**On the Jacobi Group and the Mapping Class Group of
***S*^{3}×*S*^{3}

I'll discuss a relation between the mapping class group of
*S*^{k}×*S*^{k}
where *k*=1,3 or 7 on one hand and the (full) Jacobi group
Γ^{J} and the
group of (2*k*+1)-dimensional homotopy spheres on the other. Using a
presentation of Γ^{J} and the
μ-invariant of the homotopy spheres,
I'll give a presentation of the mapping class group of
*S*^{k}×*S*^{k} with
generators and defining relations. Some corollaries of this result will be
discussed.

*Nikolai Krylov*