Any 3 manifold can be realized as a boundary of a 4 manifold?

Question

It was mentioned in Topological gauge theories and group cohomology, Robbert Dijkgraaf, Edward Witten, Comm. Math. Phys. 129(2): 393-429 (1990).

Any 3 dimensional manifold can be realized as the boundary of a 4 dimensional manifold. (In page 2 (labeled 394)).

It seems that there are some unstated assumptions given here. The paper may mean that the 3-manifold is orientable. My questions:

1. What are assumptions?

• orientable or non-orientable?

• compact or non-compact?

• smooth differentiable or not?

2. How to prove: Any 3 dimensional manifold (with what assumptions) can be realized as the boundary of a 4 dimensional manifold.

How does it apply to for example $\mathbf{RP}^2 \times S^1$?

I find some related questions, but I still hope someone can provide new and easier answers than these:

$RP^2$ as a boundary of a 3-manifold

https://mathoverflow.net/questions/63373/elegant-proof-that-any-closed-oriented-3-manifold-is-the-boundary-of-some-orien

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