Homotopy type of symplectomorphism groups of \(S^2 \times S^2\) (Q1397595)

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)] \textit{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 \(\lambda >0\). \textit{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, \textit{M. Abreu} and \textit{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 \(\{ t^2 = x_i^2 = 0, x_1x_2 = x_2x_1\}\).