On the asphericity of a symplectic

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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