### Abstract

*M*is called symplectically aspherical if it admits a symplectic form

*ω*with

*ω*|

*π*

_{2}(

*M*) = 0. It is easy to see that, unlike in the case of closed symplectic manifolds, not every finitely presented group can be realized as the fundamental group of a closed symplectically aspherical manifold. The goal of the paper is to study the fundamental groups of closed symplectically aspherical manifolds. Motivated by some results of Gompf, we introduce two classes of fundamental groups

*π*

_{1}(

*M*) of symplectically aspherical manifolds

*M*. The first one consists of fundamental groups of such

*M*with

*π*

_{2}(

*M*)=0, while the second with

*π*

_{2}(

*M*)≠0. Relations between these classes are discussed. We show that several important (classes of) groups can be realized in both classes, while some groups can be realized in the first class but not in the second one. Also, we notice that there are some interesting dimensional phenomena in the realization problem. The above results are framed by a general study of symplectically aspherical manifolds. For example, we find some conditions which imply that the Gompf sum of symplectically aspherical manifolds is symplectically aspherical, or that a total space of a bundle is symplectically aspherical.

Original language | English |
---|---|

Pages (from-to) | 805-826 |

Number of pages | 22 |

Journal | Mathematische Zeitschrift |

Volume | 248 |

Issue number | 4 |

Early online date | 6 Jul 2004 |

DOIs | |

Publication status | Published - Dec 2004 |

### Cite this

*Mathematische Zeitschrift*,

*248*(4), 805-826. https://doi.org/10.1007/s00209-004-0682-8

**On fundamental groups of symplectically aspherical manifolds.** / Ibanez, R.; Kedra, J.; Rudyak, Yu.; Tralle, A.

Research output: Contribution to journal › Article

*Mathematische Zeitschrift*, vol. 248, no. 4, pp. 805-826. https://doi.org/10.1007/s00209-004-0682-8

}

TY - JOUR

T1 - On fundamental groups of symplectically aspherical manifolds

AU - Ibanez, R.

AU - Kedra, J.

AU - Rudyak, Yu.

AU - Tralle, A.

N1 - This paper was supported by MCyT, project BFM 2002-00788, Spain. The first author was partially supported by the project UPV/EHU 00127.310-EA-7781/2000, Spain. The second author is a member of EDGE, Research Training Network HPRN-CT-2000-00101, supported by The European Human Potential Programme. The third author was partially supported by Max-Planck Institute for Mathematics, Bonn, Germany, and by the free-term research money from the University of Florida, Gainesville, USA. The fourth author acknowledges the support of the Polish Committee for the Scientific Research (KBN), grant 2P03A 036 24. We are grateful to Vladimir Chernov and Dusa McDuff for useful discussions and valuable advice. We are also grateful to the anonymous referee for many helpful comments.

PY - 2004/12

Y1 - 2004/12

N2 - A smooth manifold M is called symplectically aspherical if it admits a symplectic form ω with ω|π2(M) = 0. It is easy to see that, unlike in the case of closed symplectic manifolds, not every finitely presented group can be realized as the fundamental group of a closed symplectically aspherical manifold. The goal of the paper is to study the fundamental groups of closed symplectically aspherical manifolds. Motivated by some results of Gompf, we introduce two classes of fundamental groups π1(M) of symplectically aspherical manifolds M. The first one consists of fundamental groups of such M with π2(M)=0, while the second with π2(M)≠0. Relations between these classes are discussed. We show that several important (classes of) groups can be realized in both classes, while some groups can be realized in the first class but not in the second one. Also, we notice that there are some interesting dimensional phenomena in the realization problem. The above results are framed by a general study of symplectically aspherical manifolds. For example, we find some conditions which imply that the Gompf sum of symplectically aspherical manifolds is symplectically aspherical, or that a total space of a bundle is symplectically aspherical.

AB - A smooth manifold M is called symplectically aspherical if it admits a symplectic form ω with ω|π2(M) = 0. It is easy to see that, unlike in the case of closed symplectic manifolds, not every finitely presented group can be realized as the fundamental group of a closed symplectically aspherical manifold. The goal of the paper is to study the fundamental groups of closed symplectically aspherical manifolds. Motivated by some results of Gompf, we introduce two classes of fundamental groups π1(M) of symplectically aspherical manifolds M. The first one consists of fundamental groups of such M with π2(M)=0, while the second with π2(M)≠0. Relations between these classes are discussed. We show that several important (classes of) groups can be realized in both classes, while some groups can be realized in the first class but not in the second one. Also, we notice that there are some interesting dimensional phenomena in the realization problem. The above results are framed by a general study of symplectically aspherical manifolds. For example, we find some conditions which imply that the Gompf sum of symplectically aspherical manifolds is symplectically aspherical, or that a total space of a bundle is symplectically aspherical.

U2 - 10.1007/s00209-004-0682-8

DO - 10.1007/s00209-004-0682-8

M3 - Article

VL - 248

SP - 805

EP - 826

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 4

ER -