$3$-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.References
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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