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 \left| \Omega _ { 0 } -1 \right| , 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 manifolds 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 \left| \Omega _ { 0 } -1 \right| which is more accurate and closer to zero .