I'm reading a paper which includes the following line, and I can't find a reference anywhere to the result the authors mention: "Let M be a compact orientable embedded minimal hypersurface of a compact orientable Riemannian manifold N. Suppose we know that the first Betti number is zero. Then using that M,N are both orientable and chasing through exact sequences of homology groups, it is easy to see that M divides N into two components $\Omega_1$ and $\Omega_2$ such that $\partial \Omega_1=M=\partial \Omega_2$."

It would be great if someone could help me with this - I can't imagine the argument is too complicated but I can't see where to go. Thanks.