  Given two $2n$--dimensional symplectic ellipsoids whose symplectic sizes
satisfy certain inequalities, we show that a certain map from the $n$--torus to
the space of symplectic embeddings from one ellipsoid to the other induces an
injective map on singular homology with mod $2$ coefficients. The proof uses
parametrized moduli spaces of $J$--holomorphic cylinders in completed
symplectic cobordisms.
