Is there a non-trivial connex compact orientable [topological manifold](https://en.wikipedia.org/wiki/Topological_manifold) of [Euler characteristic](https://en.wikipedia.org/wiki/Euler_characteristic#Topological_definition) $\chi = 1$? 

*Remark*: the point has $\chi = 1$, but it is trivial. The [real projective plane](https://en.wikipedia.org/wiki/Real_projective_plane) has $\chi = 1$, but it is not orientable. The [wedge](https://en.wikipedia.org/wiki/Wedge_sum) of a sphere and a torus has $\chi = 1$, but it is not a topological manifold. I don't know if the connexity is necessary.