| The algebraic structure of the group of pseudo-isotopy classes of orientation preserving diffeomorphisms on a closed, (k - 1)-connected, smooth, almost-parallelizable 2k-manifold M is investigated. When k ≡ 3 (mod 4) it is proven that such a group modulo the group of homotopy spheres is isomorphic to a semi-direct product of a subgroup of the symplectic group and the free abelian group. In case of M = Sk x Sk, it is shown that the group is residually finite. It is also shown that for some k, the natural map pi0Diff(Sk x Sk) → Aut Hk( Sk x Sk; Z ) splits. Cohomology with trivial Z -coefficients of the Jacobi group GammaJ is computed and a complete description of the group of pseudo-isotopy classes of diffeomorphisms on S3 x S3 is given via a presentation with generators and defining relations, and by means of the theory of group extensions. The approach is based on direct surgery arguments and classical methods of algebraic and differential topology. |
