  J.H.C. Whitehead defined a map $J_r:\pi_r(SO)\rightarrow \pi_r^s$ from the
homotopy of the special orthogonal group to the stable homotopy of spheres.
Within a toy model we show how the known computation for kernel$(J)$ leads to
nonlinear $\sigma$-models with spherical source (space) and spherical target
which admit false vacua separated from the true vacuum by an energy barrier. In
this construction, the dimension of space must be at least 8 and the dimension
of the $\sigma$-model target at least 5.
