  The ordering of scalar fields after a phase transition in which a group $G$
of global symmetries is spontaneously broken to a subgroup $H$ provides a
possible explanation for the origin of structure in the universe, as well as
leading to observable effects in condensed matter systems. The field dynamics
can depend in principle on the geometry and topology of the vacuum manifold
G/H; for example, texture configurations which collapse and unwind will exist
if the third homotopy group $\pi_3(G/H)$ is nontrivial. We numerically simulate
the evolution of texture-like configurations in a number of different models,
in order to determine the extent to which the geometry and topology of the
vacuum manifold influences the field evolution. We find that the dynamics is
affected by whether or not the theory supports strings or monopoles
[characterized by $\pi_1(G/H)$ and $\pi_2(G/H)$, respectively]. In some of the
theories studied, configurations with initially spherically symmetric energy
densities are unstable to nonspherical collapse; these theories are also found
to nucleate defects during the collapse. Models that do not support monopoles
or strings behave similarly to each other, regardless of the specific vacuum
manifold.
