I'm wondering about the topological properties of non-compact spaces.
Suppose I have the symmetric space $$\frac{U(n,n)}{ U(n) \times U(n)}$$
where $U(n,n)$ is the pseudo-unitary group and $U(n)$ is the unitary group. As Moishe pointed out, this is diffeomorphic to Euclidean space, and so is topologically trivial.
But now if we impose an additional constraint on the space: $U(s) = U^*(-s)$ where $s$ parametrizes the space and $s \in T^1$ i.e., is periodic, can this space be made non-trivial?