  Recent observations suggest that the ratio of the total density to the
critical density of the universe, $\Omega_0$, is likely to be very close to
one, with a significant proportion of this energy being in the form of a dark
component with negative pressure. Motivated by this result, we study the
question of observational detection of possible non-trivial topologies in
universes with $\Omega_0 \sim 1$, which include a cosmological constant. Using
a number of indicators we find that as $\Omega_0 \to 1$, increasing families of
possible manifolds (topologies) become either undetectable or can be excluded
observationally. Furthermore, given a non-zero lower bound on $|\Omega_0 - 1|$,
we can rule out families of topologies (manifolds) as possible candidates for
the shape of our universe. We demonstrate these findings concretely by
considering families of topologies and employing bounds on cosmological
parameters from recent observations. We find that given the present bounds on
cosmological parameters, there are families of both hyperbolic and spherical
manifolds that remain undetectable and families that can be excluded as the
shape of our universe. These results are of importance in future search
strategies for the detection of the shape of our universe, given that there are
an infinite number of theoretically possible topologies and that the future
observations are expected to put a non-zero lower bound on $|\Omega_0 - 1|$
which is more accurate and closer to zero.
