannon and C. D. Feustel
Journal: Trans. Amer. Math. Soc. 215 (1976), 219-239
MSC: Primary 57A10
DOI: https://doi.org/10.1090/S0002-9947-1976-0391094-1
MathSciNet review: 0391094
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Abstract: In this paper we give conditions when the existence of an ``essential'' map of an annulus or Möbius band into a 3-manifold implies the existence of an ``essential'' embedding of an annulus or Möbius band into that 3-manifold.

Let $ {\lambda _1}$ and $ {\lambda _2}$ be disjoint simple ``orientation reversing'' loops in the boundary of a 3-manifold M and A an annulus. Let $ f:(A,\partial A) \to (M,\partial M)$ be a map such that $ {f_\ast}:{\pi _1}(A) \to {\pi _1}(M)$ is monic and $ f(\partial A) = {\lambda _1} \cup {\lambda _2}$. Then we show that there is an embedding $ g:(A,\partial A) \to (M,\partial M)$ such that $ g(\partial A) = {\lambda _1} \cup {\lambda _2}$.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0391094-1
Keywords: Essential map, essential embedding, annulus, Möbius band
Article copyright: © Copyright 1976 American Mathematical Society