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.



Homology and submanifolds...
The argument the author has in mind is probably via Poincare-Lefschetz duality. Here is a slightly different argument, using the intersection numbers. Assume that the complement $N - M$ is connected. Then you can find an embedded $S^1 \cong S \subset N$ which meets $M$ transversally at precisely one point (if $dim (N) \geq 3$). So the intersection number $S \cdot M=\pm$, whence $S$ has to represent a nonzero element in $H\_1 (N;\mathbb{Q})$, contradicting the assumption on the Betti number. So $N-M$ is disconnected and has precisely two components. The statement on the boundaries is clear if everything is smooth.


In dimension $2$, you still get a smooth map $\bS^1 \to M$ and the same argument applies.


