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On the asphericity of a symplectic $M^3\times S^1$


Author: John D. McCarthy
Journal: Proc. Amer. Math. Soc. 129 (2001), 257-264
MSC (1991): Primary 53C15; Secondary 57M50, 57N10, 57N13
DOI: https://doi.org/10.1090/S0002-9939-00-05571-4
Published electronically: August 29, 2000
MathSciNet review: 1707526
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Abstract: C. H. Taubes asked whether a closed (i.e. compact and without boundary) connected oriented three-dimensional manifold whose product with a circle admits a symplectic structure must fiber over a circle. An affirmative answer to Taubes' question would imply that any such manifold either is diffeomorphic to the product of a two-sphere with a circle or is irreducible and aspherical. In this paper, we prove that this implication holds up to connect sum with a manifold which admits no proper covering spaces with finite index. It is pointed out that Thurston's geometrization conjecture and known results in the theory of three-dimensional manifolds imply that such a manifold is a three-dimensional sphere. Hence, modulo the present conjectural picture of three-dimensional manifolds, we have shown that the stated consequence of an affirmative answer to Taubes' question holds.


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Additional Information

John D. McCarthy
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: mccarthy@math.msu.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05571-4
Keywords: Symplectic, manifold, aspherical
Received by editor(s): April 6, 1999
Published electronically: August 29, 2000
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2000 American Mathematical Society

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