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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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$3$-manifolds fibering over $S^{1}$
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by Dean A. Neumann PDF
Proc. Amer. Math. Soc. 58 (1976), 353-356 Request permission

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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Additional Information
  • © Copyright 1976 American Mathematical Society
  • 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