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A new construction of symplectic manifolds. (English) Zbl 0849.53027
As is well-known, a Kähler manifold is symplectic but the converse does not hold. The first counterexample was the so-called Kodaira-Thurston manifold KT, which is a compact symplectic nilmanifold of dimension 4 with \(b_1= 3\). Since the odd Betti numbers of a compact Kähler manifold have to be even, we conclude that KT does not admit any Kähler structure. This example was extended to arbitrary higher dimensions in [L. A. Cordero, M. Fernández and the reviewer, Proc. Am. Math. Soc. 95, 280-286 (1985; Zbl 0575.53015); L. A. Cordero, M. Fernández and A. Gray, Topology 25, 375-380 (1986; Zbl 0596.53030)] but the non-existence of Kähler structures was proved by using the techniques of minimal models [P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Invent. Math. 29, 245-274 (1975; Zbl 0312.55011)]. Simply-connected examples were exhibited by D. McDuff [J. Differ Geom. 20, 267-277 (1984; Zbl 0567.53031)] by blowing up along submanifolds techniques. Many other examples were constructed in recent years [see for instance, Ch. Benson and C. S. Gordon, Topology 27, No. 4, 513-518 (1988; Zbl 0672.53036); W. Jelonek, Math. Ann. 305, 639-649 (1996)].
There are many obstructions to the existence of Kähler structures on manifolds but things are dramatically different in the symplectic setting. Of course, the second Betti number \(b_2\) of a compact symplectic manifold does not vanish, but no other obstructions are known.
In this remarkable paper, the author constructs in each even dimension families of compact, symplectic manifolds such that all finitely presentable groups occur as fundamental groups and, moreover, these manifolds can be assumed not to be homotopy equivalent to Kähler manifolds. Compact symplectic 4-manifolds deserve special attention, and important results are obtained about simultaneously realizing the signature and Euler characteristic for any fixed fundamental group. The author has obtained his results by introducing a new surgery construction: symplectically forming connected sums along codimension 2 submanifolds.

MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57R65 Surgery and handlebodies
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