On fundamental groups of symplectically aspherical manifolds

R. Ibanez, J. Kedra, Yu. Rudyak, A. Tralle

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)805-826
Number of pages22
JournalMathematische Zeitschrift
Volume248
Issue number4
Early online date6 Jul 2004
DOIs
Publication statusPublished - Dec 2004

Cite this

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

In: Mathematische Zeitschrift, Vol. 248, No. 4, 12.2004, p. 805-826.

Research output: Contribution to journalArticle

Ibanez, R. ; Kedra, J. ; Rudyak, Yu. ; Tralle, A. / On fundamental groups of symplectically aspherical manifolds. In: Mathematische Zeitschrift. 2004 ; Vol. 248, No. 4. pp. 805-826.
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title = "On fundamental groups of symplectically aspherical manifolds",
abstract = "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.",
author = "R. Ibanez and J. Kedra and Yu. Rudyak and A. Tralle",
note = "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.",
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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.

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