  We study a class of Monte Carlo algorithms for the nonlinear $\sigma$-model,
based on a Wolff-type embedding of Ising spins into the target manifold $M$. We
argue heuristically that, at least for an asymptotically free model, such an
algorithm can have dynamic critical exponent $z \ll 2$ only if the embedding is
based on an (involutive) isometry of $M$ whose fixed-point manifold has
codimension 1. Such an isometry exists only if the manifold is a discrete
quotient of a product of spheres. Numerical simulations of the idealized
codimension-2 algorithm for the two-dimensional $O(4)$-symmetric $\sigma$-model
yield $z_{int,{\cal M}^2} = 1.5 \pm 0.5$ (subjective 68\% confidence interval),
in agreement with our heuristic argument.
