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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

$ 3$-manifolds fibering over $ S\sp{1}$


Author: Dean A. Neumann
Journal: Proc. Amer. Math. Soc. 58 (1976), 353-356
MSC: Primary 57A10
DOI: https://doi.org/10.1090/S0002-9939-1976-0413105-2
MathSciNet review: 0413105
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Abstract: Let $ M$ be a closed $ 3$-manifold that is the total space of a fiber bundle with base $ {S^1}$ and fiber the closed $ 2$-manifold $ F$. Assume that genus $ (F) \geq 2$ if $ F$ is orientable, and that genus $ (F) \geq 3$ if $ F$ is nonorientable. We say that $ M$ has unique fiber over $ {S^1}$ if, for any fibering of $ M$ over $ {S^1}$ with fiber $ F'$, we have $ F' \cong F$. We prove that $ M$ has unique fiber over $ {S^1}$ if and only if rank $ ({H_1}(M;{\mathbf{Z}})) = 1$. In the case that rank $ ({H_1}(M;{\mathbf{Z}})) \ne 1,M$ fibers over $ {S^1}$ with fiber any of infinitely many distinct closed surfaces.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0413105-2
Article copyright: © Copyright 1976 American Mathematical Society

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