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Symplectomorphism groups and almost complex structures. (English) Zbl 1010.53064

Ghys, Étienne (ed.) et al., Essays on geometry and related topics. Mémoires dédiés à André Haefliger. Vol. 2. Genève: L’Enseignement Mathématique. Monogr. Enseign. Math. 38, 527-556 (2001).
Let \((M, \omega)\) be a symplectic manifold. One of the interesting facts of symplectic geometry is that symplectic manifolds admit a large family of symplectomorphisms, i.e. diffeomorphisms that preserve the symplectic structure. The group of symplectomorphisms of any symplectic manifold is always infinite-dimensional. On the other hand, in some cases it is possible to determine the topology of the group, i.e. the homotopy type of the group can be calculated, which turns out to be not too large. This problem was first addressed by M. L. Gromov [Invent. Math. 82, 307-347 (1985; Zbl 0592.53025)], who showed that the group of compactly supported symplectomorphisms of \(R^4\) is contractible and that the group of symplectomorphisms of \(S^2\times S^2\), with the product symplectic form in which both spheres have the same size, has the homotopy type of a Lie group.
M. Abreu [Invent. Math. 131, 1-23 (1998; Zbl 0902.53025)] showed that when one sphere factor is larger than the other, the group of symplectomorphisms is no longer homotopy equivalent to a Lie group, because it does not have the right kind of rational homotopy type. M. Abreu and the present author [J. Am. Math. Soc. 13, 971-1009 (2000; Zbl 0965.57031)] calculated this rational homotopy type and showed that it changes precisely when the ratio \(\lambda\) of the size of the larger to the smaller sphere passes an integer value.
In this interesting paper, the author extends and makes more precise the previous results concerning the rational homotopy type of groups of symplectomorphisms for rational ruled surfaces. Ruled surfaces are compact smooth 4-manifolds \(M\) that fiber over a Riemann surface \(\Sigma_g\) of genus \(g\) with fiber \(S^2\). The paper studies groups of symplectomorphisms of ruled surfaces \(M\) for symplectic forms with varying cohomology class. This cohomology class is characterized by the ratio \(\lambda\) of the size of the base to that of the fiber. By considering appropriate spaces of complex structures \(J\), the author investigates how the topological type of these groups changes as \(\lambda\) increases. If the base is a sphere, it changes precisely when \(\lambda\) passes an integer, and, for general bases, it stabilizes as \(\lambda\to\infty\).
For the entire collection see [Zbl 0988.00115].

MSC:

53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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