  In this paper we discuss the topology of the symplectomorphism group of a
product of two 2-dimensional spheres when the ratio of their areas lies in the
interval (1,2]. More precisely we compute the homotopy type of this
symplectomorphism group and we also show that the group contains two finite
dimensional Lie groups generating the homotopy. A key step in this work is to
calculate the mod 2 homology of the group of symplectomorphisms. Although this
homology has a finite number of generators with respect to the Pontryagin
product, it is unexpected large containing in particular a free noncommutative
ring with 3 generators.
