On the Jacobi Group and the Mapping Class Group of $S^3\times S^3$

The paper containes 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 ${\mathrm{\Gamma }}^J$ by the group of 7-dimensional homotopy spheres. Using a presentation of the group ${\mathrm{\Gamma }}^J$ and the $\mu$-invariant of the homotopy spheres, we give a presentation of the mapping class group of $S^3\times S^3$ with generators and defining relations. We also compute cohomology of the group ${\mathrm{\Gamma }}^J$ and determine a 2-cocycle that corresponds to the mapping class group of $S^3\times S^3$.