Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
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
MathSciNet review: 0413105
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57A10

Retrieve articles in all journals with MSC: 57A10


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0413105-2
PII: S 0002-9939(1976)0413105-2
Article copyright: © Copyright 1976 American Mathematical Society