On the Jacobi group and the mapping class group of $S^3\times S^3$

The paper contains a proof that the mapping class group of the manifold $S^3\times S^3$ is isomorphic to a central extension of the (full) Jacobi group $\Gamma^J$by the group of 7-dimensional homotopy spheres. Using a presentation of the group $\Gamma^J$ and the $\mu$-invariant of the homotopy spheres, we give a presentation of this mapping class group with generators and defining relations. We also compute the cohomology of the group $\Gamma^J$ and determine 2-cocycles that correspond to the mapping class group of $S^3\times S^3$.