  We show examples which reveal influences of spatial topologies to dynamics,
using a class of spatially {\it closed} inhomogeneous cosmological models. The
models, called the {\it locally U(1)$\times$U(1) symmetric models} (or the {\it
generalized Gowdy models}), are characterized by the existence of two commuting
spatial {\it local} Killing vectors. For systematic investigations we first
present a classification of possible spatial topologies in this class. We
stress the significance of the locally homogeneous limits (i.e., the Bianchi
types or the `geometric structures') of the models. In particular, we show a
method of reduction to the natural reduced manifold, and analyze the
equivalences at the reduced level of the models as dynamical models. Based on
these fundamentals, we examine the influence of spatial topologies on dynamics
by obtaining translation and reflection operators which commute with the
dynamical flow in the phase space.
