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Homotopy type of symplectomorphism groups of \(S^2 \times S^2\). (English) Zbl 1023.57021

The author studies the homotopy type of the group \(G_{\lambda}\) of symplectomorphisms of \((S^2\times S^2,\omega_{\lambda}) = M_{\lambda}\) where \(\omega_{\lambda} = (1+\lambda) \omega_0 \oplus \omega,\) and \(\omega\) is the standard area form of \(S^2\) with the area equal 1, \(0 \leq \lambda \in \mathbb{R}\). In [Invent. Math. 82, 307-347 (1985; Zbl 0592.53025)] M. Gromov showed that \(G_0\) is connected and homotopy equivalent to its subgroup of standard isometries \(SO(3) \times SO(3)\); furthermore, that this is no longer true for \(\lambda >0\). D. McDuff in [Invent. Math. 89, 13-36 (1987; Zbl 0625.53040)] constructed an element of infinite order in \(H_1(G_{\lambda})\), \(\lambda > 0\). In 2000, M. Abreu and D. McDuff [J. Am. Math. Soc. 13, 971-1009 (2000; Zbl 0965.57031)] calculated the rational cohomology of \(G_{\lambda}\) and confirmed that these groups cannot be homotopic to Lie groups. In the paper under review, the author studies the homotopy type of \(G_{\lambda}\), \(0 < \lambda \leq 1\). The main results are the following two theorems:
Theorem 1.1 If \(0< \lambda \leq 1\), \(G_{\lambda}\) is homotopy equivalent to the product \(X = L \times S^1\times SO(3) \times SO(3)\) where \(L\) is the loop space of the suspension of the smash product \(S^1\wedge SO(3)\).
Theorem 1.2 If \(0 <\lambda \leq 1\) then there is an algebra isomorphism \[ H_{\ast}(G_{\lambda};{\pmb Z}_2) = \bigwedge(y_1,y_2) \otimes {\pmb Z}_2\langle t,x_1,x_2 \rangle / R \] where \(\deg y_i=\deg x_i=i\), \(\deg t =1\) and \(R\) is the set of relations \(\{ t^2 = x_i^2 = 0, x_1x_2 = x_2x_1\}\).

MSC:

57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
57R17 Symplectic and contact topology in high or arbitrary dimension
57T20 Homotopy groups of topological groups and homogeneous spaces
57T25 Homology and cohomology of \(H\)-spaces
55P15 Classification of homotopy type
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