  Biharmonic maps are the critical points of the bienergy functional and, from
this point of view, generalise harmonic maps. We consider the Hopf map
$\psi:\s^3\to \s^2$ and modify it into a nonharmonic biharmonic map
$\phi:\s^3\to \s^3$. We show $\phi$ to be unstable and estimate its biharmonic
index and nullity. Resolving the spectrum of the vertical Laplacian associated
to the Hopf map, we recover Urakawa's determination of its harmonic index and
nullity.

Differential Geometry