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 (2k+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.