Any N dimensional manifold as a boundary of some N+1 dimensional manifold?

Question

Is this statement true:

Question: Can any N dimensional manifold be realized as a boundary of some N+1 dimensional manifold?

If so/not, how to prove/disprove it?

I read a TQFT paper from Edward Witten said that

Any 3 dimensional manifold can be realized as the boundary of a 4 dimensional manifold.

ps. I have noticed a related question.

Answer 1

I think you are asking if every closed manifold is the boundary of a compact manifold. The answer is negative. For instance, real projective plane is not the boundary of any compact 3d manifold (since it has odd Euler characteristic and the Euler characteristic of an odd-dimensional manifold is half of the Euler characteristic of its boundary: This follows from the fact that Euler characetristic is zero for every closed odd-dimensional manifold). If you want an oriented example, take $CP^2$ (since its Euler characteristic is odd). For more references, read about "cobordism theory".

Answer 2

A trivial answer would be that given any $n$-manifold $M$, the $(n+1)$-manifold-with-boundary $M\times [0,1)$ has $M$ as its boundary. But I don't know if there's intended to be any further restrictions on the situation that make a manifold occurring as a boundary of something else more difficult.