Is there a non-trivial connected compact orientable topological manifold of Euler characteristic 1? Is there a non-trivial connex compact orientable topological manifold of Euler characteristic $\chi = 1$? _Remark_ : the point has $\chi = 1$, but it is trivial. The real projective plane has $\chi = 1$, but it is not orientable. The wedge 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.

$\Bbb CP^2$ has Euler characteristic $3$. Now try to use that $\chi(M \\# N) = \chi(M) + \chi(N) - 2$ for even-dimensional manifolds to construct a manifold with $\chi=1$.