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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Nonorientable surfaces in some non-Haken $ 3$-manifolds


Author: J. H. Rubinstein
Journal: Trans. Amer. Math. Soc. 270 (1982), 503-524
MSC: Primary 57N10; Secondary 57N37, 57R95
MathSciNet review: 645327
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Abstract: If a closed, irreducible, orientable $ 3$-manifold $ M$ does not possess any $ 2$-sided incompressible surfaces, then it can be very useful to investigate embedded one-sided surfaces in $ M$ of minimal genus. In this paper such $ 3$-manifolds $ M$ are studied which admit embeddings of the nonorientable surface $ K$ of genus $ 3$. We prove that a $ 3$-manifold $ M$ of the above type has at most $ 3$ different isotopy classes of embeddings of $ K$ representing a fixed element of $ {H_2}(M,\,{Z_2})$. If $ M$ is either a binary octahedral space, an appropriate lens space or Seifert manifold, or if $ M$ has a particular type of fibered knot, then it is shown that the embedding of $ K$ in $ M$ realizing a specific homology class is unique up to isotopy.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0645327-7
PII: S 0002-9947(1982)0645327-7
Keywords: Non-Haken $ 3$-manifold, one-sided Heegaard splitting, isotopy class of embeddings, genus $ 3$ nonorientable surface
Article copyright: © Copyright 1982 American Mathematical Society