Kernel(J) warns of false vacua

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.