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

Published electronically:
August 29, 2000

MathSciNet review:
1707526

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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.

**[A]**J.W. Alexander,*On the subdivision of -space by a polyhedron*, Proc. Nat. Acad. Sci. U.S.A.**10**(1924), 6-8.**[FS]**Ronald Fintushel and Ronald J. Stern,*Immersed spheres in 4-manifolds and the immersed Thom conjecture*, Turkish J. Math.**19**(1995), no. 2, 145–157. MR**1349567****[H]**John Hempel,*Residual finiteness for 3-manifolds*, Combinatorial group theory and topology (Alta, Utah, 1984) Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 379–396. MR**895623****[K1]**Vincent Cavalier,*Pseudogroupes complexes quasi parallélisables*, Séminaire Gaston Darboux de Géométrie et Topologie Différentielle, 1991–1992 (Montpellier), Univ. Montpellier II, Montpellier, 1993, pp. 81–99 (French). MR**1223161****[K2]**D. Kotschick,*Signatures, monopoles and mapping class groups*, Math. Res. Lett.**5**(1998), no. 1-2, 227–234. MR**1617905**, 10.4310/MRL.1998.v5.n2.a9**[K3]**D. Kotschick,*On irreducible four-manifolds*, preprint, alg-geom/950412.**[KMT]**D. Kotschick, J. W. Morgan, and C. H. Taubes,*Four-manifolds without symplectic structures but with nontrivial Seiberg-Witten invariants*, Math. Res. Lett.**2**(1995), no. 2, 119–124. MR**1324695**, 10.4310/MRL.1995.v2.n2.a1**[Mc]**Darryl McCullough,*3-manifolds and their mappings*, Lecture Notes Series, vol. 26, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1995. MR**1407311****[M]**J. Milnor,*A unique decomposition theorem for 3-manifolds*, Amer. J. Math.**84**(1962), 1–7. MR**0142125****[T]**Clifford Henry Taubes,*The Seiberg-Witten invariants and symplectic forms*, Math. Res. Lett.**1**(1994), no. 6, 809–822. MR**1306023**, 10.4310/MRL.1994.v1.n6.a15**[Th]**William P. Thurston,*Three-dimensional manifolds, Kleinian groups and hyperbolic geometry*, Bull. Amer. Math. Soc. (N.S.)**6**(1982), no. 3, 357–381. MR**648524**, 10.1090/S0273-0979-1982-15003-0

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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