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Transactions of the American Mathematical Society

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

 

 

$ 2$-sided embeddings of projective planes into $ 3$-manifolds


Author: Mitsuyuki Ochiai
Journal: Trans. Amer. Math. Soc. 274 (1982), 641-650
MSC: Primary 57N10; Secondary 57M40, 57Q25
DOI: https://doi.org/10.1090/S0002-9947-1982-0675072-3
MathSciNet review: 675072
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Abstract: Let $ M$ be a nonorientable closed $ 3$-manifold which admits a $ 2$-sided embedding of a projective plane. Then we first prove the following theorem: If $ M$ has a Heegaard splitting of genus two, then $ M$ is homeomorphic to $ {P^{2}}\times {S^{1}}$. Next, let $ M$ be a nonorientable $ 3$-manifold whose fundamental group is abelian. We verify that if $ M$ has a Heegaard splitting of genus two, then $ M$ is either the nonorientable $ 2$-sphere bundle over the circle or $ {P^{2}}\times {S^{1}}$.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0675072-3
Article copyright: © Copyright 1982 American Mathematical Society